reserve x,y for object, X for set;

theorem
  for n,m,k be non zero Nat st k = n gcd m holds support ppf
  k = (support ppf n) /\ (support ppf m)
proof
  let n,m,k be non zero Nat;
  assume
A1: k = n gcd m;
A2: support ppf n = support pfexp n by NAT_3:def 9;
A3: support ppf m = support pfexp m by NAT_3:def 9;
  support ppf k =support pfexp k by NAT_3:def 9
    .=support min(pfexp n,pfexp m) by A1,NAT_3:53;
  hence thesis by A2,A3,Th10;
end;
