reserve n for Nat;

theorem bb3a:
for X being non empty set,
    b being bag of X,
    a being Element of X holds support(b \ a) = support b \ {a}
proof
let X be non empty set, b be bag of X, a be Element of X;
A: now let o be object;
   assume X: o in support(b \ a);
   then reconsider c = o as Element of X;
   B: (b\a).o <> 0 by X,PRE_POLY:def 7;
   then D: a <> o by bb1;
   then b.c = (b\a).c by FUNCT_7:32;
   then C: o in support b by B,PRE_POLY:def 7;
   not o in {a} by D,TARSKI:def 1;
   hence o in support b \ {a} by C,XBOOLE_0:def 5;
   end;
now let o be object;
   assume X: o in support b \ {a};
   then reconsider c = o as Element of X;
   B: o in support b & not o in {a} by X,XBOOLE_0:def 5;
   then o <> a by TARSKI:def 1;
   then (b\a).c  = b.c by FUNCT_7:32;
   then (b\a).o <> 0 by B,PRE_POLY:def 7;
   hence o in support(b \ a) by PRE_POLY:def 7;
   end;
hence thesis by A,TARSKI:2;
end;
