reserve I,J for set,i,j,x for object,
  S for non empty ManySortedSign;

theorem Th6:
  for I be non empty set, S be non void non empty ManySortedSign,
  A be MSAlgebra-Family of I,S, o be OperSymbol of S holds commute (OPER A) in
  Funcs(the carrier' of S, Funcs(I,rng uncurry (OPER A)))
proof
  let I be non empty set, S be non void non empty ManySortedSign, A be
  MSAlgebra-Family of I,S, o be OperSymbol of S;
  set f = uncurry (OPER A);
  dom f = [:I,the carrier' of S:] by Th5;
  then
  [:I,the carrier' of S:] <> {} & f in Funcs([:I,the carrier' of S:],rng f
  ) by FUNCT_2:def 2,ZFMISC_1:90;
  hence thesis by FUNCT_6:10;
end;
