
theorem Th6: :: a version of 3.1 (vii, viii - left to right)
  for R, S being RelStr, a, b being object holds ([a,b] in the
  InternalRel of R implies [a,b] in the InternalRel of R [*] S) & ([a,b] in the
  InternalRel of S implies [a,b] in the InternalRel of R [*] S)
proof
  let R, S be RelStr, a, b be object;
  thus [a,b] in the InternalRel of R implies [a,b] in the InternalRel of R [*]
  S
  proof
    assume [a,b] in the InternalRel of R;
    then [a,b] in (the InternalRel of R) \/ (the InternalRel of S) by
XBOOLE_0:def 3;
    then [a,b] in (the InternalRel of R) \/ (the InternalRel of S) \/ ((the
    InternalRel of R) * the InternalRel of S) by XBOOLE_0:def 3;
    hence thesis by Def2;
  end;
  assume [a,b] in the InternalRel of S;
  then [a,b] in (the InternalRel of R) \/ (the InternalRel of S) by
XBOOLE_0:def 3;
  then [a,b] in (the InternalRel of R) \/ (the InternalRel of S) \/ ((the
  InternalRel of R) * the InternalRel of S) by XBOOLE_0:def 3;
  hence thesis by Def2;
end;
