reserve x,y,z, X,Y,Z for set,
  n for Element of NAT;
reserve A for set,
  D for non empty set,
  a,b,c,l,r for Element of D,
  o,o9 for BinOp of D,
  f,g,h for Function of A,D;
reserve G for non empty multMagma;

theorem Th27:
  x is Multiset of X iff x is Function of X,NAT
proof
A1: now
    let x be Function of X,NAT;
    dom x = X & rng x c= NAT by FUNCT_2:def 1;
    hence x is Element of Funcs(X,NAT) by FUNCT_2:def 2;
  end;
  x is Multiset of X iff x is Element of Funcs(X,NAT) by Th26;
  hence thesis by A1;
end;
