
theorem Th15:
  for R, S being non empty RelStr, a, b being Element of R, c, d
  being Element of S st a = c & b = d & R tolerates S & R is transitive & S is
  transitive holds a <= b iff c <= d
proof
  let R, S be non empty RelStr, a, b be Element of R, c, d be Element of S;
  assume that
A1: a = c & b = d and
A2: R tolerates S and
A3: R is transitive and
A4: S is transitive;
  a in (the carrier of R) \/ the carrier of S & b in (the carrier of R) \/
  the carrier of S by XBOOLE_0:def 3;
  then reconsider a9 = a, b9 = b as Element of R [*] S by Def2;
  hereby
    assume a <= b;
    then a9 <= b9 by A2,A3,Th8;
    hence c <= d by A1,A2,A4,Th9;
  end;
  assume c <= d;
  then a9 <= b9 by A1,A2,A4,Th9;
  hence thesis by A2,A3,Th8;
end;
