reserve D for non empty set,
  D1,D2,x,y for set,
  n,k for Nat,
  p,x1 ,r for Real,
  f for Function;
reserve F for Functional_Sequence of D1,D2;
reserve G,H,H1,H2,J for Functional_Sequence of D,REAL;

theorem
  r<>0 implies (r(#)H)" = r" (#) H"
proof
  assume
A1: r<>0;
  now
    let n be Element of NAT;
    thus (r(#)H)".n = ((r(#)H).n)^ by Def2
      .= (r (#) H.n)^ by Def1
      .= r" (#) ((H.n)^) by A1,RFUNCT_1:28
      .= r" (#) (H".n) by Def2
      .= (r" (#) H").n by Def1;
  end;
  hence thesis by FUNCT_2:63;
end;
