
theorem Th46:
   Prod_Measure(L-Meas 2) = Prod_Measure(L-Meas,L-Meas)
& for E1,E2 be Element of L-Field holds
   [:E1,E2:] in measurable_rectangles(L-Field,L-Field)
& (Prod_Measure(L-Meas 2)).([:E1,E2:]) = (L-Meas.E1)*(L-Meas.E2)
proof
    set X = Seg 2 --> REAL, S = L-Field 2, m = L-Meas 2;
    set X1 = SubFin(X,1), S1 = SubFin(S,1), m1 = SubFin(m,1);
    set X11 = ElmFin(X,2), S11 = ElmFin(S,2), m11 = ElmFin(m,1+1);
A1: Prod_Measure(L-Meas 2) = product_sigma_Measure(Prod_Measure m1,m11)
      by MEASUR13:25;

A2: 1 in Seg 1 & 1 in Seg 2 & 2 in Seg 2;

    S11 = S.2 & m11 = m.2 by MEASUR13:def 7,def 10; then
A3: S11 = L-Field & m11 = L-Meas by A2,FUNCOP_1:7;

A4: len X1 = 1 & len S1 = 1 & len m1 = 1 by CARD_1:def 7;
    X1 = X|1 & S1 = S|1 & m1 = m|1 by MEASUR13:def 5,def 6,def 9; then
    X1.1 = (Seg 2 --> REAL).1 & S1.1 = (L-Field 2).1 & m1.1 = (L-Meas 2).1
      by A2,FUNCT_1:49; then
    X1 = <*REAL*> & S1 = <*L-Field*> & m1 = <*L-Meas*>
      by A4,A2,FUNCOP_1:7,FINSEQ_1:40; then
A5:X1 = 1 |-> REAL & S1 = 1 |-> L-Field & m1 = 1 |-> L-Meas by FINSEQ_2:59;
    hence Prod_Measure(L-Meas 2) = Prod_Measure(L-Meas,L-Meas)
      by A1,A3,Th37,Th41,Th45,MESFUN12:def 9;

    thus for E1,E2 be Element of L-Field holds
     [:E1,E2:] in measurable_rectangles(L-Field,L-Field)
   & (Prod_Measure(L-Meas 2)).([:E1,E2:]) = (L-Meas.E1)*(L-Meas.E2)
    proof
     let E1,E2 be Element of L-Field;
A6: [:E1,E2:] in measurable_rectangles(L-Field,L-Field);

     measurable_rectangles(L-Field,L-Field)
      c= sigma measurable_rectangles(L-Field,L-Field) by PROB_1:def 9; then
     reconsider E = [:E1,E2:] as Element of
                      sigma measurable_rectangles (L-Field,L-Field) by A6;
     (product_sigma_Measure(L-Meas,L-Meas)).E = (L-Meas. E1)*(L-Meas. E2)
       by MEASUR11:16;
     hence thesis by A5,A3,Th37,Th41,Th45,MEASUR13:25;
    end;
end;
