
theorem Th11:
  for S1, S2 being non empty RelStr for a, c being Element of S1,
  b, d being Element of S2 holds a <= c & b <= d iff [a,b] <= [c,d]
proof
  let S1, S2 be non empty RelStr, a, c be Element of S1, b, d be Element of S2;
  set I1 = the InternalRel of S1, I2 = the InternalRel of S2, x = [[a,b],[c,d]
  ];
A1: x`1`1 = a & x`2`1 = c;
A2: x`1`2 = b & x`2`2 = d;
  thus a <= c & b <= d implies [a,b] <= [c,d]
  proof
    assume a <= c & b <= d;
    then [x`1`1,x`2`1] in I1 & [x`1`2,x`2`2] in I2;
    then x in ["I1,I2"] by Th10;
    hence [[a,b],[c,d]] in the InternalRel of [:S1,S2:] by Def2;
  end;
  assume [a,b] <= [c,d];
  then x in the InternalRel of [:S1,S2:];
  then x in ["I1,I2"] by Def2;
  hence [a,c] in the InternalRel of S1 & [b,d] in the InternalRel of S2 by A1
,A2,Th10;
end;
