
theorem bag2a:
for Z being non empty set
for B being bag of Z, o being object holds B.o <= card B
proof
let Z be non empty set, B be bag of Z, o be object;
consider f being FinSequence of NAT such that
A: degree B = Sum f & f = B*canFS(support B) by UPROOTS:def 4;
set cS = canFS(support B);
now assume AS: B.o > card B;
  then o in support B by PRE_POLY:def 7;
  then o in rng cS by FUNCT_2:def 3;
  then consider i being Nat such that
  B: i in dom cS & cS.i = o by FINSEQ_2:10;
  reconsider i as Element of NAT by ORDINAL1:def 12;
  f.i > card B by AS,A,B,FUNCT_1:13;
  hence contradiction by A,NEWTON04:21;
  end;
hence thesis;
end;
