
theorem Th11:
  for S1,S2 being non empty ManySortedSign st S1 tolerates S2
  holds InnerVertices (S1+*S2) = (InnerVertices S1) \/ (InnerVertices S2) &
  InputVertices (S1+*S2) c= (InputVertices S1) \/ (InputVertices S2)
proof
  let S1,S2 be non empty ManySortedSign;
  set R1 = the ResultSort of S1, R2 = the ResultSort of S2;
  assume that
  the Arity of S1 tolerates the Arity of S2 and
A1: R1 tolerates R2;
  set S = S1+*S2, R = the ResultSort of S;
A2: R = R1+*R2 by Def2;
  then R1 c= R by A1,FUNCT_4:28;
  then
A3: rng R1 c= rng R by RELAT_1:11;
  rng R2 c= rng R by A2,FUNCT_4:18;
  then
A4: (rng R1) \/ (rng R2) c= rng R by A3,XBOOLE_1:8;
A5: rng R c= (rng R1) \/ (rng R2) by A2,FUNCT_4:17;
  hence InnerVertices S = (InnerVertices S1) \/ (InnerVertices S2) by A4;
  let x be object;
  assume
A6: x in InputVertices S;
  then
A7: not x in rng R by XBOOLE_0:def 5;
A8: rng R = (rng R1) \/ (rng R2) by A5,A4;
  then
A9: not x in rng R2 by A7,XBOOLE_0:def 3;
  the carrier of S = (the carrier of S1) \/ the carrier of S2 by Def2;
  then
A10: x in the carrier of S1 or x in the carrier of S2 by A6,XBOOLE_0:def 3;
  not x in rng R1 by A8,A7,XBOOLE_0:def 3;
  then x in (the carrier of S1) \ rng R1 or x in (the carrier of S2) \ rng R2
  by A10,A9,XBOOLE_0:def 5;
  hence thesis by XBOOLE_0:def 3;
end;
