
theorem Th9: :: theorem 3.1 (viii)
  for R, S being non empty RelStr, a, b being Element of R [*] S, c
, d being Element of S st a = c & b = d & R tolerates S & S is transitive holds
  a <= b iff c <= d
proof
  let R, S be non empty RelStr, a, b be Element of R [*] S, c, d be Element of
  S;
  assume that
A1: a = c & b = d and
A2: R tolerates S & S is transitive;
  hereby
    assume a <= b;
    then [a,b] in the InternalRel of R [*] S by ORDERS_2:def 5;
    then [c,d] in the InternalRel of S by A1,A2,Th5;
    hence c <= d by ORDERS_2:def 5;
  end;
  assume c <= d;
  then [c,d] in the InternalRel of S by ORDERS_2:def 5;
  then [a,b] in the InternalRel of R [*] S by A1,Th6;
  hence thesis by ORDERS_2:def 5;
end;
