{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Map le Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "# proof of the mobil ity for the 2nd family" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "restart:with(LinearAlgebr a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "# entries of matrix R" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "r11:=e0^2+e1^2-e2^2-e3^2:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "r12:=2*(e1*e2-e0*e3):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "r13:=2*(e1*e3+e0*e2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r21:=2*(e1*e2+e0*e3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "r22:=e0^2-e1^2+e2^2-e3^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r23:=2*(e2*e3-e0*e1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r31:=2*(e1*e3-e0*e2):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "r32:=2*(e2*e3+e0*e1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "r33:=e0^2-e1^2-e2^2+e3^2:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "# tra nslational part" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "t1:=2*(f 1*e0-f0*e1+f3*e2-f2*e3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "t2:=2*(f2*e0-f0*e2+f1*e3-f3*e1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "t3:=2*(f3*e0-f0*e3+f2*e1-f1*e2):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "# direct isometry: (AA,BB,CC) are the coordinates of the point (Y 1,Y2,Y3) in the moving frame with respect to the fixed frame" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "AA:=t1+r11*Y1+r12*Y2+r13*Y3: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "BB:=t2+r21*Y1+r22*Y2+r2 3*Y3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "CC:=t3+r31*Y1+r32* Y2+r33*Y3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "# computation of the sphere condit ion Lambda with base point (X1,X2,X3), platform point (Y1,Y2,Y3) and s quard radius dd" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "N:=e0^2+ e1^2+e2^2+e3^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Sphere:= AA^2+BB^2+CC^2-2*N*(AA*X1+BB*X2+CC*X3)+N^2*(X1^2+X2^2+X3^2-dd):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "psi:=e0*f0+e1*f1+e2*f2+e3*f3 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Lambda:=factor(Sphere \+ + 4*psi^2)/N:nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#!)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "# parametrisation of the fam ily" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a1:=a:b1:=b:c1:=c:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a2:=b:b2:=c:c2:=a:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a3:=c:b3:=a:c3:=b:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a4:=A:b4:=B:c4:=C:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a5:=B:b5:=C:c5:=A:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a6:=C:b6:=A:c6:=B:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A1:=a4:B1:=b4:C1:=c4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A2:=a6:B2:=b6:C2:=c6:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A3:=a5:B3:=b5:C3:=c5:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A4:=a1:B4:=b1:C4:=c1:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A5:=a3:B5:=b3:C5:=c3:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A6:=a2:B6:=b2:C6:=c2:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "# defining Lambda1, ..., Lambda6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Lambda1:=simplify(subs(X1=A1,X2=B1,X3=C1, Y1=a1,Y2=b1,Y3=c1,dd=dd1,Lambda)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Lambda2:=simplify(subs(X1=A2,X2=B2,X3=C2,Y1=a2,Y2=b2, Y3=c2,dd=dd2,Lambda)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "L ambda3:=simplify(subs(X1=A3,X2=B3,X3=C3,Y1=a3,Y2=b3,Y3=c3,dd=dd3,Lambd a)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Lambda4:=simplify(s ubs(X1=A4,X2=B4,X3=C4,Y1=a4,Y2=b4,Y3=c4,dd=dd4,Lambda)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Lambda5:=simplify(subs(X1=A5,X2=B5, X3=C5,Y1=a5,Y2=b5,Y3=c5,dd=dd5,Lambda)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Lambda6:=simplify(subs(X1=A6,X2=B6,X3=C6,Y1=a6,Y2=b6, Y3=c6,dd=dd6,Lambda)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# defining Delata12,..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Delta12:=simplify(Lambda1 -Lambda2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Delta13:=simp lify(Lambda1-Lambda3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "D elta14:=simplify(Lambda1-Lambda4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Delta15:=simplify(Lambda1-Lambda5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Delta16:=simplify(Lambda1-Lambda6):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Delta23:=simplify(Lambda1-La mbda3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Delta24:=simplif y(Lambda1-Lambda4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Delt a25:=simplify(Lambda1-Lambda5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Delta26:=simplify(Lambda1-Lambda6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "# de fining Sij, Tij, Uij, Vij, Wij" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S12:=simplify(select(has,Lambda1-Lambda2,f0)/f0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T12:=simplify(select(has,Lambda1-La mbda2,f1)/f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "U12:=simp lify(select(has,Lambda1-Lambda2,f2)/f2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V12:=simplify(select(has,Lambda1-Lambda2,f3)/f3):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "W12:=simplify(Lambda1-Lambd a2-S12*f0-T12*f1-U12*f2-V12*f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S13:=si mplify(select(has,Lambda1-Lambda3,f0)/f0):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "T13:=simplify(select(has,Lambda1-Lambda3,f1)/f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "U13:=simplify(select(has, Lambda1-Lambda3,f2)/f2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V13:=simplify(select(has,Lambda1-Lambda3,f3)/f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "W13:=simplify(Lambda1-Lambda3-S13*f0-T13*f1 -U13*f2-V13*f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S14:=simplify(select(has,Lam bda1-Lambda4,f0)/f0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T1 4:=simplify(select(has,Lambda1-Lambda4,f1)/f1):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "U14:=simplify(select(has,Lambda1-Lambda4,f2)/f 2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V14:=simplify(select (has,Lambda1-Lambda4,f3)/f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "W14:=simplify(Lambda1-Lambda4-S14*f0-T14*f1-U14*f2-V14*f3):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "S15:=simplify(select(has,Lambda1-Lambda5,f0)/f0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T15:=simplify(select(has, Lambda1-Lambda5,f1)/f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "U15:=simplify(select(has,Lambda1-Lambda5,f2)/f2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V15:=simplify(select(has,Lambda1-Lambda5,f3 )/f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "W15:=simplify(Lam bda1-Lambda5-S15*f0-T15*f1-U15*f2-V15*f3):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S16:= simplify(select(has,Lambda1-Lambda6,f0)/f0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T16:=simplify(select(has,Lambda1-Lambda6,f1)/f1) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "U16:=simplify(select(h as,Lambda1-Lambda6,f2)/f2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V16:=simplify(select(has,Lambda1-Lambda6,f3)/f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "W16:=simplify(Lambda1-Lambda6-S16*f 0-T16*f1-U16*f2-V16*f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S23:=simplify(select(ha s,Lambda2-Lambda3,f0)/f0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T23:=simplify(select(has,Lambda2-Lambda3,f1)/f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "U23:=simplify(select(has,Lambda2-La mbda3,f2)/f2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V23:=simp lify(select(has,Lambda2-Lambda3,f3)/f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "W23:=simplify(Lambda2-Lambda3-S23*f0-T23*f1-U23*f2-V2 3*f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S24:=simplify(select(has,Lambda2-Lambda4, f0)/f0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T24:=simplify(s elect(has,Lambda2-Lambda4,f1)/f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "U24:=simplify(select(has,Lambda2-Lambda4,f2)/f2):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V24:=simplify(select(has,La mbda2-Lambda4,f3)/f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "W 24:=simplify(Lambda2-Lambda4-S24*f0-T24*f1-U24*f2-V24*f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S25:=simplify(select(has,Lambda2-Lambda5,f0)/f0):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T25:=simplify(select(has,La mbda2-Lambda5,f1)/f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "U 25:=simplify(select(has,Lambda2-Lambda5,f2)/f2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V25:=simplify(select(has,Lambda2-Lambda5,f3)/ f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "W25:=simplify(Lambd a2-Lambda5-S25*f0-T25*f1-U25*f2-V25*f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S26:=si mplify(select(has,Lambda2-Lambda6,f0)/f0):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 49 "T26:=simplify(select(has,Lambda2-Lambda6,f1)/f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "U26:=simplify(select(has, Lambda2-Lambda6,f2)/f2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V26:=simplify(select(has,Lambda2-Lambda6,f3)/f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "W26:=simplify(Lambda2-Lambda6-S26*f0-T26*f1 -U26*f2-V26*f3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "# collecting Sij, Tij, Uij, \+ Vij, Wij in a vector Hij" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H12:=:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H13:=:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H14:=:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H15:=:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 27 "H16:=:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H23:=:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H24:=: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H25:=:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H26:=:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "# The coefficients of psi are colle cted in H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "H:=:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "# computing Omega1, ..., Omega6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Omega1:=Determinant():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Omega2:=De terminant():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Omega3:=Determinant():" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 41 "Omega4:=Determinant():" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Omega5:=Determinant():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Omega6:= Determinant():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# compu ting G1, ..., G6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G1:=fac tor(Omega1/N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G2:=facto r(Omega2/N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G3:=factor( Omega3/N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G4:=factor(Om ega4/N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G5:=factor(Omeg a5/N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G6:=factor(Omega6 /N):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "# test" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(G1-G2+G3-G4+G5-G6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# s pecifying the leg lengths" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1157 "dd4:=(dd3*b^2*C^2+dd3*a^2*B^2+dd1*a^2*A^2-dd1*a*b*B^2-dd1*a^2*A* B+dd1*a^2*B*C-dd1*a*b*A^2-dd1*a*c*A^2+dd1*b^2*B^2-dd1*a^2*C*A-dd1*a*C^ 2*c+dd1*a*C^2*b+dd1*a*B^2*c-dd1*b^2*B*C-dd1*c*C^2*b+dd1*c^2*C^2-dd1*c^ 2*B*C+dd1*c*b*A^2+dd1*c^2*A*B+dd1*b^2*C*A-dd1*b^2*A*B-dd2*b^2*C*A+dd3* c^2*A^2-dd1*a*b*B*C-dd1*a*c*C*B-dd1*a*c*A*B-dd1*a*C*b*A-dd1*c^2*C*A-dd 1*c*b*B^2-dd1*c*B*A*b-dd1*c*C*b*A+3*dd1*a*B*A*b+3*dd1*a*c*C*A+3*dd1*c* b*B*C-dd2*b^2*A*B+dd2*b^2*B*C-dd2*a*C^2*c-dd2*a*C^2*b-dd2*a*B^2*c+dd2* a*b*B^2+dd2*b^2*A^2-dd2*a*b*B*C-dd2*a*B*A*b-dd2*a*c*C*A-dd2*a*c*A*B-dd 2*c^2*B*C-dd2*c*b*A^2+dd2*a^2*A*B-dd2*a^2*B*C-dd2*a*b*A^2+dd2*a*c*A^2- dd2*a^2*C*A-dd2*c*b*B^2+dd2*a^2*C^2-dd2*c*C*b*A-dd2*c*b*B*C+3*dd2*a*c* C*B+3*dd2*a*C*b*A+3*dd2*c*B*A*b-dd2*c^2*A*B+dd2*c^2*C*A+dd2*c*C^2*b-dd 3*b^2*C*A-dd3*b^2*B*C+dd3*b^2*A*B+dd2*c^2*B^2-dd3*a*B^2*c-dd3*a*b*B^2- dd3*a^2*A*B-dd3*a^2*B*C+dd3*a*b*A^2+dd3*a*C^2*c-dd3*c*C^2*b+dd3*c*b*B^ 2+dd3*c^2*B*C-dd3*a*c*C*B-dd3*a*B*A*b-dd3*a*c*C*A-dd3*a*C*b*A-dd3*c*b* B*C-dd3*a*C^2*b-dd3*a*c*A^2+dd3*a^2*C*A-dd3*c*b*A^2-dd3*c^2*A*B-dd3*c^ 2*C*A-dd3*c*B*A*b+3*dd3*a*b*B*C+3*dd3*a*c*A*B+3*dd3*c*C*b*A)/(B^2-C*A+ C^2-B*A-B*C+A^2)/(a^2-b*c-a*b+c^2+b^2-a*c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1158 "dd5:=(-dd1*a*b*B^2-dd1*a^2*A*B-dd1*a^2*B*C+dd1*a *b*A^2-dd1*a*c*A^2+dd1*a^2*C*A+dd1*a*C^2*c-dd1*a*C^2*b-dd1*a*B^2*c-dd1 *b^2*B*C-dd1*c*C^2*b+dd1*c^2*B*C-dd1*c*b*A^2-dd1*c^2*A*B-dd1*b^2*C*A+d d1*b^2*A*B+dd2*b^2*C*A+3*dd1*a*b*B*C-dd1*a*c*C*B+3*dd1*a*c*A*B-dd1*a*C *b*A-dd1*c^2*C*A+dd1*c*b*B^2-dd1*c*B*A*b+3*dd1*c*C*b*A-dd1*a*B*A*b-dd1 *a*c*C*A-dd1*c*b*B*C-dd2*b^2*A*B-dd2*b^2*B*C-dd2*a*C^2*c+dd2*a*C^2*b+d d2*a*B^2*c-dd2*a*b*B^2-dd2*a*b*B*C+3*dd2*a*B*A*b+3*dd2*a*c*C*A-dd2*a*c *A*B-dd2*c^2*B*C+dd2*c*b*A^2-dd2*a^2*A*B+dd2*a^2*B*C-dd2*a*b*A^2-dd2*a *c*A^2-dd2*a^2*C*A-dd2*c*b*B^2-dd2*c*C*b*A+3*dd2*c*b*B*C-dd2*a*c*C*B-d d2*a*C*b*A-dd2*c*B*A*b+dd2*c^2*A*B-dd2*c^2*C*A-dd2*c*C^2*b-dd3*b^2*C*A +dd3*b^2*B*C-dd3*b^2*A*B-dd3*a*B^2*c+dd3*a*b*B^2+dd3*a^2*A*B-dd3*a^2*B *C-dd3*a*b*A^2-dd3*a*C^2*c+dd3*c*C^2*b-dd3*c*b*B^2-dd3*c^2*B*C+3*dd3*a *c*C*B-dd3*a*B*A*b-dd3*a*c*C*A+3*dd3*a*C*b*A-dd3*c*b*B*C-dd3*a*C^2*b+d d3*a*c*A^2-dd3*a^2*C*A-dd3*c*b*A^2-dd3*c^2*A*B+dd3*c^2*C*A+3*dd3*c*B*A *b-dd3*a*b*B*C-dd3*a*c*A*B-dd3*c*C*b*A+dd1*c^2*A^2+dd1*b^2*C^2+dd2*a^2 *A^2+dd1*a^2*B^2+dd2*c^2*C^2+dd2*b^2*B^2+dd3*b^2*A^2+dd3*c^2*B^2+dd3*a ^2*C^2)/(B^2-C*A+C^2-B*A-B*C+A^2)/(a^2-b*c-a*b+c^2+b^2-a*c):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1157 "dd6:=(dd1*a*b*B^2+dd1*a^2* A*B-dd1*a^2*B*C-dd1*a*b*A^2+dd1*a*c*A^2-dd1*a^2*C*A-dd1*a*C^2*c-dd1*a* C^2*b-dd1*a*B^2*c+dd1*b^2*B*C+dd1*c*C^2*b-dd1*c^2*B*C-dd1*c*b*A^2-dd1* c^2*A*B-dd1*b^2*C*A-dd1*b^2*A*B-dd2*b^2*C*A-dd1*a*b*B*C+3*dd1*a*c*C*B- dd1*a*c*A*B+3*dd1*a*C*b*A+dd1*c^2*C*A-dd1*c*b*B^2+3*dd1*c*B*A*b-dd1*c* C*b*A-dd1*a*B*A*b-dd1*a*c*C*A-dd1*c*b*B*C+dd2*b^2*A*B-dd2*b^2*B*C+dd2* a*C^2*c-dd2*a*C^2*b-dd2*a*B^2*c-dd2*a*b*B^2+3*dd2*a*b*B*C-dd2*a*B*A*b- dd2*a*c*C*A+3*dd2*a*c*A*B+dd2*c^2*B*C-dd2*c*b*A^2-dd2*a^2*A*B-dd2*a^2* B*C+dd2*a*b*A^2-dd2*a*c*A^2+dd2*a^2*C*A+dd2*c*b*B^2+3*dd2*c*C*b*A-dd2* c*b*B*C-dd2*a*c*C*B-dd2*a*C*b*A-dd2*c*B*A*b-dd2*c^2*A*B-dd2*c^2*C*A-dd 2*c*C^2*b+dd3*b^2*C*A-dd3*b^2*B*C-dd3*b^2*A*B+dd3*a*B^2*c-dd3*a*b*B^2- dd3*a^2*A*B+dd3*a^2*B*C-dd3*a*b*A^2-dd3*a*C^2*c-dd3*c*C^2*b-dd3*c*b*B^ 2-dd3*c^2*B*C-dd3*a*c*C*B+3*dd3*a*B*A*b+3*dd3*a*c*C*A-dd3*a*C*b*A+3*dd 3*c*b*B*C+dd3*a*C^2*b-dd3*a*c*A^2-dd3*a^2*C*A+dd3*c*b*A^2+dd3*c^2*A*B- dd3*c^2*C*A-dd3*c*B*A*b-dd3*a*b*B*C-dd3*a*c*A*B-dd3*c*C*b*A+dd1*c^2*B^ 2+dd1*a^2*C^2+dd1*b^2*A^2+dd3*b^2*B^2+dd2*b^2*C^2+dd2*a^2*B^2+dd2*c^2* A^2+dd3*c^2*C^2+dd3*a^2*A^2)/(B^2-C*A+C^2-B*A-B*C+A^2)/(a^2-b*c-a*b+c^ 2+b^2-a*c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "# completing the proof" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "G2n:=factor(numer(simplify(G2))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "G3n:=factor(numer(simplif y(G3))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "G4n:=factor(num er(simplify(G4))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "G5n:= factor(numer(simplify(G5))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "G6n:=factor(numer(simplify(G6))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "S:=gcd( gcd(gcd(G2n,G3n),gcd(G4n,G5n)),G6n):nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "op(1 ,S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\$G\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "op(2,S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,:* \$)%\"cG\"\"#\"\"\"F(*&%\"aGF(F&F(!\"\"*&F&F(%\"bGF(F+*\$)%\"AGF'F(F+*&F 0F(%\"BGF(F(*\$)F-F'F(F(*&F2F(%\"CGF(F(*&F*F(F-F(F+*&F0F(F6F(F(*\$)F*F'F (F(*\$)F6F'F(F+*\$)F2F'F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "nops(op(3,S));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\$w&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }