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 Th12:
  a,b '||' c,d iff
 ex e,f,g,h st [[a,b],[c,d]] = [[e,f],[g,h]] & (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 &
   (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 & (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
proof
  [[a,b],[c,d]] in 4C_3(F);
  hence thesis by Th9;
end;
