Article PSCOMP_1, MML version 4.99.1005

:: PSCOMP_1:attrnot 1 => SETFAM_1:attr 1
notation
  let a1 be set;
  synonym without_zero for with_non-empty_elements;
end;

:: PSCOMP_1:attrnot 2 => SETFAM_1:attr 1
notation
  let a1 be set;
  antonym with_zero for with_non-empty_elements;
end;

:: PSCOMP_1:attrnot 3 => SETFAM_1:attr 1
definition
  let a1 be set;
  attr a1 is without_zero means
    not 0 in a1;
end;

:: PSCOMP_1:dfs 1
definiens
  let a1 be set;
To prove
     a1 is with_non-empty_elements
it is sufficient to prove
  thus not 0 in a1;

:: PSCOMP_1:def 1
theorem
for b1 being set holds
      b1 is with_non-empty_elements
   iff
      not 0 in b1;

:: PSCOMP_1:funcreg 1
registration
  cluster REAL -> non with_non-empty_elements;
end;

:: PSCOMP_1:funcreg 2
registration
  cluster omega -> non with_non-empty_elements;
end;

:: PSCOMP_1:exreg 1
registration
  cluster non empty with_non-empty_elements set;
end;

:: PSCOMP_1:exreg 2
registration
  cluster non empty non with_non-empty_elements set;
end;

:: PSCOMP_1:exreg 3
registration
  cluster non empty complex-membered ext-real-membered real-membered with_non-empty_elements Element of bool REAL;
end;

:: PSCOMP_1:exreg 4
registration
  cluster non empty complex-membered ext-real-membered real-membered non with_non-empty_elements Element of bool REAL;
end;

:: PSCOMP_1:th 1
theorem
for b1 being set
      st b1 is not empty & b1 is with_non-empty_elements & b1 is c=-linear
   holds b1 is centered;

:: PSCOMP_1:condreg 1
registration
  let a1 be set;
  cluster non empty c=-linear with_non-empty_elements -> centered (Element of bool bool a1);
end;

:: PSCOMP_1:funcreg 3
registration
  let a1, a2 be non empty set;
  let a3 be Function-like quasi_total Relation of a1,a2;
  cluster a3 .: a1 -> non empty;
end;

:: PSCOMP_1:funcnot 1 => PSCOMP_1:func 1
definition
  let a1, a2 be set;
  let a3 be Function-like quasi_total Relation of a1,a2;
  func " A3 -> Function-like quasi_total Relation of bool a2,bool a1 means
    for b1 being Element of bool a2 holds
       it . b1 = a3 " b1;
end;

:: PSCOMP_1:def 2
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Function-like quasi_total Relation of bool b2,bool b1 holds
      b4 = " b3
   iff
      for b5 being Element of bool b2 holds
         b4 . b5 = b3 " b5;

:: PSCOMP_1:th 2
theorem
for b1, b2, b3 being set
for b4 being Element of bool bool b2
for b5 being Function-like quasi_total Relation of b1,b2
      st b3 in meet ((" b5) .: b4)
   holds b5 . b3 in meet b4;

:: PSCOMP_1:th 3
theorem
for b1, b2 being real set
      st (abs b1) + abs b2 = 0
   holds b1 = 0;

:: PSCOMP_1:th 4
theorem
for b1, b2, b3 being real set
      st b1 < b2 & b2 < b3
   holds abs b2 < (abs b1) + abs b3;

:: PSCOMP_1:th 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is convergent & b1 is non-empty & lim b1 = 0
   holds b1 " is not bounded;

:: PSCOMP_1:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      rng b1 is bounded
   iff
      b1 is bounded;

:: PSCOMP_1:funcnot 2 => SEQ_4:func 2
notation
  let a1 be real-membered set;
  synonym sup a1 for upper_bound a1;
end;

:: PSCOMP_1:funcnot 3 => SEQ_4:func 3
notation
  let a1 be real-membered set;
  synonym inf a1 for lower_bound a1;
end;

:: PSCOMP_1:funcnot 4 => PSCOMP_1:func 2
definition
  let a1 be Element of bool REAL;
  redefine func sup a1 -> Element of REAL;
end;

:: PSCOMP_1:funcnot 5 => PSCOMP_1:func 3
definition
  let a1 be Element of bool REAL;
  redefine func inf a1 -> Element of REAL;
end;

:: PSCOMP_1:attrnot 4 => PSCOMP_1:attr 1
definition
  let a1 be real-membered set;
  attr a1 is with_max means
    a1 is bounded_above & upper_bound a1 in a1;
end;

:: PSCOMP_1:dfs 3
definiens
  let a1 be real-membered set;
To prove
     a1 is with_max
it is sufficient to prove
  thus a1 is bounded_above & upper_bound a1 in a1;

:: PSCOMP_1:def 3
theorem
for b1 being real-membered set holds
      b1 is with_max
   iff
      b1 is bounded_above & upper_bound b1 in b1;

:: PSCOMP_1:attrnot 5 => PSCOMP_1:attr 2
definition
  let a1 be real-membered set;
  attr a1 is with_min means
    a1 is bounded_below & lower_bound a1 in a1;
end;

:: PSCOMP_1:dfs 4
definiens
  let a1 be real-membered set;
To prove
     a1 is with_min
it is sufficient to prove
  thus a1 is bounded_below & lower_bound a1 in a1;

:: PSCOMP_1:def 4
theorem
for b1 being real-membered set holds
      b1 is with_min
   iff
      b1 is bounded_below & lower_bound b1 in b1;

:: PSCOMP_1:exreg 5
registration
  cluster non empty complex-membered ext-real-membered real-membered bounded closed Element of bool REAL;
end;

:: PSCOMP_1:attrnot 6 => PSCOMP_1:attr 3
definition
  let a1 be Element of bool bool REAL;
  attr a1 is open means
    for b1 being Element of bool REAL
          st b1 in a1
       holds b1 is open;
end;

:: PSCOMP_1:dfs 5
definiens
  let a1 be Element of bool bool REAL;
To prove
     a1 is open
it is sufficient to prove
  thus for b1 being Element of bool REAL
          st b1 in a1
       holds b1 is open;

:: PSCOMP_1:def 5
theorem
for b1 being Element of bool bool REAL holds
      b1 is open
   iff
      for b2 being Element of bool REAL
            st b2 in b1
         holds b2 is open;

:: PSCOMP_1:attrnot 7 => PSCOMP_1:attr 4
definition
  let a1 be Element of bool bool REAL;
  attr a1 is closed means
    for b1 being Element of bool REAL
          st b1 in a1
       holds b1 is closed;
end;

:: PSCOMP_1:dfs 6
definiens
  let a1 be Element of bool bool REAL;
To prove
     a1 is closed
it is sufficient to prove
  thus for b1 being Element of bool REAL
          st b1 in a1
       holds b1 is closed;

:: PSCOMP_1:def 6
theorem
for b1 being Element of bool bool REAL holds
      b1 is closed
   iff
      for b2 being Element of bool REAL
            st b2 in b1
         holds b2 is closed;

:: PSCOMP_1:funcnot 6 => PSCOMP_1:func 4
definition
  let a1 be Element of bool REAL;
  func - A1 -> Element of bool REAL equals
    {- b1 where b1 is Element of REAL: b1 in a1};
  involutiveness;
::  for a1 being Element of bool REAL holds
::     - - a1 = a1;
end;

:: PSCOMP_1:def 7
theorem
for b1 being Element of bool REAL holds
   - b1 = {- b2 where b2 is Element of REAL: b2 in b1};

:: PSCOMP_1:th 14
theorem
for b1 being real set
for b2 being Element of bool REAL holds
      b1 in b2
   iff
      - b1 in - b2;

:: PSCOMP_1:funcreg 4
registration
  let a1 be non empty Element of bool REAL;
  cluster - a1 -> non empty;
end;

:: PSCOMP_1:th 15
theorem
for b1 being Element of bool REAL holds
      b1 is bounded_above
   iff
      - b1 is bounded_below;

:: PSCOMP_1:th 16
theorem
for b1 being Element of bool REAL holds
      b1 is bounded_below
   iff
      - b1 is bounded_above;

:: PSCOMP_1:th 17
theorem
for b1 being non empty Element of bool REAL
      st b1 is bounded_below
   holds inf b1 = - sup - b1;

:: PSCOMP_1:th 18
theorem
for b1 being non empty Element of bool REAL
      st b1 is bounded_above
   holds sup b1 = - inf - b1;

:: PSCOMP_1:th 19
theorem
for b1 being Element of bool REAL holds
      b1 is closed
   iff
      - b1 is closed;

:: PSCOMP_1:funcnot 7 => MEASURE6:func 6
definition
  let a1 be real-membered set;
  let a2 be real set;
  func A2 ++ A1 -> Element of bool REAL equals
    {a2 + b1 where b1 is Element of REAL: b1 in a1};
end;

:: PSCOMP_1:def 8
theorem
for b1 being Element of bool REAL
for b2 being Element of REAL holds
   b2 ++ b1 = {b2 + b3 where b3 is Element of REAL: b3 in b1};

:: PSCOMP_1:th 20
theorem
for b1 being real set
for b2 being Element of bool REAL
for b3 being Element of REAL holds
      b1 in b2
   iff
      b3 + b1 in b3 ++ b2;

:: PSCOMP_1:funcreg 5
registration
  let a1 be non empty Element of bool REAL;
  let a2 be Element of REAL;
  cluster a2 ++ a1 -> non empty;
end;

:: PSCOMP_1:th 21
theorem
for b1 being Element of bool REAL holds
   b1 = 0 ++ b1;

:: PSCOMP_1:th 22
theorem
for b1 being Element of bool REAL
for b2, b3 being Element of REAL holds
b2 ++ (b3 ++ b1) = (b2 + b3) ++ b1;

:: PSCOMP_1:th 23
theorem
for b1 being Element of bool REAL
for b2 being Element of REAL holds
      b1 is bounded_above
   iff
      b2 ++ b1 is bounded_above;

:: PSCOMP_1:th 24
theorem
for b1 being Element of bool REAL
for b2 being Element of REAL holds
      b1 is bounded_below
   iff
      b2 ++ b1 is bounded_below;

:: PSCOMP_1:th 25
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool REAL
      st b2 is bounded_below
   holds inf (b1 ++ b2) = b1 + inf b2;

:: PSCOMP_1:th 26
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool REAL
      st b2 is bounded_above
   holds sup (b1 ++ b2) = b1 + sup b2;

:: PSCOMP_1:th 27
theorem
for b1 being Element of bool REAL
for b2 being Element of REAL holds
      b1 is closed
   iff
      b2 ++ b1 is closed;

:: PSCOMP_1:funcnot 8 => PSCOMP_1:func 5
definition
  let a1 be Element of bool REAL;
  func Inv A1 -> Element of bool REAL equals
    {1 / b1 where b1 is Element of REAL: b1 in a1};
  involutiveness;
::  for a1 being Element of bool REAL holds
::     Inv Inv a1 = a1;
end;

:: PSCOMP_1:def 9
theorem
for b1 being Element of bool REAL holds
   Inv b1 = {1 / b2 where b2 is Element of REAL: b2 in b1};

:: PSCOMP_1:th 28
theorem
for b1 being real set
for b2 being Element of bool REAL holds
      b1 in b2
   iff
      1 / b1 in Inv b2;

:: PSCOMP_1:funcreg 6
registration
  let a1 be non empty Element of bool REAL;
  cluster Inv a1 -> non empty;
end;

:: PSCOMP_1:funcreg 7
registration
  let a1 be with_non-empty_elements Element of bool REAL;
  cluster Inv a1 -> with_non-empty_elements;
end;

:: PSCOMP_1:th 29
theorem
for b1 being Element of bool REAL holds
   Inv Inv b1 = b1;

:: PSCOMP_1:th 30
theorem
for b1 being with_non-empty_elements Element of bool REAL
      st b1 is closed & b1 is bounded
   holds Inv b1 is closed;

:: PSCOMP_1:th 31
theorem
for b1 being Element of bool bool REAL
      st b1 is closed
   holds meet b1 is closed;

:: PSCOMP_1:funcnot 9 => PSCOMP_1:func 6
definition
  let a1 be Element of bool REAL;
  func Cl A1 -> Element of bool REAL equals
    meet {b1 where b1 is Element of bool REAL: a1 c= b1 & b1 is closed};
  projectivity;
::  for a1 being Element of bool REAL holds
::     Cl Cl a1 = Cl a1;
end;

:: PSCOMP_1:def 10
theorem
for b1 being Element of bool REAL holds
   Cl b1 = meet {b2 where b2 is Element of bool REAL: b1 c= b2 & b2 is closed};

:: PSCOMP_1:funcreg 8
registration
  let a1 be Element of bool REAL;
  cluster Cl a1 -> closed;
end;

:: PSCOMP_1:th 32
theorem
for b1 being Element of bool REAL
for b2 being closed Element of bool REAL
      st b1 c= b2
   holds Cl b1 c= b2;

:: PSCOMP_1:th 33
theorem
for b1 being Element of bool REAL holds
   b1 c= Cl b1;

:: PSCOMP_1:th 34
theorem
for b1 being Element of bool REAL holds
      b1 is closed
   iff
      b1 = Cl b1;

:: PSCOMP_1:th 35
theorem
Cl {} REAL = {};

:: PSCOMP_1:th 36
theorem
Cl [#] REAL = REAL;

:: PSCOMP_1:th 37
theorem
for b1, b2 being Element of bool REAL
      st b1 c= b2
   holds Cl b1 c= Cl b2;

:: PSCOMP_1:th 38
theorem
for b1 being Element of bool REAL
for b2 being Element of REAL holds
      b2 in Cl b1
   iff
      for b3 being open Element of bool REAL
            st b2 in b3
         holds b3 /\ b1 is not empty;

:: PSCOMP_1:th 39
theorem
for b1 being Element of bool REAL
for b2 being Element of REAL
      st b2 in Cl b1
   holds ex b3 being Function-like quasi_total Relation of NAT,REAL st
      rng b3 c= b1 & b3 is convergent & lim b3 = b2;

:: PSCOMP_1:attrnot 8 => PSCOMP_1:attr 5
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of a1,REAL;
  redefine attr a2 is bounded_below means
    a2 .: a1 is bounded_below;
end;

:: PSCOMP_1:dfs 11
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of a1,REAL;
To prove
     a1 is bounded_below
it is sufficient to prove
  thus a2 .: a1 is bounded_below;

:: PSCOMP_1:def 11
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of b1,REAL holds
      b2 is bounded_below
   iff
      b2 .: b1 is bounded_below;

:: PSCOMP_1:attrnot 9 => PSCOMP_1:attr 6
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of a1,REAL;
  redefine attr a2 is bounded_above means
    a2 .: a1 is bounded_above;
end;

:: PSCOMP_1:dfs 12
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of a1,REAL;
To prove
     a1 is bounded_above
it is sufficient to prove
  thus a2 .: a1 is bounded_above;

:: PSCOMP_1:def 12
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of b1,REAL holds
      b2 is bounded_above
   iff
      b2 .: b1 is bounded_above;

:: PSCOMP_1:attrnot 10 => PSCOMP_1:attr 7
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of a1,REAL;
  attr a2 is with_max means
    a2 .: a1 is with_max;
end;

:: PSCOMP_1:dfs 13
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of a1,REAL;
To prove
     a2 is with_max
it is sufficient to prove
  thus a2 .: a1 is with_max;

:: PSCOMP_1:def 14
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of b1,REAL holds
      b2 is with_max(b1)
   iff
      b2 .: b1 is with_max;

:: PSCOMP_1:attrnot 11 => PSCOMP_1:attr 8
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of a1,REAL;
  attr a2 is with_min means
    a2 .: a1 is with_min;
end;

:: PSCOMP_1:dfs 14
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of a1,REAL;
To prove
     a2 is with_min
it is sufficient to prove
  thus a2 .: a1 is with_min;

:: PSCOMP_1:def 15
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of b1,REAL holds
      b2 is with_min(b1)
   iff
      b2 .: b1 is with_min;

:: PSCOMP_1:th 40
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of b1,REAL holds
   (- b3) .: b2 = - (b3 .: b2);

:: PSCOMP_1:th 41
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,REAL holds
      b2 is with_min(b1)
   iff
      - b2 is with_max(b1);

:: PSCOMP_1:th 42
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,REAL holds
      b2 is with_max(b1)
   iff
      - b2 is with_min(b1);

:: PSCOMP_1:th 43
theorem
for b1 being set
for b2 being Element of bool REAL
for b3 being Function-like quasi_total Relation of b1,REAL holds
   (- b3) " b2 = b3 " - b2;

:: PSCOMP_1:th 44
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of b1,REAL
for b4 being Element of REAL holds
   (b4 + b3) .: b2 = b4 ++ (b3 .: b2);

:: PSCOMP_1:th 45
theorem
for b1 being set
for b2 being Element of bool REAL
for b3 being Function-like quasi_total Relation of b1,REAL
for b4 being Element of REAL holds
   (b4 + b3) " b2 = b3 " ((- b4) ++ b2);

:: PSCOMP_1:funcnot 10 => VALUED_1:func 35
notation
  let a1 be Relation-like Function-like real-valued set;
  synonym Inv a1 for a1 ";
end;

:: PSCOMP_1:funcnot 11 => PSCOMP_1:func 7
definition
  let a1 be set;
  let a2 be real-membered set;
  let a3 be Function-like Relation of a1,a2;
  redefine func Inv a3 -> Function-like Relation of a1,REAL;
  involutiveness;
::  for a1 being set
::  for a2 being real-membered set
::  for a3 being Function-like Relation of a1,a2 holds
::     Inv Inv a3 = a3;
end;

:: PSCOMP_1:th 46
theorem
for b1 being set
for b2 being with_non-empty_elements Element of bool REAL
for b3 being Function-like quasi_total Relation of b1,REAL holds
   (Inv b3) " b2 = b3 " Inv b2;

:: PSCOMP_1:modenot 1
definition
  let a1 be 1-sorted;
  mode RealMap of a1 is Function-like quasi_total Relation of the carrier of a1,REAL;
end;

:: PSCOMP_1:exreg 6
registration
  let a1 be non empty 1-sorted;
  cluster non empty Relation-like Function-like complex-valued ext-real-valued real-valued quasi_total bounded total Relation of the carrier of a1,REAL;
end;

:: PSCOMP_1:sch 1
scheme PSCOMP_1:sch 1
{F1 -> non empty TopStruct}:
ex b1 being Function-like quasi_total Relation of the carrier of F1(),REAL st
   for b2 being Element of the carrier of F1() holds
      P1[b2, b1 . b2]
provided
   for b1 being set
         st b1 in the carrier of F1()
      holds ex b2 being Element of REAL st
         P1[b1, b2];


:: PSCOMP_1:funcnot 12 => PSCOMP_1:func 8
definition
  let a1 be 1-sorted;
  let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
  func inf A2 -> Element of REAL equals
    inf (a2 .: the carrier of a1);
end;

:: PSCOMP_1:def 20
theorem
for b1 being 1-sorted
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
   inf b2 = inf (b2 .: the carrier of b1);

:: PSCOMP_1:funcnot 13 => PSCOMP_1:func 9
definition
  let a1 be 1-sorted;
  let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
  func sup A2 -> Element of REAL equals
    sup (a2 .: the carrier of a1);
end;

:: PSCOMP_1:def 21
theorem
for b1 being 1-sorted
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
   sup b2 = sup (b2 .: the carrier of b1);

:: PSCOMP_1:th 47
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total bounded_below Relation of the carrier of b1,REAL
for b3 being Element of the carrier of b1 holds
   inf b2 <= b2 . b3;

:: PSCOMP_1:th 48
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total bounded_below Relation of the carrier of b1,REAL
for b3 being Element of REAL
      st for b4 being Element of the carrier of b1 holds
           b3 <= b2 . b4
   holds b3 <= inf b2;

:: PSCOMP_1:th 49
theorem
for b1 being real set
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b2,REAL
      st (for b4 being Element of the carrier of b2 holds
            b1 <= b3 . b4) &
         (for b4 being real set
               st for b5 being Element of the carrier of b2 holds
                    b4 <= b3 . b5
            holds b4 <= b1)
   holds b1 = inf b3;

:: PSCOMP_1:th 50
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total bounded_above Relation of the carrier of b1,REAL
for b3 being Element of the carrier of b1 holds
   b2 . b3 <= sup b2;

:: PSCOMP_1:th 51
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total bounded_above Relation of the carrier of b1,REAL
for b3 being real set
      st for b4 being Element of the carrier of b1 holds
           b2 . b4 <= b3
   holds sup b2 <= b3;

:: PSCOMP_1:th 52
theorem
for b1 being real set
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b2,REAL
      st (for b4 being Element of the carrier of b2 holds
            b3 . b4 <= b1) &
         (for b4 being real set
               st for b5 being Element of the carrier of b2 holds
                    b3 . b5 <= b4
            holds b1 <= b4)
   holds b1 = sup b3;

:: PSCOMP_1:th 53
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total bounded Relation of the carrier of b1,REAL holds
   inf b2 <= sup b2;

:: PSCOMP_1:attrnot 12 => PSCOMP_1:attr 9
definition
  let a1 be TopStruct;
  let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
  attr a2 is continuous means
    for b1 being Element of bool REAL
          st b1 is closed
       holds a2 " b1 is closed(a1);
end;

:: PSCOMP_1:dfs 17
definiens
  let a1 be TopStruct;
  let a2 be Function-like quasi_total Relation of the carrier of a1,REAL;
To prove
     a2 is continuous
it is sufficient to prove
  thus for b1 being Element of bool REAL
          st b1 is closed
       holds a2 " b1 is closed(a1);

:: PSCOMP_1:def 25
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
      b2 is continuous(b1)
   iff
      for b3 being Element of bool REAL
            st b3 is closed
         holds b2 " b3 is closed(b1);

:: PSCOMP_1:exreg 7
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty Relation-like Function-like complex-valued ext-real-valued real-valued quasi_total total continuous Relation of the carrier of a1,REAL;
end;

:: PSCOMP_1:exreg 8
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty SubSpace of a1;
  cluster non empty Relation-like Function-like complex-valued ext-real-valued real-valued quasi_total total continuous Relation of the carrier of a2,REAL;
end;

:: PSCOMP_1:th 54
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL holds
      b2 is continuous(b1)
   iff
      for b3 being Element of bool REAL
            st b3 is open
         holds b2 " b3 is open(b1);

:: PSCOMP_1:th 55
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
      st b2 is continuous(b1)
   holds - b2 is continuous(b1);

:: PSCOMP_1:th 56
theorem
for b1 being Element of REAL
for b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b2,REAL
      st b3 is continuous(b2)
   holds b1 + b3 is continuous(b2);

:: PSCOMP_1:th 57
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
      st b2 is continuous(b1) & not 0 in rng b2
   holds Inv b2 is continuous(b1);

:: PSCOMP_1:th 58
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
for b3 being Element of bool bool REAL
      st b2 is continuous(b1) & b3 is open
   holds (" b2) .: b3 is open(b1);

:: PSCOMP_1:th 59
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
for b3 being Element of bool bool REAL
      st b2 is continuous(b1) & b3 is closed
   holds (" b2) .: b3 is closed(b1);

:: PSCOMP_1:funcnot 14 => PSCOMP_1:func 10
definition
  let a1 be non empty TopStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be Function-like quasi_total Relation of the carrier of a1,REAL;
  func A3 || A2 -> Function-like quasi_total Relation of the carrier of a1 | a2,REAL equals
    a3 | a2;
end;

:: PSCOMP_1:def 26
theorem
for b1 being non empty TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL holds
   b3 || b2 = b3 | b2;

:: PSCOMP_1:exreg 9
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty compact Element of bool the carrier of a1;
end;

:: PSCOMP_1:funcreg 9
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total continuous Relation of the carrier of a1,REAL;
  let a3 be Element of bool the carrier of a1;
  cluster a2 || a3 -> Function-like quasi_total continuous;
end;

:: PSCOMP_1:funcreg 10
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty compact Element of bool the carrier of a1;
  cluster a1 | a2 -> strict compact;
end;

:: PSCOMP_1:th 60
theorem
for b1 being non empty TopSpace-like TopStruct holds
      for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
            st b2 is continuous(b1)
         holds b2 is with_max(the carrier of b1)
   iff
      for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
            st b2 is continuous(b1)
         holds b2 is with_min(the carrier of b1);

:: PSCOMP_1:th 61
theorem
for b1 being non empty TopSpace-like TopStruct holds
      for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
            st b2 is continuous(b1)
         holds b2 is bounded
   iff
      for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
            st b2 is continuous(b1)
         holds b2 is with_max(the carrier of b1);

:: PSCOMP_1:attrnot 13 => PSCOMP_1:attr 10
definition
  let a1 be TopStruct;
  attr a1 is pseudocompact means
    for b1 being Function-like quasi_total Relation of the carrier of a1,REAL
          st b1 is continuous(a1)
       holds b1 is bounded;
end;

:: PSCOMP_1:dfs 19
definiens
  let a1 be TopStruct;
To prove
     a1 is pseudocompact
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of the carrier of a1,REAL
          st b1 is continuous(a1)
       holds b1 is bounded;

:: PSCOMP_1:def 27
theorem
for b1 being TopStruct holds
      b1 is pseudocompact
   iff
      for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
            st b2 is continuous(b1)
         holds b2 is bounded;

:: PSCOMP_1:condreg 2
registration
  cluster non empty TopSpace-like compact -> pseudocompact (TopStruct);
end;

:: PSCOMP_1:exreg 10
registration
  cluster non empty TopSpace-like compact TopStruct;
end;

:: PSCOMP_1:condreg 3
registration
  let a1 be non empty TopSpace-like pseudocompact TopStruct;
  cluster Function-like quasi_total continuous -> bounded with_max with_min (Relation of the carrier of a1,REAL);
end;

:: PSCOMP_1:th 62
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being compact Element of bool the carrier of b1
for b4 being Function-like quasi_total continuous Relation of the carrier of b1,REAL
      st b2 c= b3
   holds inf (b4 || b3) <= inf (b4 || b2);

:: PSCOMP_1:th 63
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being compact Element of bool the carrier of b1
for b4 being Function-like quasi_total continuous Relation of the carrier of b1,REAL
      st b2 c= b3
   holds sup (b4 || b2) <= sup (b4 || b3);

:: PSCOMP_1:th 64
theorem
for b1 being Element of NAT
for b2, b3 being compact Element of bool the carrier of TOP-REAL b1 holds
b2 /\ b3 is compact(TOP-REAL b1);

:: PSCOMP_1:funcnot 15 => PSCOMP_1:func 11
definition
  func proj1 -> Function-like quasi_total Relation of the carrier of TOP-REAL 2,REAL means
    for b1 being Element of the carrier of TOP-REAL 2 holds
       it . b1 = b1 `1;
end;

:: PSCOMP_1:def 28
theorem
for b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,REAL holds
      b1 = proj1
   iff
      for b2 being Element of the carrier of TOP-REAL 2 holds
         b1 . b2 = b2 `1;

:: PSCOMP_1:funcnot 16 => PSCOMP_1:func 12
definition
  func proj2 -> Function-like quasi_total Relation of the carrier of TOP-REAL 2,REAL means
    for b1 being Element of the carrier of TOP-REAL 2 holds
       it . b1 = b1 `2;
end;

:: PSCOMP_1:def 29
theorem
for b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,REAL holds
      b1 = proj2
   iff
      for b2 being Element of the carrier of TOP-REAL 2 holds
         b1 . b2 = b2 `2;

:: PSCOMP_1:th 65
theorem
for b1, b2 being real set holds
proj1 " ].b1,b2.[ = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b1 < b3 & b3 < b2};

:: PSCOMP_1:th 66
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of REAL
      st b1 = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: b2 < b4 & b4 < b3}
   holds b1 is open(TOP-REAL 2);

:: PSCOMP_1:th 67
theorem
for b1, b2 being real set holds
proj2 " ].b1,b2.[ = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b1 < b4 & b4 < b2};

:: PSCOMP_1:th 68
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of REAL
      st b1 = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: b2 < b5 & b5 < b3}
   holds b1 is open(TOP-REAL 2);

:: PSCOMP_1:funcreg 11
registration
  cluster proj1 -> Function-like quasi_total continuous;
end;

:: PSCOMP_1:funcreg 12
registration
  cluster proj2 -> Function-like quasi_total continuous;
end;

:: PSCOMP_1:th 69
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in b1
   holds (proj1 || b1) . b2 = b2 `1;

:: PSCOMP_1:th 70
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in b1
   holds (proj2 || b1) . b2 = b2 `2;

:: PSCOMP_1:funcnot 17 => PSCOMP_1:func 13
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func W-bound A1 -> Element of REAL equals
    inf (proj1 || a1);
end;

:: PSCOMP_1:def 30
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   W-bound b1 = inf (proj1 || b1);

:: PSCOMP_1:funcnot 18 => PSCOMP_1:func 14
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func N-bound A1 -> Element of REAL equals
    sup (proj2 || a1);
end;

:: PSCOMP_1:def 31
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   N-bound b1 = sup (proj2 || b1);

:: PSCOMP_1:funcnot 19 => PSCOMP_1:func 15
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func E-bound A1 -> Element of REAL equals
    sup (proj1 || a1);
end;

:: PSCOMP_1:def 32
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   E-bound b1 = sup (proj1 || b1);

:: PSCOMP_1:funcnot 20 => PSCOMP_1:func 16
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func S-bound A1 -> Element of REAL equals
    inf (proj2 || a1);
end;

:: PSCOMP_1:def 33
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   S-bound b1 = inf (proj2 || b1);

:: PSCOMP_1:th 71
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 in b2
   holds W-bound b2 <= b1 `1 & b1 `1 <= E-bound b2 & S-bound b2 <= b1 `2 & b1 `2 <= N-bound b2;

:: PSCOMP_1:funcnot 21 => PSCOMP_1:func 17
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func SW-corner A1 -> Element of the carrier of TOP-REAL 2 equals
    |[W-bound a1,S-bound a1]|;
end;

:: PSCOMP_1:def 34
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   SW-corner b1 = |[W-bound b1,S-bound b1]|;

:: PSCOMP_1:funcnot 22 => PSCOMP_1:func 18
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func NW-corner A1 -> Element of the carrier of TOP-REAL 2 equals
    |[W-bound a1,N-bound a1]|;
end;

:: PSCOMP_1:def 35
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   NW-corner b1 = |[W-bound b1,N-bound b1]|;

:: PSCOMP_1:funcnot 23 => PSCOMP_1:func 19
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func NE-corner A1 -> Element of the carrier of TOP-REAL 2 equals
    |[E-bound a1,N-bound a1]|;
end;

:: PSCOMP_1:def 36
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   NE-corner b1 = |[E-bound b1,N-bound b1]|;

:: PSCOMP_1:funcnot 24 => PSCOMP_1:func 20
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func SE-corner A1 -> Element of the carrier of TOP-REAL 2 equals
    |[E-bound a1,S-bound a1]|;
end;

:: PSCOMP_1:def 37
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   SE-corner b1 = |[E-bound b1,S-bound b1]|;

:: PSCOMP_1:th 80
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SW-corner b1) `1 = (NW-corner b1) `1;

:: PSCOMP_1:th 81
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SE-corner b1) `1 = (NE-corner b1) `1;

:: PSCOMP_1:th 82
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (NW-corner b1) `2 = (NE-corner b1) `2;

:: PSCOMP_1:th 83
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SW-corner b1) `2 = (SE-corner b1) `2;

:: PSCOMP_1:funcnot 25 => PSCOMP_1:func 21
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func W-most A1 -> Element of bool the carrier of TOP-REAL 2 equals
    (LSeg(SW-corner a1,NW-corner a1)) /\ a1;
end;

:: PSCOMP_1:def 38
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   W-most b1 = (LSeg(SW-corner b1,NW-corner b1)) /\ b1;

:: PSCOMP_1:funcnot 26 => PSCOMP_1:func 22
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func N-most A1 -> Element of bool the carrier of TOP-REAL 2 equals
    (LSeg(NW-corner a1,NE-corner a1)) /\ a1;
end;

:: PSCOMP_1:def 39
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   N-most b1 = (LSeg(NW-corner b1,NE-corner b1)) /\ b1;

:: PSCOMP_1:funcnot 27 => PSCOMP_1:func 23
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func E-most A1 -> Element of bool the carrier of TOP-REAL 2 equals
    (LSeg(SE-corner a1,NE-corner a1)) /\ a1;
end;

:: PSCOMP_1:def 40
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   E-most b1 = (LSeg(SE-corner b1,NE-corner b1)) /\ b1;

:: PSCOMP_1:funcnot 28 => PSCOMP_1:func 24
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func S-most A1 -> Element of bool the carrier of TOP-REAL 2 equals
    (LSeg(SW-corner a1,SE-corner a1)) /\ a1;
end;

:: PSCOMP_1:def 41
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   S-most b1 = (LSeg(SW-corner b1,SE-corner b1)) /\ b1;

:: PSCOMP_1:funcreg 13
registration
  let a1 be non empty compact Element of bool the carrier of TOP-REAL 2;
  cluster W-most a1 -> non empty compact;
end;

:: PSCOMP_1:funcreg 14
registration
  let a1 be non empty compact Element of bool the carrier of TOP-REAL 2;
  cluster N-most a1 -> non empty compact;
end;

:: PSCOMP_1:funcreg 15
registration
  let a1 be non empty compact Element of bool the carrier of TOP-REAL 2;
  cluster E-most a1 -> non empty compact;
end;

:: PSCOMP_1:funcreg 16
registration
  let a1 be non empty compact Element of bool the carrier of TOP-REAL 2;
  cluster S-most a1 -> non empty compact;
end;

:: PSCOMP_1:funcnot 29 => PSCOMP_1:func 25
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func W-min A1 -> Element of the carrier of TOP-REAL 2 equals
    |[W-bound a1,inf (proj2 || W-most a1)]|;
end;

:: PSCOMP_1:def 42
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   W-min b1 = |[W-bound b1,inf (proj2 || W-most b1)]|;

:: PSCOMP_1:funcnot 30 => PSCOMP_1:func 26
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func W-max A1 -> Element of the carrier of TOP-REAL 2 equals
    |[W-bound a1,sup (proj2 || W-most a1)]|;
end;

:: PSCOMP_1:def 43
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   W-max b1 = |[W-bound b1,sup (proj2 || W-most b1)]|;

:: PSCOMP_1:funcnot 31 => PSCOMP_1:func 27
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func N-min A1 -> Element of the carrier of TOP-REAL 2 equals
    |[inf (proj1 || N-most a1),N-bound a1]|;
end;

:: PSCOMP_1:def 44
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   N-min b1 = |[inf (proj1 || N-most b1),N-bound b1]|;

:: PSCOMP_1:funcnot 32 => PSCOMP_1:func 28
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func N-max A1 -> Element of the carrier of TOP-REAL 2 equals
    |[sup (proj1 || N-most a1),N-bound a1]|;
end;

:: PSCOMP_1:def 45
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   N-max b1 = |[sup (proj1 || N-most b1),N-bound b1]|;

:: PSCOMP_1:funcnot 33 => PSCOMP_1:func 29
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func E-max A1 -> Element of the carrier of TOP-REAL 2 equals
    |[E-bound a1,sup (proj2 || E-most a1)]|;
end;

:: PSCOMP_1:def 46
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   E-max b1 = |[E-bound b1,sup (proj2 || E-most b1)]|;

:: PSCOMP_1:funcnot 34 => PSCOMP_1:func 30
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func E-min A1 -> Element of the carrier of TOP-REAL 2 equals
    |[E-bound a1,inf (proj2 || E-most a1)]|;
end;

:: PSCOMP_1:def 47
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   E-min b1 = |[E-bound b1,inf (proj2 || E-most b1)]|;

:: PSCOMP_1:funcnot 35 => PSCOMP_1:func 31
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func S-max A1 -> Element of the carrier of TOP-REAL 2 equals
    |[sup (proj1 || S-most a1),S-bound a1]|;
end;

:: PSCOMP_1:def 48
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   S-max b1 = |[sup (proj1 || S-most b1),S-bound b1]|;

:: PSCOMP_1:funcnot 36 => PSCOMP_1:func 32
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func S-min A1 -> Element of the carrier of TOP-REAL 2 equals
    |[inf (proj1 || S-most a1),S-bound a1]|;
end;

:: PSCOMP_1:def 49
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   S-min b1 = |[inf (proj1 || S-most b1),S-bound b1]|;

:: PSCOMP_1:th 85
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SW-corner b1) `1 = (W-min b1) `1 & (SW-corner b1) `1 = (W-max b1) `1 & (W-min b1) `1 = (W-max b1) `1 & (W-min b1) `1 = (NW-corner b1) `1 & (W-max b1) `1 = (NW-corner b1) `1;

:: PSCOMP_1:th 87
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (SW-corner b1) `2 <= (W-min b1) `2 & (SW-corner b1) `2 <= (W-max b1) `2 & (SW-corner b1) `2 <= (NW-corner b1) `2 & (W-min b1) `2 <= (W-max b1) `2 & (W-min b1) `2 <= (NW-corner b1) `2 & (W-max b1) `2 <= (NW-corner b1) `2;

:: PSCOMP_1:th 88
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 in W-most b2
   holds b1 `1 = (W-min b2) `1 &
    (b2 is compact(TOP-REAL 2) implies (W-min b2) `2 <= b1 `2 & b1 `2 <= (W-max b2) `2);

:: PSCOMP_1:th 89
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   W-most b1 c= LSeg(W-min b1,W-max b1);

:: PSCOMP_1:th 90
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   LSeg(W-min b1,W-max b1) c= LSeg(SW-corner b1,NW-corner b1);

:: PSCOMP_1:th 91
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   W-min b1 in W-most b1 & W-max b1 in W-most b1;

:: PSCOMP_1:th 92
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (LSeg(SW-corner b1,W-min b1)) /\ b1 = {W-min b1} &
    (LSeg(W-max b1,NW-corner b1)) /\ b1 = {W-max b1};

:: PSCOMP_1:th 93
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st W-min b1 = W-max b1
   holds W-most b1 = {W-min b1};

:: PSCOMP_1:th 95
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (NW-corner b1) `2 = (N-min b1) `2 & (NW-corner b1) `2 = (N-max b1) `2 & (N-min b1) `2 = (N-max b1) `2 & (N-min b1) `2 = (NE-corner b1) `2 & (N-max b1) `2 = (NE-corner b1) `2;

:: PSCOMP_1:th 97
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (NW-corner b1) `1 <= (N-min b1) `1 & (NW-corner b1) `1 <= (N-max b1) `1 & (NW-corner b1) `1 <= (NE-corner b1) `1 & (N-min b1) `1 <= (N-max b1) `1 & (N-min b1) `1 <= (NE-corner b1) `1 & (N-max b1) `1 <= (NE-corner b1) `1;

:: PSCOMP_1:th 98
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 in N-most b2
   holds b1 `2 = (N-min b2) `2 &
    (b2 is compact(TOP-REAL 2) implies (N-min b2) `1 <= b1 `1 & b1 `1 <= (N-max b2) `1);

:: PSCOMP_1:th 99
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   N-most b1 c= LSeg(N-min b1,N-max b1);

:: PSCOMP_1:th 100
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   LSeg(N-min b1,N-max b1) c= LSeg(NW-corner b1,NE-corner b1);

:: PSCOMP_1:th 101
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   N-min b1 in N-most b1 & N-max b1 in N-most b1;

:: PSCOMP_1:th 102
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (LSeg(NW-corner b1,N-min b1)) /\ b1 = {N-min b1} &
    (LSeg(N-max b1,NE-corner b1)) /\ b1 = {N-max b1};

:: PSCOMP_1:th 103
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st N-min b1 = N-max b1
   holds N-most b1 = {N-min b1};

:: PSCOMP_1:th 105
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SE-corner b1) `1 = (E-min b1) `1 & (SE-corner b1) `1 = (E-max b1) `1 & (E-min b1) `1 = (E-max b1) `1 & (E-min b1) `1 = (NE-corner b1) `1 & (E-max b1) `1 = (NE-corner b1) `1;

:: PSCOMP_1:th 107
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (SE-corner b1) `2 <= (E-min b1) `2 & (SE-corner b1) `2 <= (E-max b1) `2 & (SE-corner b1) `2 <= (NE-corner b1) `2 & (E-min b1) `2 <= (E-max b1) `2 & (E-min b1) `2 <= (NE-corner b1) `2 & (E-max b1) `2 <= (NE-corner b1) `2;

:: PSCOMP_1:th 108
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 in E-most b2
   holds b1 `1 = (E-min b2) `1 &
    (b2 is compact(TOP-REAL 2) implies (E-min b2) `2 <= b1 `2 & b1 `2 <= (E-max b2) `2);

:: PSCOMP_1:th 109
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   E-most b1 c= LSeg(E-min b1,E-max b1);

:: PSCOMP_1:th 110
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   LSeg(E-min b1,E-max b1) c= LSeg(SE-corner b1,NE-corner b1);

:: PSCOMP_1:th 111
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   E-min b1 in E-most b1 & E-max b1 in E-most b1;

:: PSCOMP_1:th 112
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (LSeg(SE-corner b1,E-min b1)) /\ b1 = {E-min b1} &
    (LSeg(E-max b1,NE-corner b1)) /\ b1 = {E-max b1};

:: PSCOMP_1:th 113
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st E-min b1 = E-max b1
   holds E-most b1 = {E-min b1};

:: PSCOMP_1:th 115
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SW-corner b1) `2 = (S-min b1) `2 & (SW-corner b1) `2 = (S-max b1) `2 & (S-min b1) `2 = (S-max b1) `2 & (S-min b1) `2 = (SE-corner b1) `2 & (S-max b1) `2 = (SE-corner b1) `2;

:: PSCOMP_1:th 117
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (SW-corner b1) `1 <= (S-min b1) `1 & (SW-corner b1) `1 <= (S-max b1) `1 & (SW-corner b1) `1 <= (SE-corner b1) `1 & (S-min b1) `1 <= (S-max b1) `1 & (S-min b1) `1 <= (SE-corner b1) `1 & (S-max b1) `1 <= (SE-corner b1) `1;

:: PSCOMP_1:th 118
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 in S-most b2
   holds b1 `2 = (S-min b2) `2 &
    (b2 is compact(TOP-REAL 2) implies (S-min b2) `1 <= b1 `1 & b1 `1 <= (S-max b2) `1);

:: PSCOMP_1:th 119
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   S-most b1 c= LSeg(S-min b1,S-max b1);

:: PSCOMP_1:th 120
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   LSeg(S-min b1,S-max b1) c= LSeg(SW-corner b1,SE-corner b1);

:: PSCOMP_1:th 121
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   S-min b1 in S-most b1 & S-max b1 in S-most b1;

:: PSCOMP_1:th 122
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (LSeg(SW-corner b1,S-min b1)) /\ b1 = {S-min b1} &
    (LSeg(S-max b1,SE-corner b1)) /\ b1 = {S-max b1};

:: PSCOMP_1:th 123
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st S-min b1 = S-max b1
   holds S-most b1 = {S-min b1};

:: PSCOMP_1:th 124
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st W-max b1 = N-min b1
   holds W-max b1 = NW-corner b1;

:: PSCOMP_1:th 125
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st N-max b1 = E-max b1
   holds N-max b1 = NE-corner b1;

:: PSCOMP_1:th 126
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st E-min b1 = S-max b1
   holds E-min b1 = SE-corner b1;

:: PSCOMP_1:th 127
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st S-min b1 = W-min b1
   holds S-min b1 = SW-corner b1;

:: PSCOMP_1:th 128
theorem
for b1, b2 being real set holds
proj2 . |[b1,b2]| = b2 & proj1 . |[b1,b2]| = b1;

:: PSCOMP_1:th 129
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2
   holds N-bound b1 <= N-bound b2;

:: PSCOMP_1:th 130
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2
   holds E-bound b1 <= E-bound b2;

:: PSCOMP_1:th 131
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2
   holds S-bound b2 <= S-bound b1;

:: PSCOMP_1:th 132
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2
   holds W-bound b2 <= W-bound b1;