
theorem Th2:
  for S being non empty non void ManySortedSign for A1,A2 being
  MSAlgebra over S st the Sorts of A1 is_transformable_to the Sorts of A2 for o
  being OperSymbol of S st Args(o,A1) <> {} holds Args(o,A2) <> {}
proof
  let S be non void non empty ManySortedSign;
  let A1,A2 be MSAlgebra over S such that
A1: for i be set st i in the carrier of S & (the Sorts of A2).i = {}
  holds (the Sorts of A1).i = {};
  let o be OperSymbol of S;
  assume
A2: Args(o,A1) <> {};
  now
    let i be Element of NAT;
    assume i in dom the_arity_of o;
    then (the Sorts of A1).((the_arity_of o)/.i) <> {} by A2,MSUALG_6:3;
    hence (the Sorts of A2).((the_arity_of o)/.i) <> {} by A1;
  end;
  hence thesis by MSUALG_6:3;
end;
