reserve x,y for set;
reserve C,C9,D,E for non empty set;
reserve c,c9,c1,c2,c3 for Element of C;
reserve B,B9,B1,B2 for Element of Fin C;
reserve A for Element of Fin C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve f,f9 for Function of C,D;
reserve g for Function of C9,D;
reserve H for BinOp of E;
reserve h for Function of D,E;
reserve i,j for Nat;
reserve s for Function;
reserve p,q for FinSequence of D;
reserve T1,T2 for Element of i-tuples_on D;

theorem
  F is commutative associative & F is having_a_unity & F is
having_an_inverseOp & G is_distributive_wrt F implies G[;](d,id D).(F$$(B,f)) =
  F $$(B,G[;](d,id D)*f)
proof
  assume that
A1: F is commutative associative & F is having_a_unity and
A2: F is having_an_inverseOp and
A3: G is_distributive_wrt F;
  set e = the_unity_wrt F;
  set u = G[;](d,id D);
  u is_distributive_wrt F by A3,FINSEQOP:54;
  then
A4: for d1,d2 holds u.(F.(d1,d2)) = F.(u.d1,u.d2);
  G[;](d,id D).e = e by A1,A2,A3,FINSEQOP:69;
  hence thesis by A1,A4,Th16;
end;
