reserve x,y,z,x1,x2,y1,y2,z1,z2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  K for Ring;

theorem
  for I, J be non empty set for F,G be BinOp of D for f be Function of I
  ,D for g being Function of J,D st F is commutative associative & G is
commutative holds for x being Element of I for y being Element of J holds
  F $$([:{.x.},{.y.}:],G*(f,g))=F$$({.y.},G[;](F$$({.x.},f),g))
proof
  let I, J be non empty set;
  let F,G be BinOp of D;
  let f be Function of I,D;
  let g be Function of J,D;
  assume that
A1: F is commutative associative and
A2: G is commutative;
  now
    let x be Element of I;
    let y be Element of J;
    thus F $$ ([:{.x.},{.y.}:],G*(f,g))=F$$([:{.y.},{.x.}:],G*(g,f)) by A1,A2
,Th23
      .=F$$({.y.},G[:](g,F$$({.x.},f))) by A1,Th24
      .=F$$({.y.},G[;] (F$$({.x.},f),g)) by A2,FUNCOP_1:64;
  end;
  hence thesis;
end;
