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 Th8:
  not a,p '||' a,b & not a,p '||' a,c & a,p '||' b,q & a,p '||' c,r
  & a,b '||' p,q & a,c '||' p,r implies b,c '||' q,r
proof
  assume that
A1: not a,p '||' a,b and
A2: not a,p '||' a,c and
A3: a,p '||' b,q and
A4: a,p '||' c,r and
A5: a,b '||' p,q and
A6: a,c '||' p,r;
  consider i,j,k,l such that
A7: [[a,p],[c,r]]=[[i,j],[k,l]] and
  (ex L st L*(i`1_3-j`1_3) = k`1_3-l`1_3 & L*(i`2_3-j`2_3) = k`2_3-l`2_3 &
  L*(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 A4,Th2;
  consider e,f,g,h such that
A8: [[a,b],[p,q]]=[[e,f],[g,h]] and
  (ex K st K*(e`1_3-f`1_3) = g`1_3-h`1_3 & K*(e`2_3-f`2_3) = g`2_3-h`2_3 &
 K*(e`3_3-f`3_3 ) =
  g`3_3-h`3_3) or e`1_3-f`1_3 = 0.F & e`2_3-f`2_3 = 0.F & e`3_3-f`3_3 = 0.F
  by A5,Th2;
A9: a=e & p=g by A8,MCART_1:93;
A10: c =k by A7,MCART_1:93;
  then
A11: [[a,p],[c,r]]=[[e,g],[k,l]] by A9,A7,MCART_1:93;
  then
A12: l`1_3=g`1_3+k`1_3-e`1_3 by A2,A4,A6,Th5;
A13: b=f by A8,MCART_1:93;
  then
A14: [[a,p],[b,q]]=[[e,g],[f,h]] by A8,A9,MCART_1:93;
  then h`1_3=g`1_3+f`1_3-e`1_3 by A1,A3,A5,Th5;
  then
A15: 1_F*(f`1_3-k`1_3)=h`1_3-l`1_3 by A12,Lm10;
A16: l`3_3=g`3_3+k`3_3-e`3_3 by A2,A4,A6,A11,Th5;
A17: l`2_3=g`2_3+k`2_3-e`2_3 by A2,A4,A6,A11,Th5;
  h`3_3=g`3_3+f`3_3-e`3_3 by A1,A3,A5,A14,Th5;
  then
A18: 1_F*(f`3_3-k`3_3)=h`3_3-l`3_3 by A16,Lm10;
  h`2_3=g`2_3+f`2_3-e`2_3 by A1,A3,A5,A14,Th5;
  then
A19: 1_F*(f`2_3-k`2_3)=h`2_3-l`2_3 by A17,Lm10;
  q=h by A8,MCART_1:93;
  then [[b,c],[q,r]]=[[f,k],[h,l]] by A13,A7,A10,MCART_1:93;
  hence thesis by A15,A19,A18,Th2;
end;
