
theorem Th3:
  for R, S being RelStr st R is reflexive & S is reflexive holds R
  [*] S is reflexive
proof
  let R, S be RelStr;
  assume R is reflexive & S is reflexive;
  then
A1: the InternalRel of R is_reflexive_in the carrier of R & the InternalRel
  of S is_reflexive_in the carrier of S by ORDERS_2:def 2;
A2: the InternalRel of R c= the InternalRel of R [*] S & the InternalRel of
  S c= the InternalRel of R [*] S by Th2;
  the InternalRel of R [*] S is_reflexive_in the carrier of R [*] S
  proof
    let x be object;
    assume x in the carrier of R [*] S;
    then x in (the carrier of R) \/ (the carrier of S) by Def2;
    then x in the carrier of R or x in the carrier of S by XBOOLE_0:def 3;
    then
    [x,x] in the InternalRel of R or [x,x] in the InternalRel of S by A1;
    hence thesis by A2;
  end;
  hence thesis by ORDERS_2:def 2;
end;
