reserve k for Element of NAT;
reserve r,r1 for Real;
reserve i for Integer;
reserve q for Rational;
reserve X for set;
reserve f for PartFunc of X,ExtREAL;
reserve S for SigmaField of X;
reserve F for sequence of S;
reserve A for set;
reserve a for ExtReal;
reserve r,s for Real;
reserve n,m for Element of NAT;
reserve X for non empty set;
reserve x for Element of X;
reserve f,g for PartFunc of X,ExtREAL;
reserve S for SigmaField of X;
reserve A,B for Element of S;

theorem
  for X,S,f,A,r st f is A-measurable & A c= dom f
  holds r(#)f is A-measurable
proof
  let X,S,f,A,r;
  assume that
A1: f is A-measurable and
A2: A c= dom f;
 for r1 being Real holds A /\ less_dom(r(#)f,r1) in S
  proof
    let r1 being Real;
 now per cases;
      suppose
A3:     r<>0;
        reconsider r0=r1/r as Real;
A4:     r1=r*r0 by A3,XCMPLX_1:87;
     now per cases by A3;
          suppose
A5:         r > 0;
        for x being Element of X st x in less_dom(f,r0)
            holds x in less_dom(r(#)f,r1)
            proof
              let x be Element of X;
              assume
A6:          x in less_dom(f,r0);
then           x in dom f by Def11;
then A7:          x in dom(r(#)f) by Def6;
A8:          f.x < r0 by A6,Def11;
f.x < r1 / r by A8;
then A9: f.x * r < r1 / r * (r qua ExtReal) by A5,XXREAL_3:72;
A10:         r1 / r * r = (r1/r)*r
                .= r1/(r/r) by XCMPLX_1:81
                .= r1/1 by A3,XCMPLX_1:60
                .= r1;
          (r(#)f).x = r * f.x by A7,Def6;
              hence thesis by A7,A9,A10,Def11;
            end;
            then A11:        less_dom
(f,r0) c= less_dom(r(#)f,r1);
        for x being Element of X st x in less_dom(r(#)f,r1)
            holds x in less_dom(f,r0)
            proof
              let x being Element of X;
              assume
A12:          x in less_dom(r(#)f,r1);
          then
A13:          x in dom(r(#)f) by Def11;
          (r(#)f).x < r1 by A12,Def11;
              then (r(#)f).x < r * r0 by A4;
              then
A14:          (r(#)f).x / r < r * r0 / (r qua ExtReal)
              by A5,XXREAL_3:80;
A15:          r * r0 / r = (r*r0)/r
                .= r0/(r/r) by XCMPLX_1:77
                .= r0/1 by A3,XCMPLX_1:60
                .= r0;
            x in dom f & f.x = (r(#)f).x / r by A3,A13,Def6,Th6;
              hence thesis by A14,A15,Def11;
            end;
then
        less_dom(r(#)f,r1) c= less_dom(f,r0);
then         less_dom
(f,r0) = less_dom(r(#)f,r1) by A11,XBOOLE_0:def 10;
            hence thesis by A1;
          end;
          suppose
A16:        r < 0;
        for x being Element of X st x in great_dom(f,r0)
            holds x in less_dom(r(#)f,r1)
            proof
              let x be Element of X;
              assume
A17:          x in great_dom(f,r0);
then           x in dom f by Def13;
then A18:          x in dom(r(#)f) by Def6;
           r0 < f.x by A17,Def13;
then           r1 / r < f.x;
then A19:          f .x * r < r1 / r * (r qua ExtReal) by A16,
XXREAL_3:102;
A20:          r1 / r * r = (r1/r)*r
                .= r1/(r/r) by XCMPLX_1:81
                .= r1/1 by A3,XCMPLX_1:60
                .= r1;
          (r(#)f).x = r * f.x by A18,Def6;
              hence thesis by A18,A19,A20,Def11;
            end;
            then A21:        great_dom
(f,r0) c= less_dom(r(#)f,r1);
        for x being Element of X st x in less_dom(r(#)f,r1)
            holds x in great_dom(f,r0)
            proof
              let x being Element of X;
              assume
A22:          x in less_dom(r(#)f,r1);
then A23:          x in dom(r(#)f) by Def11;
          (r(#)f).x < r1 by A22,Def11;
then           (r(#)f).x < r * r0 by A4;
              then
A24:          r * r0 / (r qua ExtReal) < (r(#)f).x / r
              by A16,XXREAL_3:104;
A25:        r * r0 / r = (r*r0)/r
                .= r0/(r/r) by XCMPLX_1:77
                .= r0/1 by A3,XCMPLX_1:60
                .= r0;
          x in dom f & f.x = (r(#)f).x / r by A3,A23,Def6,Th6;
              hence thesis by A24,A25,Def13;
            end;
then         less_dom
(r(#)f,r1) c= great_dom(f,r0);
then         great_dom(f,r0) = less_dom(r(#)f,r1) by A21,
XBOOLE_0:def 10;
            hence thesis by A1,A2,Th29;
          end;
        end;
        hence thesis;
      end;
      suppose
A26:    r=0;
    now per cases;
          suppose
A27:        r1>0;
        for x1 being object holds x1 in A implies x1 in A /\
            less_dom(r(#)f,r1)
            proof
              let x1 being object;
              assume
A28:          x1 in A;
              then reconsider x1 as Element of X;
          x1 in dom f by A2,A28;
then A29:          x1 in dom (r(#)f) by Def6;
              reconsider y = (r(#)f).x1 as R_eal;
          y = (r) * f.x1 by A29,Def6
                .= 0. by A26;
then           x1 in less_dom(r(#)f,r1) by A27,A29,Def11;
              hence thesis by A28,XBOOLE_0:def 4;
            end;
            then         A
 /\ less_dom(r(#)f,r1) c= A & A c= A /\ less_dom(r(#)f,r1) by
XBOOLE_1:17;
then         A /\ less_dom(r(#)f,r1) = A by XBOOLE_0:def 10;
            hence thesis;
          end;
          suppose
A30:        r1<=0;
        less_dom(r(#)f,r1) = {}
            proof
              assume less_dom(r(#)f,r1) <> {};
              then consider x1 being Element of X such that
A31:          x1 in less_dom(r(#)f,r1) by SUBSET_1:4;
A32:          x1 in dom (r(#)f) by A31,Def11;
A33:          (r(#)f).x1 < r1 by A31,Def11;
A34:          (r(#)f).x1 in rng(r(#)f) by A32,FUNCT_1:def 3;
                        for
 y being R_eal st y in rng(r(#)f) holds not y < r1
              proof
                let y being R_eal;
                assume y in rng(r(#)f);
                then consider x such that
A35:            x in dom(r(#)f) & y = (r(#)f).x by PARTFUN1:3;
            y = (r) * f.x by A35,Def6
                  .= 0. by A26;
                hence thesis by A30;
              end;
              hence contradiction by A33,A34;
            end;
            hence thesis by PROB_1:4;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  hence thesis;
end;
