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 Th15:
  F is idempotent & F is commutative & F is associative implies
  for a,b being Element of X holds F $$ ({.a,b.},f) = F.(f.a, f.b)
proof
  assume
A1: F is idempotent & F is commutative & F is associative;
  let a,b be Element of X;
  consider G being Function of Fin X, Y such that
A2: F $$ ({.a,b.},f) = G.{a,b} and
  for e being Element of Y st e is_a_unity_wrt F holds G.{} = e and
A3: for x being Element of X holds G.{x} = f.x and
A4: for B9 being Element of Fin X st B9 c= {a,b} & B9 <> {} for x being
  Element of X st x in {a,b} holds G.(B9 \/ {x}) = F.(G.B9,f.x) by A1,Th13;
A5: b in {a,b} by TARSKI:def 2;
  thus F $$ ({.a,b.},f) = G.({.a.} \/ {.b.}) by A2,ENUMSET1:1
    .= F.(G.{.a.}, f.b) by A4,A5,ZFMISC_1:7
    .= F.(f.a, f.b) by A3;
end;
