
theorem Th12:
  for S1,S2 being non empty ManySortedSign for v2 being Vertex of
  S2 st v2 in InputVertices (S1+*S2) holds v2 in InputVertices S2
proof
  let S1,S2 be non empty ManySortedSign;
  set R1 = the ResultSort of S1, R2 = the ResultSort of S2;
  set S = S1+*S2, R = the ResultSort of S;
  let v2 be Vertex of S2;
  R = R1+*R2 by Def2;
  then
A1: rng R2 c= rng R by FUNCT_4:18;
  assume v2 in InputVertices S;
  then not v2 in rng R2 by A1,XBOOLE_0:def 5;
  hence thesis by XBOOLE_0:def 5;
end;
