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 Th12:
  for f being Function holds support (EmptyBag X +* f) = support f
  proof
    let f be Function;
    set F = EmptyBag X +* f;
    support F c= support EmptyBag X \/ support f by GLIB_004:1;
    hence support F c= support f;
    let x be object;
    assume
A1: x in support f;
    then
A2: f.x <> 0 by PRE_POLY:def 7;
    support f c= dom f by PRE_POLY:37;
    then F.x = f.x by A1,FUNCT_4:13;
    hence thesis by A2,PRE_POLY:def 7;
  end;
