reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th27:
  for X, Y, W be RealNormSpace,
      I be Function of X, Y,
      f be PartFunc of Y, W,
      r be Real holds
    r(#)(f*I) = (r(#)f)*I
  proof
    let X, Y, W be RealNormSpace,
        I be Function of X, Y,
        f be PartFunc of Y, W,
        r be Real;
    set DI = the carrier of X;
    A1: dom(r(#)f) = dom f by VFUNCT_1:def 4;
    A2: dom(r(#)(f*I)) = dom(f*I) by VFUNCT_1:def 4;
    A3: dom(I) = DI by FUNCT_2:def 1;
    A4b: for s be Element of DI holds s in dom((r(#)f)*I) iff s in dom(f*I)
    proof
      let s be Element of DI;
      s in dom((r(#)f)*I) iff I.s in dom(r(#)f) by A3,FUNCT_1:11;
      hence thesis by A1,A3,FUNCT_1:11;
    end; then
    A4: for s be object holds s in dom(r(#)(f*I))
    iff s in dom((r(#)f)*I) by A2; then
    A4a: dom(r(#)(f*I)) = dom((r(#)f)*I) by TARSKI:2;
    A5: for s be Element of DI holds s in dom((r(#)f)*I) iff I.s in dom(r(#)f)
    proof
      let s be Element of DI;
      dom(I)= DI by FUNCT_2:def 1;
      hence thesis by FUNCT_1:11;
    end;
    for z being Element of DI st z in dom(r(#)(f*I)) holds
    (r(#)(f*I)).z = ((r(#)f)*I).z
    proof
      let z be Element of DI;
      assume A6: z in dom(r(#)(f*I)); then
      A7: z in dom(f*I) by VFUNCT_1:def 4;
      A9: f/.(I.z) = f.(I.z) by A1,A5,A4a,A6,PARTFUN1:def 6
      .= (f*I).z by A7,FUNCT_1:12
      .= (f*I)/.z by A7,PARTFUN1:def 6;
      A10: (r(#)(f*I)).z =(r(#)(f*I))/.z by A6,PARTFUN1:def 6
      .= r * f/.(I.z) by A6,A9,VFUNCT_1:def 4;
      ((r(#)f)*I).z = (r(#)f).(I.z) by A2,A4b,A6,FUNCT_1:12
      .= (r(#)f)/.(I.z) by A5,A4a,A6,PARTFUN1:def 6
      .= r * f/.(I.z) by A5,A4a,A6,VFUNCT_1:def 4;
      hence thesis by A10;
    end;
    hence thesis by A4,TARSKI:2,PARTFUN1:5;
  end;
