Article MESFUNC4, MML version 4.99.1005

:: MESFUNC4:th 1
theorem
for b1, b2, b3 being FinSequence of ExtREAL
      st (for b4 being natural set
               st b4 in dom b1
            holds 0. <= b1 . b4) &
         (for b4 being natural set
               st b4 in dom b2
            holds 0. <= b2 . b4) &
         dom b1 = dom b2 &
         b3 = b1 + b2
   holds Sum b3 = (Sum b1) + Sum b2;

:: MESFUNC4:th 2
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being natural set
for b5 being Function-like Relation of b1,ExtREAL
for b6 being disjoint_valued FinSequence of b2
for b7, b8 being FinSequence of ExtREAL
      st b5 is_simple_func_in b2 &
         proj1 b5 <> {} &
         (for b9 being set
               st b9 in proj1 b5
            holds 0. <= b5 . b9) &
         b6,b7 are_Re-presentation_of b5 &
         dom b8 = dom b6 &
         (for b9 being natural set
               st b9 in dom b8
            holds b8 . b9 = (b7 . b9) * ((b3 * b6) . b9)) &
         len b6 = b4
   holds integral(b1,b2,b3,b5) = Sum b8;

:: MESFUNC4:th 3
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
for b4 being sigma_Measure of b2
for b5 being disjoint_valued FinSequence of b2
for b6, b7 being FinSequence of ExtREAL
      st b3 is_simple_func_in b2 &
         proj1 b3 <> {} &
         (for b8 being set
               st b8 in proj1 b3
            holds 0. <= b3 . b8) &
         b5,b6 are_Re-presentation_of b3 &
         dom b7 = dom b5 &
         (for b8 being natural set
               st b8 in dom b7
            holds b7 . b8 = (b6 . b8) * ((b4 * b5) . b8))
   holds integral(b1,b2,b4,b3) = Sum b7;

:: MESFUNC4:th 4
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
for b4 being sigma_Measure of b2
      st b3 is_simple_func_in b2 &
         proj1 b3 <> {} &
         (for b5 being set
               st b5 in proj1 b3
            holds 0. <= b3 . b5)
   holds ex b5 being disjoint_valued FinSequence of b2 st
      ex b6, b7 being FinSequence of ExtREAL st
         b5,b6 are_Re-presentation_of b3 &
          dom b7 = dom b5 &
          (for b8 being natural set
                st b8 in dom b7
             holds b7 . b8 = (b6 . b8) * ((b4 * b5) . b8)) &
          integral(b1,b2,b4,b3) = Sum b7;

:: MESFUNC4:th 5
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,ExtREAL
      st b4 is_simple_func_in b2 &
         proj1 b4 <> {} &
         (for b6 being set
               st b6 in proj1 b4
            holds 0. <= b4 . b6) &
         b5 is_simple_func_in b2 &
         proj1 b5 = proj1 b4 &
         (for b6 being set
               st b6 in proj1 b5
            holds 0. <= b5 . b6)
   holds b4 + b5 is_simple_func_in b2 &
    proj1 (b4 + b5) <> {} &
    (for b6 being set
          st b6 in proj1 (b4 + b5)
       holds 0. <= (b4 + b5) . b6) &
    integral(b1,b2,b3,b4 + b5) = (integral(b1,b2,b3,b4)) + integral(b1,b2,b3,b5);

:: MESFUNC4:th 6
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,ExtREAL
for b6 being Element of ExtREAL
      st b4 is_simple_func_in b2 &
         proj1 b4 <> {} &
         (for b7 being set
               st b7 in proj1 b4
            holds 0. <= b4 . b7) &
         0. <= b6 &
         b6 < +infty &
         proj1 b5 = proj1 b4 &
         (for b7 being set
               st b7 in proj1 b5
            holds b5 . b7 = b6 * (b4 . b7))
   holds integral(b1,b2,b3,b5) = b6 * integral(b1,b2,b3,b4);