
theorem
  Prod_Measure(L-Meas 3) = Prod_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas)
& for E1,E2,E3 be Element of L-Field holds
   [:E1,E2,E3:] in measurable_rectangles(
                     sigma measurable_rectangles(L-Field,L-Field),L-Field)
& (Prod_Measure(L-Meas 3)).([:E1,E2,E3:]) = (L-Meas.E1)*(L-Meas.E2)*(L-Meas.E3)
proof
    set X = Seg 3 --> REAL, S = L-Field 3, m = L-Meas 3;
    set X2 = SubFin(X,2), S2 = SubFin(S,2), m2 = SubFin(m,2);
    set X21 = ElmFin(X,3), S21 = ElmFin(S,3), m21 = ElmFin(m,2+1);
A1: Prod_Measure(L-Meas 3)
     = product_sigma_Measure(Prod_Measure m2,m21) by MEASUR13:25;

A2: 1 in Seg 2 & 2 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3;

    X21 = X.3 & S21 = S.3 & m21 = m.3 by MEASUR13:def 1,def 7,def 10; then
A3: X21 = REAL & S21 = L-Field & m21 = L-Meas by A2,FUNCOP_1:7;

A4: len X2 = 2 & len S2 = 2 & len m2 = 2 by CARD_1:def 7;

A5: SubFin(X,2) = X|2 by MEASUR13:def 5; then
    X2.1 = (Seg 3 --> REAL).1 by A2,FUNCT_1:49; then
A6:X2.1 = REAL by FUNCOP_1:7,A2;

    X2.2 = (Seg 3 --> REAL).2 by A2,A5,FUNCT_1:49; then
    X2 = <*REAL,REAL*> by A4,A6,A2,FUNCOP_1:7,FINSEQ_1:44; then
A7:X2 = 2 |-> REAL by FINSEQ_2:61;

A8: SubFin(S,2) = S|2 by MEASUR13:def 6; then
    S2.1 = (L-Field 3).1 by A2,FUNCT_1:49; then
A9:S2.1 = L-Field by A2,FUNCOP_1:7;

    S2.2 = (L-Field 3).2 by A2,A8,FUNCT_1:49; then
    S2 = <*L-Field,L-Field*> by A4,A9,A2,FUNCOP_1:7,FINSEQ_1:44; then
A10:S2 = 2 |-> L-Field by FINSEQ_2:61;

A11: SubFin(m,2) = m|2 by MEASUR13:def 9; then
    m2.1 = (L-Meas 3).1 by A2,FUNCT_1:49; then
A12:m2.1 = L-Meas by A2,FUNCOP_1:7;

    m2.2 = (L-Meas 3).2 by A2,A11,FUNCT_1:49; then
    m2 = <*L-Meas,L-Meas*> by A4,A12,A2,FUNCOP_1:7,FINSEQ_1:44; then
    m2 = 2 |-> L-Meas by FINSEQ_2:61;
    hence
A13:Prod_Measure m = Prod_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas)
      by A1,A7,A10,A3,Th37,Th42,Th46,MESFUN12:def 9;

    thus for E1,E2,E3 be Element of L-Field holds
      [:E1,E2,E3:] in measurable_rectangles(
          sigma measurable_rectangles (L-Field,L-Field),L-Field)
    & (Prod_Measure(L-Meas 3)).([:E1,E2,E3:])
         = (L-Meas.E1)*(L-Meas.E2)*(L-Meas.E3)
    proof
     let E1,E2,E3 be Element of L-Field;
     measurable_rectangles(L-Field,L-Field)
      c= sigma measurable_rectangles (L-Field,L-Field) by PROB_1:def 9; then
     reconsider F = [:E1,E2:] as Element of
        sigma measurable_rectangles (L-Field,L-Field) by Th46;

A14: [:F,E3:] in measurable_rectangles
        (sigma measurable_rectangles (L-Field,L-Field),L-Field);
     hence [:E1,E2,E3:] in measurable_rectangles
        (sigma measurable_rectangles (L-Field,L-Field),L-Field)
          by ZFMISC_1:def 3;

     measurable_rectangles(sigma measurable_rectangles(L-Field,L-Field),
       L-Field) c= sigma measurable_rectangles(sigma
         measurable_rectangles (L-Field,L-Field),L-Field) by PROB_1:def 9; then
     (product_sigma_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas)).([:F,E3:])
      = ((Prod_Measure(L-Meas,L-Meas)).F)*(L-Meas.E3) by A14,MEASUR11:16
     .= (L-Meas.E1)*(L-Meas.E2)*(L-Meas. E3) by Th46; then
     (Prod_Measure(Prod_Measure(L-Meas,L-Meas),L-Meas)).([:F,E3:])
      = (L-Meas.E1)*(L-Meas.E2)*(L-Meas. E3) by MESFUN12:def 9;
     hence thesis by A13,ZFMISC_1:def 3;
    end;
end;
