
theorem Th5:
  for S1,S2 being non empty ManySortedSign st S1 tolerates S2 holds
  S1+*S2 = S2+*S1
proof
  let S1,S2 be non empty ManySortedSign;
  set S = S1+*S2;
  assume that
A1: the Arity of S1 tolerates the Arity of S2 and
A2: the ResultSort of S1 tolerates the ResultSort of S2;
A3: the carrier' of S = (the carrier' of S1) \/ (the carrier' of S2) by Def2;
  (the ResultSort of S1)+*the ResultSort of S2 = (the ResultSort of S2)+*
  the ResultSort of S1 by A2,FUNCT_4:34;
  then
A4: the ResultSort of S = (the ResultSort of S2) +* (the ResultSort of S1)
  by Def2;
  (the Arity of S1)+*the Arity of S2 = (the Arity of S2)+*the Arity of S1
  by A1,FUNCT_4:34;
  then
A5: the Arity of S = (the Arity of S2) +* (the Arity of S1) by Def2;
  the carrier of S = (the carrier of S1) \/ (the carrier of S2) by Def2;
  hence thesis by A3,A5,A4,Def2;
end;
