
theorem Th13:
  for L being reflexive RelStr, S1, S2 being full SubRelStr of L
  st the carrier of S1 c= the carrier of S2 holds S1 is SubRelStr of S2
proof
  let L be reflexive RelStr, S1,S2 be full SubRelStr of L;
  assume
A1: the carrier of S1 c= the carrier of S2;
  hence the carrier of S1 c= the carrier of S2;
  let x,y be object;
  assume
A2: [x,y] in the InternalRel of S1;
  then
A3: x in the carrier of S1 by ZFMISC_1:87;
  reconsider x,y as Element of S1 by A2,ZFMISC_1:87;
  the carrier of S1 c= the carrier of L by YELLOW_0:def 13;
  then reconsider a = x, b = y as Element of L by A3;
  reconsider x9 = x, y9 = y as Element of S2 by A1,A3;
  x <= y by A2;
  then a <= b by YELLOW_0:59;
  then x9 <= y9 by A1,A3,YELLOW_0:60;
  hence thesis;
end;
