reserve n for Nat;

theorem
for X being non empty set,
    b being bag of X,
    a being Element of X holds b \ a = b iff not a in support b
proof
let X be non empty set, b be bag of X, a be Element of X;
A: now assume B: not a in support b;
   now let o be object;
     assume o in X;
     per cases;
     suppose D: o = a;
       hence (b\a).o = 0 by bb1 .= b.o by B,D,PRE_POLY:def 7;
       end;
     suppose o <> a;
       hence (b\a).o = b.o by FUNCT_7:32;
       end;
     end;
   hence b \ a = b by PBOOLE:3;
   end;
now assume b \ a = b;
   then b.a = 0 by bb1;
   hence not a in support b by PRE_POLY:def 7;
   end;
hence thesis by A;
end;
