
theorem Th8: :: theorem 3.1 (vii)
  for R, S being non empty RelStr for x, y being Element of R, a, b
  being Element of R [*] S st x = a & y = b & R tolerates S & R is transitive
  holds x <= y iff a <= b
proof
  let R, S be non empty RelStr, x, y be Element of R, a, b be Element of R [*]
  S;
  assume
A1: x = a & y = b;
  assume
A2: R tolerates S & R is transitive;
  hereby
    assume x <= y;
    then [x,y] in the InternalRel of R by ORDERS_2:def 5;
    then [a,b] in the InternalRel of R [*] S by A1,Th6;
    hence a <= b by ORDERS_2:def 5;
  end;
  assume a <= b;
  then [a,b] in the InternalRel of R [*] S by ORDERS_2:def 5;
  then [x,y] in the InternalRel of R by A1,A2,Th4;
  hence thesis by ORDERS_2:def 5;
end;
