reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem Th60:
  for X being set, f being finite-support Function of X, NAT holds
  NatMinor f c= Bags X
proof
  let X be set, f be finite-support Function of X, NAT;
  let x be object;
  assume x in NatMinor f;
  then reconsider x9 = x as Element of NatMinor f;
A1: dom x9 = X by FUNCT_2:92;
  then
A2: x9 is ManySortedSet of X by PARTFUN1:def 2,RELAT_1:def 18;
  support x9 c= support f
  proof
    let a be object;
A3: support x9 c= dom x9 by Th36;
    assume
A4: a in support x9;
    then x9.a <> 0 by Def7;
    then f.a <> 0 by A1,A2,A4,A3,Def14;
    hence thesis by Def7;
  end;
  then x is bag of X by A1,Def8,PARTFUN1:def 2,RELAT_1:def 18;
  hence thesis by Def12;
end;
