reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;

theorem
  F is commutative & F is associative & F is idempotent implies for x
  being Element of X st x in B holds F.(f.x,F$$(B,f)) = F$$(B,f)
proof
  assume that
A1: F is commutative & F is associative and
A2: F is idempotent;
  let x be Element of X;
  assume
A3: x in B;
  thus F.(f.x,F$$(B,f)) = F.(F$$({.x.},f), F$$(B,f)) by A1,Th14
    .= F$$({.x.} \/ B, f) by A1,A2,A3,Th18
    .= F$$(B,f) by A3,ZFMISC_1:40;
end;
