reserve F for Field,
  a,b,c,d,e,f,g,h for Element of F;
reserve x,y for Element of [:the carrier of F,the carrier of F,the carrier of
  F:];
reserve F for Field;
reserve PS for non empty ParStr;
reserve x for set,
  a,b,c,d,e,f,g,h,i,j,k,l for Element of [:the carrier of F,
  the carrier of F,the carrier of F:];
reserve a,b,c,d,p,q,r,s for Element of MPS(F);

theorem Th15:
  a,b '||' p,q & a,b '||' r,s implies p,q '||' r,s or a=b
proof
  defpred CB[(Element of 3F(F)),(Element of 3F(F)), (Element of 3F(F)),Element
of 3F(F)] means ($1`1_3-$2`1_3)*($3`2_3-$4`2_3) -
  ($3`1_3-$4`1_3)*($1`2_3-$2`2_3) = 0.F & ($1`1_3
-$2`1_3)*($3`3_3-$4`3_3) - ($3`1_3-$4`1_3)*($1`3_3-$2`3_3) = 0.F &
   ($1`2_3-$2`2_3)*($3`3_3-$4`3_3) -
  ($3`2_3-$4`2_3)*($1`3_3-$2`3_3) = 0.F;
  assume that
A1: a,b '||' p,q and
A2: a,b '||' r,s;
  consider e,f,g,h such that
A3: [[e,f],[g,h]] = [[a,b],[p,q]] and
A4: CB[e,f,g,h] by A1,Th12;
  consider i,j,k,l such that
A5: [[i,j],[k,l]] = [[a,b],[r,s]] and
A6: CB[i,j,k,l] by A2,Th12;
A7: i = a & j = b by A5,MCART_1:93;
A8: k = r & l = s by A5,MCART_1:93;
  set A = e`1_3-f`1_3, B = e`2_3-f`2_3, C = e`3_3-f`3_3, D = g`1_3-h`1_3,
   E = g`2_3-h`2_3, K = g`3_3
  -h`3_3, G = k`1_3-l`1_3, H = k`2_3-l`2_3, I = k`3_3-l`3_3;
A9: e = a & f = b by A3,MCART_1:93;
A10: g = p & h = q by A3,MCART_1:93;
  now
    assume
A11: a <> b;
    now
AA: for X1,X2,X3 being non empty set
  for x being Element of [:X1,X2,
  X3:] holds x = [x`1_3,x`2_3,x`3_3];
      e = [e`1_3,e`2_3,e`3_3];
      then
A12:  e`1_3 <> f`1_3 or e`2_3 <> f`2_3 or e`3_3 <> f`3_3 by A9,A11,AA;
      per cases by A12,RLVECT_1:21;
      case
A13:    A <> 0.F;
        hence D*H-G*E = 0.F by A4,A6,A9,A7,Lm4;
        thus
A14:    D*I-G*K = 0.F by A4,A6,A9,A7,A13,Lm4;
        E*I = ((D*B)*A")*I & H*K = ((G*B)*A")*K by A4,A6,A9,A7,A13,VECTSP_1:30;
        hence E*I-H*K = 0.F by A14,Lm8;
      end;
      case
A15:    B <> 0.F;
        hence D*H-G*E = 0.F by A4,A6,A9,A7,Lm6;
        thus
A16:    E*I-H*K = 0.F by A4,A6,A9,A7,A15,Lm4;
        D*I = ((E*A)*B")*I & G*K = ((H*A)*B")*K by A4,A6,A9,A7,A15,VECTSP_1:30;
        hence D*I-G*K = 0.F by A16,Lm8;
      end;
      case
A17:    C <> 0.F;
        hence E*I-H*K = 0.F by A4,A6,A9,A7,Lm6;
A18:    D*H = ((K*A)*C")*H & G*E = ((I*A)*C")*E by A4,A6,A9,A7,A17,VECTSP_1:30;
        K*H - I*E = 0.F by A4,A6,A9,A7,A17,Lm6;
        hence D*H-G*E = 0.F by A18,Lm8;
        thus D*I-G*K = 0.F by A4,A6,A9,A7,A17,Lm6;
      end;
    end;
    hence ex g,h,k,l st [[g,h],[k,l]] = [[p,q],[r,s]] & CB[g,h,k,l] by A10,A8;
  end;
  hence thesis by Th12;
end;
