theorem
  for A being disjoint_valued non-empty MSAlgebra over S
  for B being non-empty MSAlgebra over S
  for f1,f2 being ManySortedFunction of A,B st
    for a being Element of A holds f1.a = f2.a
    holds f1 = f2
  proof
    let A be disjoint_valued non-empty MSAlgebra over S;
    let B be non-empty MSAlgebra over S;
    let f1,f2 be ManySortedFunction of A,B;
    assume Z0: for a being Element of A holds f1.a = f2.a;
    let s;
    now
      thus f1.s is Function of (the Sorts of A).s, (the Sorts of B).s &
      f2.s is Function of (the Sorts of A).s, (the Sorts of B).s;
      let a be Element of (the Sorts of A).s;
      thus f1.s.a = f1.a by Th9 .= f2.a by Z0 .= f2.s.a by Th9;
    end;
    hence thesis by FUNCT_2:def 8;
  end;
