reserve n for Nat;

theorem bb7a:
for X being non empty set,
    b being bag of X,
    x being Element of X holds support b = {x} implies b = ({x},b.x)-bag
proof
let X be non empty set, b be bag of X, x be Element of X;
assume AS: support b = {x};
now let o be object;
  assume o in X;
  per cases;
  suppose A: o = x;
    then o in {x} by TARSKI:def 1;
    hence b.o = (({x},b.x)-bag).o by A,UPROOTS:7;
    end;
  suppose o <> x;
    then B: not o in {x} by TARSKI:def 1;
    hence (({x},b.x)-bag).o = 0 by UPROOTS:6
                           .= b.o by AS,B,PRE_POLY:def 7;
    end;
  end;
hence b = ({x},b.x)-bag by PBOOLE:3;
end;
