
theorem Th4:
  for ps,pt,f be bag of SetPrimes,
  q being Nat
  st (support ps) misses (support pt) & f = ps + pt & q in (support pt) holds
  pt.q = f.q
  proof
    let ps,pt,f be bag of SetPrimes,
    q be Nat;
    assume A1: (support ps) misses (support pt)
    & f = ps + pt & q in (support pt); then
    (support ps) /\ (support pt) = {} by XBOOLE_0:def 7; then
    A2:not q in (support ps) by A1,XBOOLE_0:def 4;
    thus f.q =ps.q + pt.q by A1,PRE_POLY:def 5
    .= 0 + pt.q by A2,PRE_POLY:def 7
    .= pt.q;
  end;
