
theorem :: theorem 1 (ix)
  for x, y being set, R, S being RelStr st [x,y] in the InternalRel of R
  [*] S & (the carrier of R) /\ (the carrier of S) is upper Subset of R holds x
in the carrier of R & y in the carrier of R or x in the carrier of S & y in the
  carrier of S or x in (the carrier of R) \ (the carrier of S) & y in (the
  carrier of S) \ (the carrier of R)
proof
  let x, y be set, R, S be RelStr;
  assume that
A1: [x,y] in the InternalRel of R [*] S and
A2: (the carrier of R) /\ (the carrier of S) is upper Subset of R;
  x in the carrier of R [*] S by A1,ZFMISC_1:87;
  then
A3: x in (the carrier of R) \/ (the carrier of S) by Def2;
  y in the carrier of R [*] S by A1,ZFMISC_1:87;
  then
A4: y in (the carrier of R) \/ (the carrier of S) by Def2;
  per cases by A3,A4,XBOOLE_0:def 3;
  suppose
    x in the carrier of R & y in the carrier of R;
    hence thesis;
  end;
  suppose
    x in the carrier of S & y in the carrier of S;
    hence thesis;
  end;
  suppose
    x in the carrier of R & y in the carrier of S;
    hence thesis by XBOOLE_0:def 5;
  end;
  suppose
    x in the carrier of S & y in the carrier of R;
    hence thesis by A1,A2,Th17;
  end;
end;
