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 Th41:
  for f being natural-valued finite-support Function
  for F being real-valued FinSequence st
  F = ((EmptyBag SetPrimes)+*f)*canFS(support((EmptyBag SetPrimes)+*f))
  holds F is positive-yielding
  proof
    let f be natural-valued finite-support Function;
    set b = B+*f;
    set C = canFS(support b);
    let F be real-valued FinSequence such that
A1: F = b*C;
A2: dom(b*C) = dom C by Th13;
A3: rng C = support b by FUNCT_2:def 3;
    let r;
    assume r in rng F;
    then consider x being object such that
A4: x in dom F and
A5: F.x = r by FUNCT_1:def 3;
     C.x in rng C by A1,A2,A4,FUNCT_1:def 3;
     then b.(C.x) <> 0 by A3,PRE_POLY:def 7;
     hence 0 < r by A1,A4,A5,FUNCT_1:12;
   end;
