
theorem Th47:
  for S1,S2 being unsplit gate`1=arity non empty ManySortedSign
  holds S1 tolerates S2
proof
  let S1,S2 be unsplit gate`1=arity non empty ManySortedSign;
  set A1 = the Arity of S1, A2 = the Arity of S2;
  set R1 = the ResultSort of S1, R2 = the ResultSort of S2;
  thus A1 tolerates A2
  proof
    let x be object;
    assume
A1: x in dom A1 /\ dom A2;
    then x in dom A2 by XBOOLE_0:def 4;
    then x in the carrier' of S2 by FUNCT_2:def 1;
    then
A2: x = [A2.x, x`2] by Def8;
    x in dom A1 by A1,XBOOLE_0:def 4;
    then x in the carrier' of S1 by FUNCT_2:def 1;
    then x = [A1.x, x`2] by Def8;
    hence thesis by A2,XTUPLE_0:1;
  end;
  let x be object;
  assume
A3: x in dom R1 /\ dom R2;
  then x in dom R1 by XBOOLE_0:def 4;
  then x in the carrier' of S1 by FUNCT_2:def 1;
  then
A4: R1.x = x by Th44;
  x in dom R2 by A3,XBOOLE_0:def 4;
  then x in the carrier' of S2 by FUNCT_2:def 1;
  hence thesis by A4,Th44;
end;
