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 Th6:
  ex a,b,c st not a,b '||' a,c
proof
  consider e,f,g being Element of [:the carrier of F,the carrier of F,the
  carrier of F:] such that
A1: e=[1_F,1_F,0.F] and
A2: f=[-0.F,1_F,0.F] and
A3: g=[1_F,-0.F,0.F];
A4: f`1_3=-0.F & f`2_3=1_F by A2;
A5: g`1_3=1_F & g`2_3=-0.F by A3;
  the carrier of MPS(F) = [:the carrier of F,the carrier of F,the carrier
  of F:] by PARSP_1:10;
  then consider a,b,c being Element of MPS(F) such that
A6: [[a,b],[a,c]]=[[e,f],[e,g]];
  take a,b,c;
  e`1_3=1_F & e`2_3=1_F by A1;
  then
A7: (e`1_3-f`1_3)*(e`2_3-g`2_3)-(e`1_3-g`1_3)*(e`2_3-f`2_3)
    = (1.F+(-(-0.F)))*(1.F-(-0.F))-
  (1.F-1.F)*(1.F-1.F) by A4,A5,RLVECT_1:def 11
    .= (1.F+(-(-0.F)))*(1.F+(-(-0.F)))-(1.F-1.F)*(1.F-1.F) by RLVECT_1:def 11
    .= (1.F+0.F)*(1.F+(-(-0.F)))-(1.F-1.F)*(1.F-1.F) by RLVECT_1:17
    .= (1.F+0.F)*(1.F+0.F)-(1.F-1.F)*(1.F-1.F) by RLVECT_1:17
    .= 1.F*(1.F+0.F)-(1.F-1.F)*(1.F-1.F) by RLVECT_1:4
    .= 1.F*1.F-(1.F-1.F)*(1.F-1.F) by RLVECT_1:4
    .= 1.F*1.F-0.F*(1.F-1.F) by RLVECT_1:15
    .= 1.F*1.F-0.F
    .= 1.F-0.F;
  now
    let e9,f9,g9,h9 be Element of [:the carrier of F,the carrier of F,the
    carrier of F:];
    assume
A8: [[e9,f9],[g9,h9]]=[[a,b],[a,c]];
    then
A9: g9=e & h9=g by A6,MCART_1:93;
    e9=e & f9=f by A6,A8,MCART_1:93;
    hence (e9`1_3-f9`1_3)*(g9`2_3-h9`2_3) - (g9`1_3-h9`1_3)*(e9`2_3-f9`2_3)
    <> 0.F or (e9`1_3-f9
`1_3)*(g9`3_3-h9`3_3) - (g9`1_3-h9`1_3)*(e9`3_3-f9`3_3) <> 0.F or
  (e9`2_3-f9`2_3)*(g9`3_3-h9`3_3) - (
    g9`2_3-h9`2_3)*(e9`3_3-f9`3_3) <> 0.F by A7,A9,Lm2;
  end;
  hence thesis by PARSP_1:12;
end;
