reserve x,y for object, X for set;

theorem Th4:
  for X be set,p,q be bag of X st support p = {} & support q={} holds p = q
proof
  let X be set,p,q be bag of X;
  assume that
A1: support p = {} and
A2: support q={};
A3: {} = dom q /\ {} .= dom (q| (support q)) by A2;
A4: dom (p| (support p)) =dom p /\ (support p) by RELAT_1:61
    .= {} by A1;
  then
  for x be object st x in dom (p| (support p)) holds (p| (support p)).x = (q|
  (support q)).x;
  hence thesis by A4,A3,Th3,FUNCT_1:2;
end;
