
theorem Th50:
  for S1,S2 being gate`1=arity non empty ManySortedSign holds S1+*
  S2 is gate`1=arity
proof
  let S1,S2 be gate`1=arity non empty ManySortedSign;
  set S = S1+*S2;
  let g be set;
A1: dom the Arity of S1 = the carrier' of S1 by FUNCT_2:def 1;
A2: the Arity of S = (the Arity of S1)+*the Arity of S2 by Def2;
  assume
A3: g in the carrier' of S;
  then reconsider g as Gate of S;
A4: dom the Arity of S2 = the carrier' of S2 by FUNCT_2:def 1;
A5: the carrier' of S = (the carrier' of S1) \/ the carrier' of S2 by Def2;
  then
A6: g in the carrier' of S1 or g in the carrier' of S2 by A3,XBOOLE_0:def 3;
A7: now
    assume
A8: not g in the carrier' of S2;
    then reconsider g1 = g as Gate of S1 by A5,A3,XBOOLE_0:def 3;
    thus g = [(the Arity of S1).g1, g`2] by A6,A8,Def8
      .= [(the Arity of S).g, g`2] by A5,A2,A3,A1,A4,A8,FUNCT_4:def 1;
  end;
  now
    assume
A9: g in the carrier' of S2;
    then reconsider g2 = g as Gate of S2;
    thus g = [(the Arity of S2).g2, g`2] by A9,Def8
      .= [(the Arity of S).g, g`2] by A5,A2,A1,A4,A9,FUNCT_4:def 1;
  end;
  hence thesis by A7;
end;
