
theorem Th22:
  for S1,S2 being non empty ManySortedSign, A1 being non-empty
  MSAlgebra over S1, A2 being non-empty MSAlgebra over S2 st A1 tolerates A2
  holds A1+*A2 = A2+*A1
proof
  let S1,S2 be non empty ManySortedSign;
  let A1 be non-empty MSAlgebra over S1, A2 be non-empty MSAlgebra over S2;
  set A = A1+*A2;
  assume that
A1: S1 tolerates S2 and
A2: the Sorts of A1 tolerates the Sorts of A2 and
A3: the Charact of A1 tolerates the Charact of A2;
  (the Sorts of A1)+*the Sorts of A2 = (the Sorts of A2)+*the Sorts of A1
  by A2,FUNCT_4:34;
  then
A4: the Sorts of A = (the Sorts of A2) +* (the Sorts of A1) by A2,Def4;
  (the Charact of A1)+*the Charact of A2 = (the Charact of A2)+*the
  Charact of A1 by A3,FUNCT_4:34;
  then
A5: the Charact of A = (the Charact of A2) +* (the Charact of A1) by A2,Def4;
  S1+*S2 = S2+*S1 by A1,Th5;
  hence thesis by A2,A4,A5,Def4;
end;
