
theorem Th12:
  for S1, S2 being non empty RelStr, x, y being Element of [:S1,S2
  :] holds x <= y iff x`1 <= y`1 & x`2 <= y`2
proof
  let S1, S2 be non empty RelStr, x, y be Element of [:S1,S2:];
A1: the carrier of [:S1,S2:] = [:the carrier of S1, the carrier of S2:] by Def2
;
  then
  ex c, d being object st c in the carrier of S1 & d in the carrier of S2 & y
  = [c,d] by ZFMISC_1:def 2;
  then
A2: y = [y`1,y`2];
  ex a, b being object st a in the carrier of S1 & b in the carrier of S2 & x
  = [a,b] by A1,ZFMISC_1:def 2;
  then x = [x`1,x`2];
  hence thesis by A2,Th11;
end;
