reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th4:
  for f1,f2,f3 being complex-valued ManySortedSet of X st
  for x being object st x in X holds f1.x = f2.x * f3.x holds
  f1 = f2(#)f3
  proof
    let f1,f2,f3 be complex-valued ManySortedSet of X such that
A1: for x being object st x in X holds f1.x = f2.x * f3.x;
A2: dom f1 = X & dom f2 = X & dom f3 = X by PARTFUN1:def 2;
    dom(f2(#)f3) = dom f2 /\ dom f3 by VALUED_1:def 4;
    hence thesis by A1,A2,VALUED_1:def 4;
  end;
