reserve A for non empty set,
  S for non void non empty ManySortedSign;
reserve x for set;
reserve o for OperSymbol of S;

theorem Th3:
  for U1,U2 be MSAlgebra over S st the Sorts of U1
is_transformable_to the Sorts of U2 & Args (o,U1) <> {} holds Args (o,U2) <> {}
proof
  let U1,U2 be MSAlgebra over S;
  assume that
A1: the Sorts of U1 is_transformable_to the Sorts of U2 and
A2: Args (o,U1) <> {};
A3: ((the Sorts of U1)# * the Arity of S).o <> {} by A2,MSUALG_1:def 4;
  dom (the Sorts of U1) = the carrier of S by PARTFUN1:def 2;
  then rng (the_arity_of o) c= dom (the Sorts of U1);
  then
A4: dom ((the Sorts of U1) * (the_arity_of o)) = dom (the_arity_of o) by
RELAT_1:27;
A5: dom ((the Sorts of U2) * (the_arity_of o)) c= dom (the_arity_of o) by
RELAT_1:25;
A6: dom (the Arity of S) = the carrier' of S by FUNCT_2:def 1;
  then ((the Sorts of U1)# * the Arity of S).o = (the Sorts of U1)# . ((the
  Arity of S).o) by FUNCT_1:13
    .= (the Sorts of U1)# . (the_arity_of o) by MSUALG_1:def 1;
  then product ((the Sorts of U1) * (the_arity_of o)) <> {} by A3,
FINSEQ_2:def 5;
  then
A7: not {} in rng ((the Sorts of U1) * (the_arity_of o)) by CARD_3:26;
  now
    let x be object;
    assume
A8: x in dom ((the Sorts of U2) * ( the_arity_of o ) );
    then (the_arity_of o).x in rng (the_arity_of o) by A5,FUNCT_1:def 3;
    then
A9: (the Sorts of U1).((the_arity_of o).x) <> {} implies (the Sorts of U2
    ).((the_arity_of o).x) <> {} by A1,PZFMISC1:def 3;
    ((the Sorts of U1) * (the_arity_of o)).x = (the Sorts of U1).((
    the_arity_of o).x) by A5,A8,FUNCT_1:13;
    hence ((the Sorts of U2)*(the_arity_of o)).x <> {} by A7,A4,A5,A8,A9,
FUNCT_1:13,def 3;
  end;
  then not {} in rng ((the Sorts of U2) * (the_arity_of o)) by FUNCT_1:def 3;
  then product ((the Sorts of U2)*(the_arity_of o)) <> {} by CARD_3:26;
  then (the Sorts of U2)# . (the_arity_of o) <> {} by FINSEQ_2:def 5;
  then (the Sorts of U2)# . ((the Arity of S).o) <> {} by MSUALG_1:def 1;
  then ((the Sorts of U2)# * the Arity of S).o <> {} by A6,FUNCT_1:13;
  hence thesis by MSUALG_1:def 4;
end;
