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 Th3:
  not a,b '||' a,c & [[a,b],[a,c]]=[[e,f],[e,g]] implies e<>f & e<>g & f<>g
proof
  assume
A1: ( not a,b '||' a,c)& [[a,b],[a,c]]=[[e,f],[e,g]];
  then
  0.F*(e`1_3-f`1_3) <> e`1_3-g`1_3 or 0.F*(e`2_3-f`2_3) <> e`2_3-g`2_3 or
  0.F*(e`3_3-f`3_3) <>
  e`3_3-g`3_3 by Th2;
  then
A2: 0.F <> e`1_3-g`1_3 or 0.F <> e`2_3-g`2_3 or 0.F <> e`3_3-g`3_3;
A3: f<>g
  proof
    assume
A4: not thesis;
    (e`1_3-f`1_3)*1_F <> e`1_3-g`1_3 or (e`2_3-f`2_3)*1_F <> e`2_3-g`2_3 or
(e`3_3-f`3_3)*1_F
    <> e`3_3-g`3_3 by A1,Th2;
    hence contradiction by A4;
  end;
  e`1_3-f`1_3 <> 0.F or e`2_3-f`2_3 <> 0.F or e`3_3-f`3_3 <> 0.F by A1,Th2;
  hence thesis by A2,A3,RLVECT_1:15;
end;
