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 Th7:
  1_F+1_F<>0.F & b,c '||' a,d & a,b '||' c,d & a,c '||' b,d implies
  a,b '||' a,c
proof
  assume that
A1: 1_F+1_F<>0.F and
A2: b,c '||' a,d and
A3: a,b '||' c,d and
A4: a,c '||' b,d;
  assume
A5: not thesis;
  consider i,j,k,l such that
A6: [[b,c],[a,d]]=[[i,j],[k,l]] and
A7: (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 A2,Th2;
A8: b=i & c =j by A6,MCART_1:93;
A9: a=k & d=l by A6,MCART_1:93;
  consider e,f,g,h such that
A10: [[a,b],[c,d]]=[[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 A3,Th2;
A11: b=f by A10,MCART_1:93;
A12: d=h by A10,MCART_1:93;
A13: c =g by A10,MCART_1:93;
A14: a=e by A10,MCART_1:93;
  then
A15: [[a,b],[a,c]]=[[e,f],[e,g]] by A10,A11,MCART_1:93;
  f=[f`1_3,f`2_3,f`3_3] & g=[g`1_3,g`2_3,g`3_3];
  then i`1_3<>j`1_3 or i`2_3<>j`2_3 or i`3_3<>j`3_3 by A5,A11,A13,A15,A8,Th3;
  then consider L such that
A16: L*(f`1_3-g`1_3) = e`1_3-h`1_3 and
A17: L*(f`2_3-g`2_3) = e`2_3-h`2_3 and
A18: L*(f`3_3-g`3_3) = e`3_3-h`3_3 by A14,A11,A13,A12,A7,A8,A9,Lm2;
  h`2_3=f`2_3+g`2_3-e`2_3 by A3,A4,A5,A10,Th5;
  then
A19: (L-1_F)*(e`2_3-g`2_3)=(L+1_F)*(e`2_3-f`2_3) by A17,Lm9;
  h`3_3=f`3_3+g`3_3-e`3_3 by A3,A4,A5,A10,Th5;
  then
A20: (L-1_F)*(e`3_3-g`3_3)=(L+1_F)*(e`3_3-f`3_3) by A18,Lm9;
  h`1_3=f`1_3+g`1_3-e`1_3 by A3,A4,A5,A10,Th5;
  then (L-1_F)*(e`1_3-g`1_3)=(L+1_F)*(e`1_3-f`1_3) by A16,Lm9;
  then L+1_F=0.F & L-1_F=0.F by A5,A15,A19,A20,Th4;
  then L+1_F-(L-1_F)=0.F+(-0.F) by RLVECT_1:def 11;
  then L+1_F-(L-1_F)=0.F by RLVECT_1:5;
  then L+1_F+(-(L-1_F))=0.F by RLVECT_1:def 11;
  then L+1_F+(1_F+-L)=0.F by RLVECT_1:33;
  then (L+1_F+1_F)+(-L)=0.F by RLVECT_1:def 3;
  then (1_F+1_F+L)+(-L)=0.F by RLVECT_1:def 3;
  then 1_F+1_F+(L+(-L))=0.F by RLVECT_1:def 3;
  then 1_F+1_F+0.F=0.F by RLVECT_1:5;
  hence contradiction by A1,RLVECT_1:4;
end;
