
theorem Th2:
  for R, S being RelStr holds the InternalRel of R c= the
  InternalRel of R [*] S & the InternalRel of S c= the InternalRel of R [*] S
proof
  let R, S be RelStr;
  the InternalRel of R c= (the InternalRel of R) \/ (the InternalRel of S)
  & ( the InternalRel of R) \/ (the InternalRel of S) c= (the InternalRel of R)
  \/ ( the InternalRel of S) \/ ((the InternalRel of R) * the InternalRel of S)
  by XBOOLE_1:7;
  then
  the InternalRel of R c= (the InternalRel of R) \/ (the InternalRel of S)
  \/ ((the InternalRel of R) * the InternalRel of S) by XBOOLE_1:1;
  hence the InternalRel of R c= the InternalRel of R [*] S by Def2;
  the InternalRel of S c= (the InternalRel of R) \/ (the InternalRel of S)
  & ( the InternalRel of R) \/ (the InternalRel of S) c= (the InternalRel of R)
  \/ ( the InternalRel of S) \/ ((the InternalRel of R) * the InternalRel of S)
  by XBOOLE_1:7;
  then
  the InternalRel of S c= (the InternalRel of R) \/ (the InternalRel of S)
  \/ ((the InternalRel of R) * the InternalRel of S) by XBOOLE_1:1;
  hence thesis by Def2;
end;
