
theorem
  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,
      E1,E2 be Element of measurable_rectangles(S1,S2)
  st E1 misses E2 & E1 \/ E2 in measurable_rectangles(S1,S2)
  holds product-pre-Measure(m1,m2).(E1 \/ E2)
         = product-pre-Measure(m1,m2).E1 + product-pre-Measure(m1,m2).E2
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,
       E1,E2 be Element of measurable_rectangles(S1,S2);
   assume that
A1:E1 misses E2 and
A2:E1 \/ E2 in measurable_rectangles(S1,S2);
   set S = measurable_rectangles(S1,S2);
   set P = product-pre-Measure(m1,m2);
   reconsider E = E1 \/ E2 as Element of S by A2;
   consider A be Element of S1, B be Element of S2 such that
A3: E = [:A,B:] & P.E = m1.A * m2.B by Def6;
   consider A1 be Element of S1, B1 be Element of S2 such that
A4: E1 = [:A1,B1:] & P.E1 = m1.A1 * m2.B1 by Def6;
   consider A2 be Element of S1, B2 be Element of S2 such that
A5: E2 = [:A2,B2:] & P.E2 = m1.A2 * m2.B2 by Def6;
   per cases;
   suppose E1 = {} or E2 = {}; then
    (P.E1 = 0 & P.E = P.E2) or (P.E2 = 0 & P.E = P.E1) by VALUED_0:def 19;
    hence P.(E1 \/ E2) = P.E1 + P.E2 by XXREAL_3:4;
   end;
   suppose E1 <> {} & E2 <> {}; then
A11:A <> {} & B <> {} & A1 <> {} & B1 <> {} & A2 <> {} & B2 <> {}
      by A3,A4,A5;
    per cases by A1,A4,A5,A3,A11,Th23;
    suppose A13: A1 misses A2 & A = A1 \/ A2 & B = B1 & B = B2; then
     [:A1,B1:] \/ [:A2,B2:] = [:A1 \/ A2,B:] by ZFMISC_1:97; then
     P.(E1 \/ E2) = m1.(A1\/A2) * m2.B by A4,A5,Th20; then
A14: P.(E1 \/ E2) = (m1.A1 + m1.A2) * m2.B by A13,MEASURE1:def 3;
     m1.A1 >= 0 & m1.A2 >= 0 by MEASURE1:def 2;
     hence P.(E1 \/ E2) = P.E1 + P.E2 by A4,A5,A13,A14,XXREAL_3:96;
    end;
    suppose A15: B1 misses B2 & B = B1 \/ B2 & A = A1 & A = A2; then
     [:A1,B1:] \/ [:A2,B2:] = [:A, B1 \/ B2:] by ZFMISC_1:97; then
     P.(E1 \/ E2) = m1.A * m2.(B1\/B2) by A4,A5,Th20; then
A16: P.(E1 \/ E2) = m1.A * (m2.B1 + m2.B2) by A15,MEASURE1:def 3;
     m2.B1 >= 0 & m2.B2 >= 0 by MEASURE1:def 2;
     hence P.(E1 \/ E2) = P.E1 + P.E2 by A4,A5,A15,A16,XXREAL_3:96;
    end;
   end;
end;
