Article TOPREAL1, MML version 4.99.1005

:: TOPREAL1:sch 1
scheme TOPREAL1:sch 1
{F1 -> non empty set}:
{b1 where b1 is Element of F1(): (P1[b1] or P2[b1])} = {b1 where b1 is Element of F1(): P1[b1]} \/ {b1 where b1 is Element of F1(): P2[b1]}


:: TOPREAL1:prednot 1 => TOPREAL1:pred 1
definition
  let a1 be TopSpace-like TopStruct;
  let a2, a3 be Element of the carrier of a1;
  let a4 be Element of bool the carrier of a1;
  pred A4 is_an_arc_of A2,A3 means
    ex b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of a1 | a4 st
       b1 is being_homeomorphism(I[01], a1 | a4) & b1 . 0 = a2 & b1 . 1 = a3;
end;

:: TOPREAL1:dfs 1
definiens
  let a1 be TopSpace-like TopStruct;
  let a2, a3 be Element of the carrier of a1;
  let a4 be Element of bool the carrier of a1;
To prove
     a4 is_an_arc_of a2,a3
it is sufficient to prove
  thus ex b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of a1 | a4 st
       b1 is being_homeomorphism(I[01], a1 | a4) & b1 . 0 = a2 & b1 . 1 = a3;

:: TOPREAL1:def 2
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
      b4 is_an_arc_of b2,b3
   iff
      ex b5 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 | b4 st
         b5 is being_homeomorphism(I[01], b1 | b4) & b5 . 0 = b2 & b5 . 1 = b3;

:: TOPREAL1:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
      st b2 is_an_arc_of b3,b4
   holds b3 in b2 & b4 in b2;

:: TOPREAL1:th 5
theorem
for b1 being TopSpace-like being_T2 TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4, b5, b6 being Element of the carrier of b1
      st b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b5,b6 & b2 /\ b3 = {b5}
   holds b2 \/ b3 is_an_arc_of b4,b6;

:: TOPREAL1:funcnot 1 => TOPREAL1:func 1
definition
  let a1 be natural set;
  let a2, a3 be Element of the carrier of TOP-REAL a1;
  func LSeg(A2,A3) -> Element of bool the carrier of TOP-REAL a1 equals
    {((1 - b1) * a2) + (b1 * a3) where b1 is Element of REAL: 0 <= b1 & b1 <= 1};
end;

:: TOPREAL1:def 3
theorem
for b1 being natural set
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
LSeg(b2,b3) = {((1 - b4) * b2) + (b4 * b3) where b4 is Element of REAL: 0 <= b4 & b4 <= 1};

:: TOPREAL1:funcnot 2 => TOPREAL1:func 2
definition
  func R^2-unit_square -> Element of bool the carrier of TOP-REAL 2 equals
    ((LSeg(|[0,0]|,|[0,1]|)) \/ LSeg(|[0,1]|,|[1,1]|)) \/ ((LSeg(|[1,1]|,|[1,0]|)) \/ LSeg(|[1,0]|,|[0,0]|));
end;

:: TOPREAL1:def 4
theorem
R^2-unit_square = ((LSeg(|[0,0]|,|[0,1]|)) \/ LSeg(|[0,1]|,|[1,1]|)) \/ ((LSeg(|[1,1]|,|[1,0]|)) \/ LSeg(|[1,0]|,|[0,0]|));

:: TOPREAL1:funcreg 1
registration
  let a1 be natural set;
  let a2, a3 be Element of the carrier of TOP-REAL a1;
  cluster LSeg(a2,a3) -> non empty;
end;

:: TOPREAL1:th 6
theorem
for b1 being natural set
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
b2 in LSeg(b2,b3) & b3 in LSeg(b2,b3);

:: TOPREAL1:th 7
theorem
for b1 being natural set
for b2 being Element of the carrier of TOP-REAL b1 holds
   LSeg(b2,b2) = {b2};

:: TOPREAL1:th 8
theorem
for b1 being natural set
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
LSeg(b2,b3) = LSeg(b3,b2);

:: TOPREAL1:funcnot 3 => TOPREAL1:func 3
definition
  let a1 be natural set;
  let a2, a3 be Element of the carrier of TOP-REAL a1;
  redefine func LSeg(a2,a3) -> Element of bool the carrier of TOP-REAL a1;
  commutativity;
::  for a1 being natural set
::  for a2, a3 being Element of the carrier of TOP-REAL a1 holds
::  LSeg(a2,a3) = LSeg(a3,a2);
end;

:: TOPREAL1:th 9
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 `1 <= b2 `1 & b3 in LSeg(b1,b2)
   holds b1 `1 <= b3 `1 & b3 `1 <= b2 `1;

:: TOPREAL1:th 10
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 `2 <= b2 `2 & b3 in LSeg(b1,b2)
   holds b1 `2 <= b3 `2 & b3 `2 <= b2 `2;

:: TOPREAL1:th 11
theorem
for b1 being natural set
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 in LSeg(b3,b4)
   holds LSeg(b3,b4) = (LSeg(b3,b2)) \/ LSeg(b2,b4);

:: TOPREAL1:th 12
theorem
for b1 being natural set
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st b4 in LSeg(b2,b3) & b5 in LSeg(b2,b3)
   holds LSeg(b4,b5) c= LSeg(b2,b3);

:: TOPREAL1:th 13
theorem
for b1 being natural set
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st b2 in LSeg(b4,b5) & b3 in LSeg(b4,b5)
   holds LSeg(b4,b5) = ((LSeg(b4,b2)) \/ LSeg(b2,b3)) \/ LSeg(b3,b5);

:: TOPREAL1:th 14
theorem
for b1 being natural set
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 in LSeg(b3,b4)
   holds (LSeg(b3,b2)) /\ LSeg(b2,b4) = {b2};

:: TOPREAL1:th 15
theorem
for b1 being natural set
for b2, b3 being Element of the carrier of TOP-REAL b1
      st b2 <> b3
   holds LSeg(b2,b3) is_an_arc_of b2,b3;

:: TOPREAL1:funcreg 2
registration
  let a1 be natural set;
  cluster TOP-REAL a1 -> strict TopSpace-like being_T2;
end;

:: TOPREAL1:th 16
theorem
for b1 being natural set
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b3,b4 & b2 /\ LSeg(b4,b5) = {b4}
   holds b2 \/ LSeg(b4,b5) is_an_arc_of b3,b5;

:: TOPREAL1:th 17
theorem
for b1 being natural set
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b4,b3 & (LSeg(b5,b4)) /\ b2 = {b4}
   holds (LSeg(b5,b4)) \/ b2 is_an_arc_of b5,b3;

:: TOPREAL1:th 18
theorem
for b1 being natural set
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
      st (b2 = b3 implies b3 <> b4) &
         (LSeg(b2,b3)) /\ LSeg(b3,b4) = {b3}
   holds (LSeg(b2,b3)) \/ LSeg(b3,b4) is_an_arc_of b2,b4;

:: TOPREAL1:th 19
theorem
LSeg(|[0,0]|,|[0,1]|) = {b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `1 = 0 & b1 `2 <= 1 & 0 <= b1 `2} &
 LSeg(|[0,1]|,|[1,1]|) = {b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `1 <= 1 & 0 <= b1 `1 & b1 `2 = 1} &
 LSeg(|[0,0]|,|[1,0]|) = {b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `1 <= 1 & 0 <= b1 `1 & b1 `2 = 0} &
 LSeg(|[1,0]|,|[1,1]|) = {b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `1 = 1 & b1 `2 <= 1 & 0 <= b1 `2};

:: TOPREAL1:th 20
theorem
R^2-unit_square = {b1 where b1 is Element of the carrier of TOP-REAL 2: ((b1 `1 = 0 & b1 `2 <= 1 implies b1 `2 < 0) &
 (b1 `1 <= 1 & 0 <= b1 `1 implies b1 `2 <> 1) &
 (b1 `1 <= 1 & 0 <= b1 `1 implies b1 `2 <> 0) implies b1 `1 = 1 & b1 `2 <= 1 & 0 <= b1 `2)};

:: TOPREAL1:funcreg 3
registration
  cluster R^2-unit_square -> non empty;
end;

:: TOPREAL1:th 21
theorem
(LSeg(|[0,0]|,|[0,1]|)) /\ LSeg(|[0,1]|,|[1,1]|) = {|[0,1]|};

:: TOPREAL1:th 22
theorem
(LSeg(|[0,0]|,|[1,0]|)) /\ LSeg(|[1,0]|,|[1,1]|) = {|[1,0]|};

:: TOPREAL1:th 23
theorem
(LSeg(|[0,0]|,|[0,1]|)) /\ LSeg(|[0,0]|,|[1,0]|) = {|[0,0]|};

:: TOPREAL1:th 24
theorem
(LSeg(|[0,1]|,|[1,1]|)) /\ LSeg(|[1,0]|,|[1,1]|) = {|[1,1]|};

:: TOPREAL1:th 25
theorem
LSeg(|[0,0]|,|[1,0]|) misses LSeg(|[0,1]|,|[1,1]|);

:: TOPREAL1:th 26
theorem
LSeg(|[0,0]|,|[0,1]|) misses LSeg(|[1,0]|,|[1,1]|);

:: TOPREAL1:funcnot 4 => TOPREAL1:func 4
definition
  let a1 be natural set;
  let a2 be FinSequence of the carrier of TOP-REAL a1;
  let a3 be natural set;
  func LSeg(A2,A3) -> Element of bool the carrier of TOP-REAL a1 equals
    LSeg(a2 /. a3,a2 /. (a3 + 1))
    if 1 <= a3 & a3 + 1 <= len a2
    otherwise {};
end;

:: TOPREAL1:def 5
theorem
for b1 being natural set
for b2 being FinSequence of the carrier of TOP-REAL b1
for b3 being natural set holds
   (1 <= b3 & b3 + 1 <= len b2 implies LSeg(b2,b3) = LSeg(b2 /. b3,b2 /. (b3 + 1))) &
    (1 <= b3 & b3 + 1 <= len b2 or LSeg(b2,b3) = {});

:: TOPREAL1:th 27
theorem
for b1, b2 being natural set
for b3 being FinSequence of the carrier of TOP-REAL b1
      st 1 <= b2 & b2 + 1 <= len b3
   holds b3 /. b2 in LSeg(b3,b2) & b3 /. (b2 + 1) in LSeg(b3,b2);

:: TOPREAL1:funcnot 5 => TOPREAL1:func 5
definition
  let a1 be natural set;
  let a2 be FinSequence of the carrier of TOP-REAL a1;
  func L~ A2 -> Element of bool the carrier of TOP-REAL a1 equals
    union {LSeg(a2,b1) where b1 is Element of NAT: 1 <= b1 & b1 + 1 <= len a2};
end;

:: TOPREAL1:def 6
theorem
for b1 being natural set
for b2 being FinSequence of the carrier of TOP-REAL b1 holds
   L~ b2 = union {LSeg(b2,b3) where b3 is Element of NAT: 1 <= b3 & b3 + 1 <= len b2};

:: TOPREAL1:th 28
theorem
for b1 being natural set
for b2 being FinSequence of the carrier of TOP-REAL b1 holds
      (len b2 = 0 or len b2 = 1)
   iff
      L~ b2 = {};

:: TOPREAL1:th 29
theorem
for b1 being natural set
for b2 being FinSequence of the carrier of TOP-REAL b1
      st 2 <= len b2
   holds L~ b2 <> {};

:: TOPREAL1:attrnot 1 => TOPREAL1:attr 1
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  attr a1 is special means
    for b1 being natural set
          st 1 <= b1 &
             b1 + 1 <= len a1 &
             (a1 /. b1) `1 <> (a1 /. (b1 + 1)) `1
       holds (a1 /. b1) `2 = (a1 /. (b1 + 1)) `2;
end;

:: TOPREAL1:dfs 6
definiens
  let a1 be FinSequence of the carrier of TOP-REAL 2;
To prove
     a1 is special
it is sufficient to prove
  thus for b1 being natural set
          st 1 <= b1 &
             b1 + 1 <= len a1 &
             (a1 /. b1) `1 <> (a1 /. (b1 + 1)) `1
       holds (a1 /. b1) `2 = (a1 /. (b1 + 1)) `2;

:: TOPREAL1:def 7
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
      b1 is special
   iff
      for b2 being natural set
            st 1 <= b2 &
               b2 + 1 <= len b1 &
               (b1 /. b2) `1 <> (b1 /. (b2 + 1)) `1
         holds (b1 /. b2) `2 = (b1 /. (b2 + 1)) `2;

:: TOPREAL1:attrnot 2 => TOPREAL1:attr 2
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  attr a1 is unfolded means
    for b1 being natural set
          st 1 <= b1 & b1 + 2 <= len a1
       holds (LSeg(a1,b1)) /\ LSeg(a1,b1 + 1) = {a1 /. (b1 + 1)};
end;

:: TOPREAL1:dfs 7
definiens
  let a1 be FinSequence of the carrier of TOP-REAL 2;
To prove
     a1 is unfolded
it is sufficient to prove
  thus for b1 being natural set
          st 1 <= b1 & b1 + 2 <= len a1
       holds (LSeg(a1,b1)) /\ LSeg(a1,b1 + 1) = {a1 /. (b1 + 1)};

:: TOPREAL1:def 8
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
      b1 is unfolded
   iff
      for b2 being natural set
            st 1 <= b2 & b2 + 2 <= len b1
         holds (LSeg(b1,b2)) /\ LSeg(b1,b2 + 1) = {b1 /. (b2 + 1)};

:: TOPREAL1:attrnot 3 => TOPREAL1:attr 3
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  attr a1 is s.n.c. means
    for b1, b2 being natural set
          st b1 + 1 < b2
       holds LSeg(a1,b1) misses LSeg(a1,b2);
end;

:: TOPREAL1:dfs 8
definiens
  let a1 be FinSequence of the carrier of TOP-REAL 2;
To prove
     a1 is s.n.c.
it is sufficient to prove
  thus for b1, b2 being natural set
          st b1 + 1 < b2
       holds LSeg(a1,b1) misses LSeg(a1,b2);

:: TOPREAL1:def 9
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
      b1 is s.n.c.
   iff
      for b2, b3 being natural set
            st b2 + 1 < b3
         holds LSeg(b1,b2) misses LSeg(b1,b3);

:: TOPREAL1:attrnot 4 => TOPREAL1:attr 4
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  attr a1 is being_S-Seq means
    a1 is one-to-one & 2 <= len a1 & a1 is unfolded & a1 is s.n.c. & a1 is special;
end;

:: TOPREAL1:dfs 9
definiens
  let a1 be FinSequence of the carrier of TOP-REAL 2;
To prove
     a1 is being_S-Seq
it is sufficient to prove
  thus a1 is one-to-one & 2 <= len a1 & a1 is unfolded & a1 is s.n.c. & a1 is special;

:: TOPREAL1:def 10
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
      b1 is being_S-Seq
   iff
      b1 is one-to-one & 2 <= len b1 & b1 is unfolded & b1 is s.n.c. & b1 is special;

:: TOPREAL1:prednot 2 => TOPREAL1:attr 4
notation
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  synonym a1 is_S-Seq for being_S-Seq;
end;

:: TOPREAL1:th 30
theorem
ex b1, b2 being FinSequence of the carrier of TOP-REAL 2 st
   b1 is being_S-Seq &
    b2 is being_S-Seq &
    R^2-unit_square = (L~ b1) \/ L~ b2 &
    (L~ b1) /\ L~ b2 = {|[0,0]|,|[1,1]|} &
    b1 /. 1 = |[0,0]| &
    b1 /. len b1 = |[1,1]| &
    b2 /. 1 = |[0,0]| &
    b2 /. len b2 = |[1,1]|;

:: TOPREAL1:th 31
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
      st b1 is being_S-Seq
   holds L~ b1 is_an_arc_of b1 /. 1,b1 /. len b1;

:: TOPREAL1:attrnot 5 => TOPREAL1:attr 5
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  attr a1 is being_S-P_arc means
    ex b1 being FinSequence of the carrier of TOP-REAL 2 st
       b1 is being_S-Seq & a1 = L~ b1;
end;

:: TOPREAL1:dfs 10
definiens
  let a1 be Element of bool the carrier of TOP-REAL 2;
To prove
     a1 is being_S-P_arc
it is sufficient to prove
  thus ex b1 being FinSequence of the carrier of TOP-REAL 2 st
       b1 is being_S-Seq & a1 = L~ b1;

:: TOPREAL1:def 11
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
      b1 is being_S-P_arc
   iff
      ex b2 being FinSequence of the carrier of TOP-REAL 2 st
         b2 is being_S-Seq & b1 = L~ b2;

:: TOPREAL1:prednot 3 => TOPREAL1:attr 5
notation
  let a1 be Element of bool the carrier of TOP-REAL 2;
  synonym a1 is_S-P_arc for being_S-P_arc;
end;

:: TOPREAL1:th 32
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is being_S-P_arc
   holds b1 <> {};

:: TOPREAL1:condreg 1
registration
  cluster being_S-P_arc -> non empty (Element of bool the carrier of TOP-REAL 2);
end;

:: TOPREAL1:th 34
theorem
ex b1, b2 being non empty Element of bool the carrier of TOP-REAL 2 st
   b1 is being_S-P_arc &
    b2 is being_S-P_arc &
    R^2-unit_square = b1 \/ b2 &
    b1 /\ b2 = {|[0,0]|,|[1,1]|};

:: TOPREAL1:th 35
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is being_S-P_arc
   holds ex b2, b3 being Element of the carrier of TOP-REAL 2 st
      b1 is_an_arc_of b2,b3;

:: TOPREAL1:th 36
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is being_S-P_arc
   holds ex b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1 st
      b2 is being_homeomorphism(I[01], (TOP-REAL 2) | b1);

:: TOPREAL1:sch 2
scheme TOPREAL1:sch 2
{F1 -> natural set}:
ex b1 being Element of bool the carrier of TOP-REAL F1() st
   for b2 being Element of the carrier of TOP-REAL F1() holds
         b2 in b1
      iff
         P1[b2]


:: TOPREAL1:sch 3
scheme TOPREAL1:sch 3
{F1 -> natural set}:
for b1, b2 being Element of bool the carrier of TOP-REAL F1()
      st (for b3 being Element of the carrier of TOP-REAL F1() holds
               b3 in b1
            iff
               P1[b3]) &
         (for b3 being Element of the carrier of TOP-REAL F1() holds
               b3 in b2
            iff
               P1[b3])
   holds b1 = b2


:: TOPREAL1:funcnot 6 => TOPREAL1:func 6
definition
  let a1 be Element of the carrier of TOP-REAL 2;
  func north_halfline A1 -> Element of bool the carrier of TOP-REAL 2 means
    for b1 being Element of the carrier of TOP-REAL 2 holds
          b1 in it
       iff
          b1 `1 = a1 `1 & a1 `2 <= b1 `2;
end;

:: TOPREAL1:def 12
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2 holds
      b2 = north_halfline b1
   iff
      for b3 being Element of the carrier of TOP-REAL 2 holds
            b3 in b2
         iff
            b3 `1 = b1 `1 & b1 `2 <= b3 `2;

:: TOPREAL1:funcnot 7 => TOPREAL1:func 7
definition
  let a1 be Element of the carrier of TOP-REAL 2;
  func east_halfline A1 -> Element of bool the carrier of TOP-REAL 2 means
    for b1 being Element of the carrier of TOP-REAL 2 holds
          b1 in it
       iff
          a1 `1 <= b1 `1 & b1 `2 = a1 `2;
end;

:: TOPREAL1:def 13
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2 holds
      b2 = east_halfline b1
   iff
      for b3 being Element of the carrier of TOP-REAL 2 holds
            b3 in b2
         iff
            b1 `1 <= b3 `1 & b3 `2 = b1 `2;

:: TOPREAL1:funcnot 8 => TOPREAL1:func 8
definition
  let a1 be Element of the carrier of TOP-REAL 2;
  func south_halfline A1 -> Element of bool the carrier of TOP-REAL 2 means
    for b1 being Element of the carrier of TOP-REAL 2 holds
          b1 in it
       iff
          b1 `1 = a1 `1 & b1 `2 <= a1 `2;
end;

:: TOPREAL1:def 14
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2 holds
      b2 = south_halfline b1
   iff
      for b3 being Element of the carrier of TOP-REAL 2 holds
            b3 in b2
         iff
            b3 `1 = b1 `1 & b3 `2 <= b1 `2;

:: TOPREAL1:funcnot 9 => TOPREAL1:func 9
definition
  let a1 be Element of the carrier of TOP-REAL 2;
  func west_halfline A1 -> Element of bool the carrier of TOP-REAL 2 means
    for b1 being Element of the carrier of TOP-REAL 2 holds
          b1 in it
       iff
          b1 `1 <= a1 `1 & b1 `2 = a1 `2;
end;

:: TOPREAL1:def 15
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2 holds
      b2 = west_halfline b1
   iff
      for b3 being Element of the carrier of TOP-REAL 2 holds
            b3 in b2
         iff
            b3 `1 <= b1 `1 & b3 `2 = b1 `2;

:: TOPREAL1:th 37
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   north_halfline b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `1 = b1 `1 & b1 `2 <= b2 `2};

:: TOPREAL1:th 38
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   north_halfline b1 = {|[b1 `1,b2]| where b2 is Element of REAL: b1 `2 <= b2};

:: TOPREAL1:th 39
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   east_halfline b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: b1 `1 <= b2 `1 & b2 `2 = b1 `2};

:: TOPREAL1:th 40
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   east_halfline b1 = {|[b2,b1 `2]| where b2 is Element of REAL: b1 `1 <= b2};

:: TOPREAL1:th 41
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   south_halfline b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `1 = b1 `1 & b2 `2 <= b1 `2};

:: TOPREAL1:th 42
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   south_halfline b1 = {|[b1 `1,b2]| where b2 is Element of REAL: b2 <= b1 `2};

:: TOPREAL1:th 43
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   west_halfline b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `1 <= b1 `1 & b2 `2 = b1 `2};

:: TOPREAL1:th 44
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   west_halfline b1 = {|[b2,b1 `2]| where b2 is Element of REAL: b2 <= b1 `1};

:: TOPREAL1:funcreg 4
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster north_halfline a1 -> non empty;
end;

:: TOPREAL1:funcreg 5
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster east_halfline a1 -> non empty;
end;

:: TOPREAL1:funcreg 6
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster south_halfline a1 -> non empty;
end;

:: TOPREAL1:funcreg 7
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster west_halfline a1 -> non empty;
end;

:: TOPREAL1:th 45
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   b1 in west_halfline b1 & b1 in east_halfline b1 & b1 in north_halfline b1 & b1 in south_halfline b1;