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 being non empty set for F,G being BinOp of D for f being
Function of I,D for g being Function of J,D for X being Element of Fin I for Y
being Element of Fin J st F is having_a_unity & F is commutative associative &
F is having_an_inverseOp & G is_distributive_wrt F & G is commutative holds F$$
  ([:X,Y:],G*(f,g))=F$$(Y,G[;](F$$(X,f),g))
proof
  let I, J be non empty set;
  let F,G be BinOp of D;
  let f be Function of I,D;
  let g be Function of J,D;
  let X be Element of Fin I;
  let Y be Element of Fin J;
  assume that
A1: F is having_a_unity & F is commutative & F is associative and
A2: F is having_an_inverseOp & G is_distributive_wrt F and
A3: G is commutative;
  thus F $$ ([:X,Y:],G*(f,g))=F$$([:Y,X:],G*(g,f)) by A1,A3,Th23
    .=F$$(Y,G[:](g,F$$(X,f))) by A1,A2,Th26
    .=F$$(Y,G[;](F$$(X,f),g)) by A3,FUNCOP_1:64;
end;
