
theorem Th20:
  for X1,X2 be set, S1 be Field_Subset of X1, S2 be Field_Subset of X2,
      m1 be Measure of S1, m2 be Measure of S2, A,B be set
  st A in S1 & B in S2 holds
    product-pre-Measure(m1,m2).([:A,B:]) = m1.A * m2.B
proof
   let X1,X2 be set, S1 be Field_Subset of X1, S2 be Field_Subset of X2,
     m1 be Measure of S1, m2 be Measure of S2, A,B be set;
   assume A in S1 & B in S2; then
   [:A,B:] in measurable_rectangles(S1,S2); then
   consider A1 be Element of S1, B1 be Element of S2 such that
A2: [:A,B:] = [:A1,B1:] & product-pre-Measure(m1,m2).([:A,B:]) = m1.A1 * m2.B1
      by Def6;
   per cases;
   suppose A3: A = {} or B = {}; then
    [:A,B:] = {} by ZFMISC_1:90; then
A4: product-pre-Measure(m1,m2).([:A,B:]) = 0 by VALUED_0:def 19;
    m1.A = 0 or m2.B = 0 by A3,VALUED_0:def 19;
    hence thesis by A4;
   end;
   suppose A <> {} & B <> {}; then
    A = A1 & B = B1 by A2,ZFMISC_1:110;
    hence thesis by A2;
   end;
end;
