reserve n for Nat;

theorem bb4:
for X being non empty set,
    a being Element of X,
    n being Element of NAT holds card(({a},n)-bag) = n
proof
let X be non empty set, a be Element of X, n be Element of NAT;
reconsider b = ({a},n)-bag as bag of X;
consider F being FinSequence of NAT such that
H: degree b = Sum F & F = b*canFS(support b) by UPROOTS:def 4;
I: a in {a} by TARSKI:def 1;
per cases;
suppose X: n = 0;
  then b = EmptyBag X by UPROOTS:9;
  then support b = {};
  hence thesis by X,bag1a;
  end;
suppose n <> 0;
  then A: support b = {a} by UPROOTS:8;
  then C: a in support b by TARSKI:def 1;
  B: support b c= dom b by PRE_POLY:37;
  F = b * <*a*> by A,H,FINSEQ_1:94
   .= <*b.a*> by C,B,FINSEQ_2:34
   .= <*n*> by I,UPROOTS:7;
  hence thesis by H,RVSUM_1:73;
  end;
end;
