reserve F for Field;
reserve a,b,c,d,p,q,r for Element of MPS(F);
reserve e,f,g,h,i,j,k,l,m,n,o,w for Element of [:the carrier of F,the carrier
  of F,the carrier of F:];
reserve K,L,M,N,R,S for Element of F;

theorem Th5:
  not a,b '||' a,c & a,b '||' c,d & a,c '||' b,d &
    [[a,b],[c,d]] = [[e,f],[g,h]]
  implies h`1_3=f`1_3+g`1_3-e`1_3 & h`2_3=f`2_3+g`2_3-e`2_3 &
 h`3_3=f`3_3+g`3_3-e`3_3
proof
  assume that
A1: not a,b '||' a,c and
A2: a,b '||' c,d and
A3: a,c '||' b,d and
A4: [[a,b],[c,d]]=[[e,f],[g,h]];
  consider m,n,o,w such that
A6: [[a,c],[b,d]]=[[m,n],[o,w]] and
A7: ( ex L st L*(m`1_3-n`1_3) = o`1_3-w`1_3 & L*(m`2_3-n`2_3) = o`2_3-w`2_3 &
 L*(m`3_3-n
  `3_3) = o`3_3-w`3_3) or m`1_3-n`1_3=0.F & m`2_3-n`2_3=0.F &
 m`3_3-n`3_3=0.F by A3,Th2;
A8: b=f by A4,MCART_1:93;
  then
A9: o=f by A6,MCART_1:93;
  d=h by A4,MCART_1:93;
  then
A10: w=h by A6,MCART_1:93;
  c =g by A4,MCART_1:93;
  then
A11: n=g by A6,MCART_1:93;
A12: a=e by A4,MCART_1:93;
  then
A13: [[a,b],[a,c]]=[[e,f],[e,g]] by A4,A8,MCART_1:93;
  consider i,j,k,l such that
A14: [[a,b],[c,d]]=[[i,j],[k,l]] and
A15: ( ex K st K*(i`1_3-j`1_3) = k`1_3-l`1_3 & K*(i`2_3-j`2_3) = k`2_3-l`2_3 &
  K*(i`3_3-j
  `3_3) = k`3_3-l`3_3) or i`1_3-j`1_3=0.F & i`2_3-j`2_3=0.F &
  i`3_3-j`3_3=0.F by A2,Th2;
A16: e=i & f=j by A4,A14,MCART_1:93;
A17: g=k & h=l by A4,A14,MCART_1:93;
A18: e=m by A12,A6,MCART_1:93;
  f=[f`1_3,f`2_3,f`3_3];
  then e`1_3<>f`1_3 or e`2_3<>f`2_3 or e`3_3<>f`3_3 by A1,A13,Th3;
  then consider K such that
A19: K*(e`1_3-f`1_3) = g`1_3-h`1_3 and
A20: K*(e`2_3-f`2_3) = g`2_3-h`2_3 and
A21: K*(e`3_3-f`3_3) = g`3_3-h`3_3 by A15,A16,A17,Lm2;
  g=[g`1_3,g`2_3,g`3_3];
  then e`1_3<>g`1_3 or e`2_3<>g`2_3 or e`3_3<>g`3_3 by A1,A13,Th3;
  then consider L such that
A22: L*(e`1_3-g`1_3) = f`1_3-h`1_3 and
A23: L*(e`2_3-g`2_3) = f`2_3-h`2_3 and
A24: L*(e`3_3-g`3_3) = f`3_3-h`3_3 by A7,A18,A11,A9,A10,Lm2;
  K*(e`2_3-f`2_3)-L*(e`2_3-g`2_3) = g`2_3-f`2_3 by A20,A23,Lm5;
  then
A25: (K+(-1_F))*(e`2_3-f`2_3) = (L+(-1_F))*(e`2_3-g`2_3) by Lm6;
  K*(e`3_3-f`3_3)-L*(e`3_3-g`3_3) = g`3_3-f`3_3 by A21,A24,Lm5;
  then
A26: (K+(-1_F))*(e`3_3-f`3_3) = (L+(-1_F))*(e`3_3-g`3_3) by Lm6;
  K*(e`1_3-f`1_3)-L*(e`1_3-g`1_3) = g`1_3-f`1_3 by A19,A22,Lm5;
  then (K+(-1_F))*(e`1_3-f`1_3) = (L+(-1_F))*(e`1_3-g`1_3) by Lm6;
  then
A27: K+(-1_F)=0.F by A1,A13,A25,A26,Th4;
  then (e`2_3-f`2_3)*1_F = g`2_3-h`2_3 by A20,Lm2;
  then
A28: e`2_3-f`2_3 = g`2_3-h`2_3;
  (e`3_3-f`3_3)*1_F = g`3_3-h`3_3 by A21,A27,Lm2;
  then
A29: e`3_3-f`3_3 = g`3_3-h`3_3;
  (e`1_3-f`1_3)*1_F = g`1_3- h`1_3 by A19,A27,Lm2;
  then e`1_3-f`1_3 = g`1_3-h`1_3;
  hence thesis by A28,A29,Lm7;
end;
