reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;
reserve i,j for natural Number;
reserve F for Function of [:D,D9:],E;
reserve p,q for FinSequence of D,
  p9,q9 for FinSequence of D9;
reserve f,f9 for Function of C,D,
  h for Function of D,E;
reserve T,T1,T2,T3 for Tuple of i,D;
reserve T9 for Tuple of i, D9;
reserve S for Tuple of j, D;
reserve S9 for Tuple of j, D9;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve H for BinOp of E;

theorem
  G is_distributive_wrt F & u = G[;](d,id D) implies u is_distributive_wrt F
proof
  assume that
A1: G is_distributive_wrt F and
A2: u = G[;](d,id D);
  let d1,d2;
  thus u.(F.(d1,d2)) = G.(d,(id D).(F.(d1,d2))) by A2,FUNCOP_1:53
    .= G.(d,F.(d1,d2))
    .= F.(G.(d,d1),G.(d,d2)) by A1,BINOP_1:11
    .= F.(G.(d,(id D).d1),G.(d,d2))
    .= F.(G.(d,(id D).d1),G.(d,(id D).d2))
    .= F.(u.d1,G.(d,(id D).d2)) by A2,FUNCOP_1:53
    .= F.(u.d1,u.d2) by A2,FUNCOP_1:53;
end;
