theorem
  f is_simple_func_in S implies r(#)f is_simple_func_in S
proof
  set g = r(#)f;
  assume f is_simple_func_in S;
  then consider G be Finite_Sep_Sequence of S such that
A1: dom f = union rng G and
A2: for n be Nat, x,y be Element of X st n in dom G & x in G.n & y in G.
  n holds f.x = f.y;
A3: dom g = dom f by VALUED_1:def 5;
  now
    let n be Nat;
    let x,y be Element of X;
    assume that
A4: n in dom G and
A5: x in G.n and
A6: y in G.n;
A7: G.n in rng G by A4,FUNCT_1:3;
    then y in dom g by A3,A1,A6,TARSKI:def 4;
    then
A8: g.y = r*f.y by VALUED_1:def 5;
    x in dom g by A3,A1,A5,A7,TARSKI:def 4;
    then g.x = r*f.x by VALUED_1:def 5;
    hence g.x = g.y by A2,A4,A5,A6,A8;
  end;
  hence thesis by A3,A1;
end;
