reserve S for non empty non void ManySortedSign,
  A for MSAlgebra over S;

theorem Th8:
  for A1,A2 being MSAlgebra over S, h being ManySortedFunction of
A1,A2 for o being OperSymbol of S st Args(o,A1) <> {} & Args(o,A2) <> {} for i
  being Element of NAT st i in dom the_arity_of o for x being Element of A1,(
  the_arity_of o)/.i holds h.((the_arity_of o)/.i).x in (the Sorts of A2).((
  the_arity_of o)/.i)
proof
  let A1,A2 be MSAlgebra over S, h be ManySortedFunction of A1,A2;
  let o be OperSymbol of S such that
A1: Args(o,A1) <> {} and
A2: Args(o,A2) <> {};
  let i be Element of NAT;
  assume
A3: i in dom the_arity_of o;
  then
A4: (the Sorts of A2).((the_arity_of o)/.i) <> {} by A2,Th3;
  (the Sorts of A1).((the_arity_of o)/.i) <> {} by A1,A3,Th3;
  hence thesis by A4,FUNCT_2:5;
end;
