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 Th4:
  not a,b '||' a,c & [[a,b],[a,c]] = [[e,f],[e,g]] &
 K*(e`1_3-f`1_3)=L*(e`1_3-g`1_3
  ) & K*(e`2_3-f`2_3)=L*(e`2_3-g`2_3) & K*(e`3_3-f`3_3)=L*(e`3_3-g`3_3)
  implies K=0.F & L=0.F
proof
  assume that
A1: ( not a,b '||' a,c)& [[a,b],[a,c]] = [[e,f],[e,g]] and
A2: K*(e`1_3-f`1_3)=L*(e`1_3-g`1_3) and
A3: K*(e`2_3-f`2_3)=L*(e`2_3-g`2_3) and
A4: K*(e`3_3-f`3_3)=L*(e`3_3-g`3_3);
  e=[e`1_3,e`2_3,e`3_3] & 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,Th3;
  then
A5: e`1_3-g`1_3 <> 0.F or e`2_3-g`2_3 <> 0.F or e`3_3-g`3_3 <> 0.F by Lm2;
  K*(e`1_3-f`1_3)*(e`2_3-g`2_3)=L*((e`1_3-g`1_3)*(e`2_3-g`2_3)) &
 K*(e`2_3-f`2_3)*(e`1_3-g`1_3)=L*
  ((e `2_3-g`2_3)*(e`1_3-g`1_3)) by A2,A3,GROUP_1:def 3;
  then K*(e`1_3-f`1_3)*(e`2_3-g`2_3)-K*(e`2_3-f`2_3)*(e`1_3-g`1_3) = 0.F
by RLVECT_1:15;
  then K*((e`1_3-f`1_3)*(e`2_3-g`2_3))-K*(e`2_3-f`2_3)*(e`1_3-g`1_3) = 0.F
by GROUP_1:def 3;
  then K*((e`1_3-f`1_3)*(e`2_3-g`2_3))-K*((e`2_3-f`2_3)*(e`1_3-g`1_3)) = 0.F
by GROUP_1:def 3;
  then
A6: K*((e`1_3-f`1_3)*(e`2_3-g`2_3)-(e`1_3-g`1_3)*(e`2_3-f`2_3)) = 0.F
by VECTSP_1:11;
  K*(e`1_3-f`1_3)*(e`3_3-g`3_3)=L*((e`1_3-g`1_3)*(e`3_3-g`3_3)) &
K*(e`3_3-f`3_3)*(e`1_3-g`1_3)=L
  *((e `3_3-g`3_3)*(e`1_3-g`1_3)) by A2,A4,GROUP_1:def 3;
  then K*(e`1_3-f`1_3)*(e`3_3-g`3_3)-K*(e`3_3-f`3_3)*(e`1_3-g`1_3) = 0.F
by RLVECT_1:15;
  then K*((e`1_3-f`1_3)*(e`3_3-g`3_3))-K*(e`3_3-f`3_3)*(e`1_3-g`1_3) = 0.F
by GROUP_1:def 3;
  then K*((e`1_3-f`1_3)*(e`3_3-g`3_3))-K*((e`3_3-f`3_3)*(e`1_3-g`1_3)) = 0.F
by GROUP_1:def 3;
  then
A7: K*((e`1_3-f`1_3)*(e`3_3-g`3_3)-(e`1_3-g`1_3)*(e`3_3-f`3_3)) = 0.F
by VECTSP_1:11;
  K*(e`2_3-f`2_3)*(e`3_3-g`3_3)=L*((e`2_3-g`2_3)*(e`3_3-g`3_3)) &
  K*(e`3_3-f`3_3)*(e`2_3-g`2_3)=L*
  ((e `3_3-g`3_3)*(e`2_3-g`2_3)) by A3,A4,GROUP_1:def 3;
  then K*(e`2_3-f`2_3)*(e`3_3-g`3_3)-K*(e`3_3-f`3_3)*(e`2_3-g`2_3) = 0.F
 by RLVECT_1:15;
  then K*((e`2_3-f`2_3)*(e`3_3-g`3_3))-K*(e`3_3-f`3_3)*(e`2_3-g`2_3) = 0.F
 by GROUP_1:def 3;
  then K*((e`2_3-f`2_3)*(e`3_3-g`3_3))-K*((e`3_3-f`3_3)*(e`2_3-g`2_3)) = 0.F
 by GROUP_1:def 3;
  then
A8: K*((e`2_3-f`2_3)*(e`3_3-g`3_3)-(e`2_3-g`2_3)*(e`3_3-f`3_3)) = 0.F
 by VECTSP_1:11;
A9: (e`1_3-f`1_3)*(e`2_3-g`2_3)-(e`1_3-g`1_3)*(e`2_3-f`2_3) <> 0.F or
 (e`2_3-f`2_3)*(e`3_3-g`3_3)-(
e`2_3-g`2_3)*(e`3_3-f`3_3) <> 0.F or
(e`1_3-f`1_3)*(e`3_3-g`3_3)-(e`1_3-g`1_3)*(e`3_3-f`3_3) <> 0.F by A1,PARSP_1:12
;
  then
A10: K=0.F by A6,A8,A7,VECTSP_1:12;
  then
A11: 0.F=L*(e`3_3-g`3_3) by A4;
  0.F=L*(e`1_3-g`1_3) & 0.F=L*(e`2_3-g`2_3) by A2,A3,A10;
  hence thesis by A6,A8,A7,A9,A11,A5,VECTSP_1:12;
end;
