Article MESFUNC1, MML version 4.99.1005

:: MESFUNC1:funcnot 1 => MESFUNC1:func 1
definition
  func INT- -> Element of bool REAL means
    for b1 being Element of REAL holds
          b1 in it
       iff
          ex b2 being Element of NAT st
             b1 = - b2;
end;

:: MESFUNC1:def 1
theorem
for b1 being Element of bool REAL holds
      b1 = INT-
   iff
      for b2 being Element of REAL holds
            b2 in b1
         iff
            ex b3 being Element of NAT st
               b2 = - b3;

:: MESFUNC1:funcreg 1
registration
  cluster INT- -> non empty;
end;

:: MESFUNC1:th 1
theorem
NAT,INT- are_equipotent;

:: MESFUNC1:th 2
theorem
INT = INT- \/ NAT;

:: MESFUNC1:th 3
theorem
NAT,INT are_equipotent;

:: MESFUNC1:funcnot 2 => MESFUNC1:func 2
definition
  redefine func INT -> Element of bool REAL;
end;

:: MESFUNC1:funcnot 3 => MESFUNC1:func 3
definition
  let a1 be natural set;
  func RAT_with_denominator A1 -> Element of bool RAT means
    for b1 being rational set holds
          b1 in it
       iff
          ex b2 being integer set st
             b1 = b2 / a1;
end;

:: MESFUNC1:def 2
theorem
for b1 being natural set
for b2 being Element of bool RAT holds
      b2 = RAT_with_denominator b1
   iff
      for b3 being rational set holds
            b3 in b2
         iff
            ex b4 being integer set st
               b3 = b4 / b1;

:: MESFUNC1:funcreg 2
registration
  let a1 be natural set;
  cluster RAT_with_denominator (a1 + 1) -> non empty;
end;

:: MESFUNC1:th 4
theorem
for b1 being natural set holds
   INT,RAT_with_denominator (b1 + 1) are_equipotent;

:: MESFUNC1:th 5
theorem
NAT,RAT are_equipotent;

:: MESFUNC1:funcnot 4 => MESFUNC1:func 4
definition
  let a1 be non empty set;
  let a2, a3 be Function-like Relation of a1,ExtREAL;
  func A2 + A3 -> Function-like Relation of a1,ExtREAL means
    proj1 it = ((proj1 a2) /\ proj1 a3) \ (((a2 " {-infty}) /\ (a3 " {+infty})) \/ ((a2 " {+infty}) /\ (a3 " {-infty}))) &
     (for b1 being Element of a1
           st b1 in proj1 it
        holds it . b1 = (a2 . b1) + (a3 . b1));
end;

:: MESFUNC1:def 3
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,ExtREAL holds
   b4 = b2 + b3
iff
   proj1 b4 = ((proj1 b2) /\ proj1 b3) \ (((b2 " {-infty}) /\ (b3 " {+infty})) \/ ((b2 " {+infty}) /\ (b3 " {-infty}))) &
    (for b5 being Element of b1
          st b5 in proj1 b4
       holds b4 . b5 = (b2 . b5) + (b3 . b5));

:: MESFUNC1:funcnot 5 => MESFUNC1:func 5
definition
  let a1 be non empty set;
  let a2, a3 be Function-like Relation of a1,ExtREAL;
  func A2 - A3 -> Function-like Relation of a1,ExtREAL means
    proj1 it = ((proj1 a2) /\ proj1 a3) \ (((a2 " {+infty}) /\ (a3 " {+infty})) \/ ((a2 " {-infty}) /\ (a3 " {-infty}))) &
     (for b1 being Element of a1
           st b1 in proj1 it
        holds it . b1 = (a2 . b1) - (a3 . b1));
end;

:: MESFUNC1:def 4
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,ExtREAL holds
   b4 = b2 - b3
iff
   proj1 b4 = ((proj1 b2) /\ proj1 b3) \ (((b2 " {+infty}) /\ (b3 " {+infty})) \/ ((b2 " {-infty}) /\ (b3 " {-infty}))) &
    (for b5 being Element of b1
          st b5 in proj1 b4
       holds b4 . b5 = (b2 . b5) - (b3 . b5));

:: MESFUNC1:funcnot 6 => MESFUNC1:func 6
definition
  let a1 be non empty set;
  let a2, a3 be Function-like Relation of a1,ExtREAL;
  func A2 (#) A3 -> Function-like Relation of a1,ExtREAL means
    proj1 it = (proj1 a2) /\ proj1 a3 &
     (for b1 being Element of a1
           st b1 in proj1 it
        holds it . b1 = (a2 . b1) * (a3 . b1));
end;

:: MESFUNC1:def 5
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,ExtREAL holds
   b4 = b2 (#) b3
iff
   proj1 b4 = (proj1 b2) /\ proj1 b3 &
    (for b5 being Element of b1
          st b5 in proj1 b4
       holds b4 . b5 = (b2 . b5) * (b3 . b5));

:: MESFUNC1:funcnot 7 => MESFUNC1:func 7
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  let a3 be Element of REAL;
  func A3 (#) A2 -> Function-like Relation of a1,ExtREAL means
    proj1 it = proj1 a2 &
     (for b1 being Element of a1
           st b1 in proj1 it
        holds it . b1 = (R_EAL a3) * (a2 . b1));
end;

:: MESFUNC1:def 6
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of REAL
for b4 being Function-like Relation of b1,ExtREAL holds
      b4 = b3 (#) b2
   iff
      proj1 b4 = proj1 b2 &
       (for b5 being Element of b1
             st b5 in proj1 b4
          holds b4 . b5 = (R_EAL b3) * (b2 . b5));

:: MESFUNC1:th 6
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of REAL
   st b3 <> 0
for b4 being Element of b1
      st b4 in proj1 (b3 (#) b2)
   holds b2 . b4 = ((b3 (#) b2) . b4) / R_EAL b3;

:: MESFUNC1:funcnot 8 => MESFUNC1:func 8
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  func - A2 -> Function-like Relation of a1,ExtREAL means
    proj1 it = proj1 a2 &
     (for b1 being Element of a1
           st b1 in proj1 it
        holds it . b1 = - (a2 . b1));
end;

:: MESFUNC1:def 7
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
   b3 = - b2
iff
   proj1 b3 = proj1 b2 &
    (for b4 being Element of b1
          st b4 in proj1 b3
       holds b3 . b4 = - (b2 . b4));

:: MESFUNC1:funcnot 9 => MESFUNC1:func 9
definition
  func 1. -> Element of ExtREAL equals
    1;
end;

:: MESFUNC1:def 8
theorem
1. = 1;

:: MESFUNC1:funcnot 10 => MESFUNC1:func 10
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  let a3 be Element of REAL;
  func A3 / A2 -> Function-like Relation of a1,ExtREAL means
    proj1 it = (proj1 a2) \ (a2 " {0.}) &
     (for b1 being Element of a1
           st b1 in proj1 it
        holds it . b1 = (R_EAL a3) / (a2 . b1));
end;

:: MESFUNC1:def 9
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of REAL
for b4 being Function-like Relation of b1,ExtREAL holds
      b4 = b3 / b2
   iff
      proj1 b4 = (proj1 b2) \ (b2 " {0.}) &
       (for b5 being Element of b1
             st b5 in proj1 b4
          holds b4 . b5 = (R_EAL b3) / (b2 . b5));

:: MESFUNC1:th 7
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
   proj1 (1 / b2) = (proj1 b2) \ (b2 " {0.}) &
    (for b3 being Element of b1
          st b3 in proj1 (1 / b2)
       holds (1 / b2) . b3 = 1. / (b2 . b3));

:: MESFUNC1:funcnot 11 => MESFUNC1:func 11
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  func |.A2.| -> Function-like Relation of a1,ExtREAL means
    proj1 it = proj1 a2 &
     (for b1 being Element of a1
           st b1 in proj1 it
        holds it . b1 = |.a2 . b1.|);
end;

:: MESFUNC1:def 10
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
   b3 = |.b2.|
iff
   proj1 b3 = proj1 b2 &
    (for b4 being Element of b1
          st b4 in proj1 b3
       holds b3 . b4 = |.b2 . b4.|);

:: MESFUNC1:th 9
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
b2 + b3 = b3 + b2;

:: MESFUNC1:th 10
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
b2 (#) b3 = b3 (#) b2;

:: MESFUNC1:funcnot 12 => MESFUNC1:func 12
definition
  let a1 be non empty set;
  let a2, a3 be Function-like Relation of a1,ExtREAL;
  redefine func a2 + a3 -> Function-like Relation of a1,ExtREAL;
  commutativity;
::  for a1 being non empty set
::  for a2, a3 being Function-like Relation of a1,ExtREAL holds
::  a2 + a3 = a3 + a2;
end;

:: MESFUNC1:funcnot 13 => MESFUNC1:func 13
definition
  let a1 be non empty set;
  let a2, a3 be Function-like Relation of a1,ExtREAL;
  redefine func a2 (#) a3 -> Function-like Relation of a1,ExtREAL;
  commutativity;
::  for a1 being non empty set
::  for a2, a3 being Function-like Relation of a1,ExtREAL holds
::  a2 (#) a3 = a3 (#) a2;
end;

:: MESFUNC1:th 11
theorem
for b1 being Element of REAL holds
   ex b2 being Element of NAT st
      b1 <= b2;

:: MESFUNC1:th 12
theorem
for b1 being Element of REAL holds
   ex b2 being Element of NAT st
      - b2 <= b1;

:: MESFUNC1:th 13
theorem
for b1, b2 being real set
      st b1 < b2
   holds ex b3 being Element of NAT st
      1 / (b3 + 1) < b2 - b1;

:: MESFUNC1:th 14
theorem
for b1, b2 being real set
      st for b3 being Element of NAT holds
           b1 - (1 / (b3 + 1)) <= b2
   holds b1 <= b2;

:: MESFUNC1:th 15
theorem
for b1 being Element of ExtREAL
      st for b2 being Element of REAL holds
           R_EAL b2 < b1
   holds b1 = +infty;

:: MESFUNC1:th 16
theorem
for b1 being Element of ExtREAL
      st for b2 being Element of REAL holds
           b1 < R_EAL b2
   holds b1 = -infty;

:: MESFUNC1:prednot 1 => HIDDEN:pred 2
notation
  let a1, a2 be set;
  synonym a1 is_measurable_on a2 for a1 in a2;
end;

:: MESFUNC1:funcnot 14 => MESFUNC1:func 14
definition
  let a1 be Relation-like Function-like ext-real-valued set;
  let a2 be ext-real set;
  func less_dom(A1,A2) -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 in proj1 a1 & a1 . b1 < a2;
end;

:: MESFUNC1:def 12
theorem
for b1 being Relation-like Function-like ext-real-valued set
for b2 being ext-real set
for b3 being set holds
      b3 = less_dom(b1,b2)
   iff
      for b4 being set holds
            b4 in b3
         iff
            b4 in proj1 b1 & b1 . b4 < b2;

:: MESFUNC1:funcnot 15 => MESFUNC1:func 15
definition
  let a1 be Relation-like Function-like ext-real-valued set;
  let a2 be ext-real set;
  func less_eq_dom(A1,A2) -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 in proj1 a1 & a1 . b1 <= a2;
end;

:: MESFUNC1:def 13
theorem
for b1 being Relation-like Function-like ext-real-valued set
for b2 being ext-real set
for b3 being set holds
      b3 = less_eq_dom(b1,b2)
   iff
      for b4 being set holds
            b4 in b3
         iff
            b4 in proj1 b1 & b1 . b4 <= b2;

:: MESFUNC1:funcnot 16 => MESFUNC1:func 16
definition
  let a1 be Relation-like Function-like ext-real-valued set;
  let a2 be ext-real set;
  func great_dom(A1,A2) -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 in proj1 a1 & a2 < a1 . b1;
end;

:: MESFUNC1:def 14
theorem
for b1 being Relation-like Function-like ext-real-valued set
for b2 being ext-real set
for b3 being set holds
      b3 = great_dom(b1,b2)
   iff
      for b4 being set holds
            b4 in b3
         iff
            b4 in proj1 b1 & b2 < b1 . b4;

:: MESFUNC1:funcnot 17 => MESFUNC1:func 17
definition
  let a1 be Relation-like Function-like ext-real-valued set;
  let a2 be ext-real set;
  func great_eq_dom(A1,A2) -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 in proj1 a1 & a2 <= a1 . b1;
end;

:: MESFUNC1:def 15
theorem
for b1 being Relation-like Function-like ext-real-valued set
for b2 being ext-real set
for b3 being set holds
      b3 = great_eq_dom(b1,b2)
   iff
      for b4 being set holds
            b4 in b3
         iff
            b4 in proj1 b1 & b2 <= b1 . b4;

:: MESFUNC1:funcnot 18 => MESFUNC1:func 18
definition
  let a1 be Relation-like Function-like ext-real-valued set;
  let a2 be ext-real set;
  func eq_dom(A1,A2) -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 in proj1 a1 & a1 . b1 = a2;
end;

:: MESFUNC1:def 16
theorem
for b1 being Relation-like Function-like ext-real-valued set
for b2 being ext-real set
for b3 being set holds
      b3 = eq_dom(b1,b2)
   iff
      for b4 being set holds
            b4 in b3
         iff
            b4 in proj1 b1 & b1 . b4 = b2;

:: MESFUNC1:funcnot 19 => MESFUNC1:func 19
definition
  let a1 be set;
  let a2 be Function-like Relation of a1,ExtREAL;
  let a3 be ext-real set;
  redefine func less_dom(a2,a3) -> Element of bool a1;
end;

:: MESFUNC1:funcnot 20 => MESFUNC1:func 20
definition
  let a1 be set;
  let a2 be Function-like Relation of a1,ExtREAL;
  let a3 be ext-real set;
  redefine func less_eq_dom(a2,a3) -> Element of bool a1;
end;

:: MESFUNC1:funcnot 21 => MESFUNC1:func 21
definition
  let a1 be set;
  let a2 be Function-like Relation of a1,ExtREAL;
  let a3 be ext-real set;
  redefine func great_dom(a2,a3) -> Element of bool a1;
end;

:: MESFUNC1:funcnot 22 => MESFUNC1:func 22
definition
  let a1 be set;
  let a2 be Function-like Relation of a1,ExtREAL;
  let a3 be ext-real set;
  redefine func great_eq_dom(a2,a3) -> Element of bool a1;
end;

:: MESFUNC1:funcnot 23 => MESFUNC1:func 23
definition
  let a1 be set;
  let a2 be Function-like Relation of a1,ExtREAL;
  let a3 be ext-real set;
  redefine func eq_dom(a2,a3) -> Element of bool a1;
end;

:: MESFUNC1:th 18
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being set
for b4 being Element of ExtREAL
      st b3 c= proj1 b2
   holds b3 /\ great_eq_dom(b2,b4) = b3 \ (b3 /\ less_dom(b2,b4));

:: MESFUNC1:th 19
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being set
for b4 being Element of ExtREAL
      st b3 c= proj1 b2
   holds b3 /\ great_dom(b2,b4) = b3 \ (b3 /\ less_eq_dom(b2,b4));

:: MESFUNC1:th 20
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being set
for b4 being Element of ExtREAL
      st b3 c= proj1 b2
   holds b3 /\ less_eq_dom(b2,b4) = b3 \ (b3 /\ great_dom(b2,b4));

:: MESFUNC1:th 21
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being set
for b4 being Element of ExtREAL
      st b3 c= proj1 b2
   holds b3 /\ less_dom(b2,b4) = b3 \ (b3 /\ great_eq_dom(b2,b4));

:: MESFUNC1:th 22
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being set
for b4 being Element of ExtREAL holds
   b3 /\ eq_dom(b2,b4) = (b3 /\ great_eq_dom(b2,b4)) /\ less_eq_dom(b2,b4);

:: MESFUNC1:th 23
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being set
for b6 being Element of REAL
      st for b7 being Element of NAT holds
           b3 . b7 = b5 /\ great_dom(b4,R_EAL (b6 - (1 / (b7 + 1))))
   holds b5 /\ great_eq_dom(b4,R_EAL b6) = meet rng b3;

:: MESFUNC1:th 24
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Function-like quasi_total Relation of NAT,b3
for b5 being set
for b6 being real set
      st for b7 being Element of NAT holds
           b4 . b7 = b5 /\ less_dom(b2,R_EAL (b6 + (1 / (b7 + 1))))
   holds b5 /\ less_eq_dom(b2,R_EAL b6) = meet rng b4;

:: MESFUNC1:th 25
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Function-like quasi_total Relation of NAT,b3
for b5 being set
for b6 being real set
      st for b7 being Element of NAT holds
           b4 . b7 = b5 /\ less_eq_dom(b2,R_EAL (b6 - (1 / (b7 + 1))))
   holds b5 /\ less_dom(b2,R_EAL b6) = union rng b4;

:: MESFUNC1:th 26
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being set
for b6 being Element of REAL
      st for b7 being Element of NAT holds
           b3 . b7 = b5 /\ great_eq_dom(b4,R_EAL (b6 + (1 / (b7 + 1))))
   holds b5 /\ great_dom(b4,R_EAL b6) = union rng b3;

:: MESFUNC1:th 27
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being set
      st for b6 being Element of NAT holds
           b3 . b6 = b5 /\ great_dom(b4,R_EAL b6)
   holds b5 /\ eq_dom(b4,+infty) = meet rng b3;

:: MESFUNC1:th 28
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being set
      st for b6 being Element of NAT holds
           b3 . b6 = b5 /\ less_dom(b4,R_EAL b6)
   holds b5 /\ less_dom(b4,+infty) = union rng b3;

:: MESFUNC1:th 29
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being set
      st for b6 being Element of NAT holds
           b3 . b6 = b5 /\ less_dom(b4,R_EAL - b6)
   holds b5 /\ eq_dom(b4,-infty) = meet rng b3;

:: MESFUNC1:th 30
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like quasi_total Relation of NAT,b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being set
      st for b6 being Element of NAT holds
           b3 . b6 = b5 /\ great_dom(b4,R_EAL - b6)
   holds b5 /\ great_dom(b4,-infty) = union rng b3;

:: MESFUNC1:prednot 2 => MESFUNC1:pred 1
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like Relation of a1,ExtREAL;
  let a4 be Element of a2;
  pred A3 is_measurable_on A4 means
    for b1 being real set holds
       a4 /\ less_dom(a3,R_EAL b1) in a2;
end;

:: MESFUNC1:dfs 16
definiens
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like Relation of a1,ExtREAL;
  let a4 be Element of a2;
To prove
     a3 is_measurable_on a4
it is sufficient to prove
  thus for b1 being real set holds
       a4 /\ less_dom(a3,R_EAL b1) in a2;

:: MESFUNC1:def 17
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 Element of b2 holds
      b3 is_measurable_on b4
   iff
      for b5 being real set holds
         b4 /\ less_dom(b3,R_EAL b5) in b2;

:: MESFUNC1:th 31
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 Element of b2
      st b4 c= proj1 b3
   holds    b3 is_measurable_on b4
   iff
      for b5 being real set holds
         b4 /\ great_eq_dom(b3,R_EAL b5) in b2;

:: MESFUNC1:th 32
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 Element of b2 holds
      b3 is_measurable_on b4
   iff
      for b5 being real set holds
         b4 /\ less_eq_dom(b3,R_EAL b5) in b2;

:: MESFUNC1:th 33
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 Element of b2
      st b4 c= proj1 b3
   holds    b3 is_measurable_on b4
   iff
      for b5 being real set holds
         b4 /\ great_dom(b3,R_EAL b5) in b2;

:: MESFUNC1:th 34
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, b5 being Element of b2
      st b5 c= b4 & b3 is_measurable_on b4
   holds b3 is_measurable_on b5;

:: MESFUNC1:th 35
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, b5 being Element of b2
      st b3 is_measurable_on b4 & b3 is_measurable_on b5
   holds b3 is_measurable_on b4 \/ b5;

:: MESFUNC1:th 36
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 Element of b2
for b5, b6 being Element of REAL
      st b3 is_measurable_on b4 & b4 c= proj1 b3
   holds (b4 /\ great_dom(b3,R_EAL b5)) /\ less_dom(b3,R_EAL b6) in b2;

:: MESFUNC1:th 37
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 Element of b2
      st b3 is_measurable_on b4 & b4 c= proj1 b3
   holds b4 /\ eq_dom(b3,+infty) in b2;

:: MESFUNC1:th 38
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 Element of b2
      st b3 is_measurable_on b4
   holds b4 /\ eq_dom(b3,-infty) in b2;

:: MESFUNC1:th 39
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 Element of b2
      st b3 is_measurable_on b4 & b4 c= proj1 b3
   holds (b4 /\ great_dom(b3,-infty)) /\ less_dom(b3,+infty) in b2;

:: MESFUNC1:th 40
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of b2
for b6 being Element of REAL
      st b3 is_measurable_on b5 & b4 is_measurable_on b5 & b5 c= proj1 b4
   holds (b5 /\ less_dom(b3,R_EAL b6)) /\ great_dom(b4,R_EAL b6) in b2;

:: MESFUNC1:th 41
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 Element of b2
for b5 being Element of REAL
      st b3 is_measurable_on b4 & b4 c= proj1 b3
   holds b5 (#) b3 is_measurable_on b4;