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 Th17:
  ex d st a,b '||' c,d & a,c '||' b,d
proof
  consider e,f,g such that
A1: e = a & f = b & g = c;
  set h = g+f+-e;
  reconsider d = h as Element of MPS(F);
A2: [[e,f],[g,h]] = [[a,b],[c,d]] by A1;
  take d;
  g+f = [g`1_3+f`1_3,g`2_3+f`2_3,g`3_3+f`3_3] & -e = [-e`1_3,-e`2_3,-e`3_3]
    by Def1,Def3;
  then
A3: h = [g`1_3+f`1_3+-e`1_3,g`2_3+f`2_3+-e`2_3,g`3_3+f`3_3+-e`3_3] by Th2;
  then
A4: h`1_3 = g`1_3+f`1_3+-e`1_3;
A5: h`3_3 = g`3_3+f`3_3+(-e`3_3) by A3;
  then
A6: (e`1_3-f`1_3)*(g`3_3-h`3_3) - (g`1_3-h`1_3)*(e`3_3-f`3_3) = 0.F by A4,Lm15;
A7: (e`1_3-g`1_3)*(f`3_3-h`3_3) - (f`1_3-h`1_3)*(e`3_3-g`3_3) = 0.F
     by A4,A5,Lm15;
A8: h`2_3 = g`2_3+f`2_3+-e`2_3 by A3;
  then
A9: (e`2_3-f`2_3)*(g`3_3-h`3_3) - (g`2_3-h`2_3)*(e`3_3-f`3_3) = 0.F by A5,Lm15;
  (e`1_3-f`1_3)*(g`2_3-h`2_3) - (g`1_3-h`1_3)*(e`2_3-f`2_3) = 0.F
     by A4,A8,Lm15;
  hence a,b '||' c,d by A2,A6,A9,Th12;
A10: [[e,g],[f,h]] = [[a,c],[b,d]] by A1;
A11: (e`2_3-g`2_3)*(f`3_3-h`3_3) - (f`2_3-h`2_3)*(e`3_3-g`3_3) = 0.F
    by A8,A5,Lm15;
  (e`1_3-g`1_3)*(f`2_3-h`2_3) - (f`1_3-h`1_3)*(e`2_3-g`2_3) = 0.F
     by A4,A8,Lm15;
  hence thesis by A10,A7,A11,Th12;
end;
