
theorem Th4:
  for R being non empty RelStr holds R is connected iff
  the InternalRel of R is_strongly_connected_in the carrier of R
proof
  let R be non empty RelStr;
  set IR = the InternalRel of R, CR = the carrier of R;
  hereby
    assume
A1: R is connected;
    now
      let x, y be object such that
A2:   x in CR and
A3:   y in CR;
      reconsider x9=x, y9=y as Element of R by A2,A3;
      x9 <= y9 or y9 <= x9 by A1,WAYBEL_0:def 29;
      hence [x,y] in IR or [y,x] in IR;
    end;
    hence IR is_strongly_connected_in CR;
  end;
  assume IR is_strongly_connected_in CR;
  then for x, y be Element of R holds x <= y or y <= x;
  hence thesis by WAYBEL_0:def 29;
end;
