reserve x,r,a,x0,p for Real;
reserve n,i,m for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve k for Nat;

theorem
  (r*p)(#)(( #Z n)^)=r(#)(p(#)(( #Z n)^))
proof
A1: dom ((r*p) (#) (( #Z n)^)) = dom((( #Z n)^)) by VALUED_1:def 5
    .= dom (p(#)(( #Z n)^)) by VALUED_1:def 5
    .= dom (r(#)(p(#)(( #Z n)^))) by VALUED_1:def 5;
  now
    let c be object;
    assume
A2: c in dom ((r*p)(#)(( #Z n)^));
    then
A3: c in dom (p(#)(( #Z n)^)) by A1,VALUED_1:def 5;
    thus ((r*p)(#)(( #Z n)^)).c = r*p * (( #Z n)^).c by A2,VALUED_1:def 5
      .= r*(p * (( #Z n)^).c)
      .= r * (p(#)(( #Z n)^)).c by A3,VALUED_1:def 5
      .= (r(#)(p(#)(( #Z n)^))).c by A1,A2,VALUED_1:def 5;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
