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 Th45:
  for n being set, d, b being bag of n
   st for k being object st k in n holds d.k <= b.k
  holds d divides b
proof
  let n be set, d, b be bag of n;
  assume
A1: for k being object st k in n holds d.k <= b.k;
  let k be object;
  per cases;
  suppose
    k in dom d;
    then k in n;
    hence thesis by A1;
  end;
  suppose
    not k in dom d;
    hence thesis by FUNCT_1:def 2;
  end;
end;
