Article JORDAN6, MML version 4.99.1005

:: JORDAN6:th 2
theorem
for b1, b2 being real set
      st b1 <= b2
   holds b1 <= (b1 + b2) / 2 & (b1 + b2) / 2 <= b2;

:: JORDAN6:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 | b2
for b4 being Element of bool the carrier of b1
      st b4 is closed(b1) & b3 = b4 /\ b2
   holds b3 is closed(b1 | b2);

:: JORDAN6:th 4
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty Element of bool the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 | b3 holds
   b4 is Function-like quasi_total Relation of the carrier of b1,the carrier of b2 &
    (for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
          st b5 = b4 & b4 is continuous(b1, b2 | b3)
       holds b5 is continuous(b1, b2));

:: JORDAN6:th 5
theorem
for b1 being real set
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b1 <= b3 `1}
   holds b2 is closed(TOP-REAL 2);

:: JORDAN6:th 6
theorem
for b1 being real set
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 <= b1}
   holds b2 is closed(TOP-REAL 2);

:: JORDAN6:th 7
theorem
for b1 being real set
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 = b1}
   holds b2 is closed(TOP-REAL 2);

:: JORDAN6:th 8
theorem
for b1 being real set
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b1 <= b3 `2}
   holds b2 is closed(TOP-REAL 2);

:: JORDAN6:th 9
theorem
for b1 being real set
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 <= b1}
   holds b2 is closed(TOP-REAL 2);

:: JORDAN6:th 10
theorem
for b1 being real set
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 = b1}
   holds b2 is closed(TOP-REAL 2);

:: JORDAN6:th 11
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b3,b4
   holds b2 is connected(TOP-REAL b1);

:: JORDAN6:th 12
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3
   holds b1 is closed(TOP-REAL 2);

:: JORDAN6:th 13
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3
   holds ex b4 being Element of the carrier of TOP-REAL 2 st
      b4 in b1 &
       b4 `1 = (b2 `1 + (b3 `1)) / 2;

:: JORDAN6:th 14
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b3,b4 &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 = (b3 `1 + (b4 `1)) / 2}
   holds b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2);

:: JORDAN6:th 15
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b3,b4 &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `2 = (b3 `2 + (b4 `2)) / 2}
   holds b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2);

:: JORDAN6:funcnot 1 => JORDAN6:func 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3 be Element of the carrier of TOP-REAL 2;
  func x_Middle(A1,A2,A3) -> Element of the carrier of TOP-REAL 2 means
    for b1 being Element of bool the carrier of TOP-REAL 2
          st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `1 = (a2 `1 + (a3 `1)) / 2}
       holds it = First_Point(a1,a2,a3,b1);
end;

:: JORDAN6:def 1
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2 holds
   b4 = x_Middle(b1,b2,b3)
iff
   for b5 being Element of bool the carrier of TOP-REAL 2
         st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: b6 `1 = (b2 `1 + (b3 `1)) / 2}
      holds b4 = First_Point(b1,b2,b3,b5);

:: JORDAN6:funcnot 2 => JORDAN6:func 2
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3 be Element of the carrier of TOP-REAL 2;
  func y_Middle(A1,A2,A3) -> Element of the carrier of TOP-REAL 2 means
    for b1 being Element of bool the carrier of TOP-REAL 2
          st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `2 = (a2 `2 + (a3 `2)) / 2}
       holds it = First_Point(a1,a2,a3,b1);
end;

:: JORDAN6:def 2
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2 holds
   b4 = y_Middle(b1,b2,b3)
iff
   for b5 being Element of bool the carrier of TOP-REAL 2
         st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: b6 `2 = (b2 `2 + (b3 `2)) / 2}
      holds b4 = First_Point(b1,b2,b3,b5);

:: JORDAN6:th 16
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3
   holds x_Middle(b1,b2,b3) in b1 & y_Middle(b1,b2,b3) in b1;

:: JORDAN6:th 17
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3
   holds    b2 = x_Middle(b1,b2,b3)
   iff
      b2 `1 = b3 `1;

:: JORDAN6:th 18
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3
   holds    b2 = y_Middle(b1,b2,b3)
   iff
      b2 `2 = b3 `2;

:: JORDAN6:th 19
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3
   holds LE b5,b4,b1,b3,b2;

:: JORDAN6:funcnot 3 => JORDAN6:func 3
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
  func L_Segment(A1,A2,A3,A4) -> Element of bool the carrier of TOP-REAL 2 equals
    {b1 where b1 is Element of the carrier of TOP-REAL 2: LE b1,a4,a1,a2,a3};
end;

:: JORDAN6:def 3
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2 holds
L_Segment(b1,b2,b3,b4) = {b5 where b5 is Element of the carrier of TOP-REAL 2: LE b5,b4,b1,b2,b3};

:: JORDAN6:funcnot 4 => JORDAN6:func 4
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
  func R_Segment(A1,A2,A3,A4) -> Element of bool the carrier of TOP-REAL 2 equals
    {b1 where b1 is Element of the carrier of TOP-REAL 2: LE a4,b1,a1,a2,a3};
end;

:: JORDAN6:def 4
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2 holds
R_Segment(b1,b2,b3,b4) = {b5 where b5 is Element of the carrier of TOP-REAL 2: LE b4,b5,b1,b2,b3};

:: JORDAN6:th 20
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2 holds
L_Segment(b1,b2,b3,b4) c= b1;

:: JORDAN6:th 21
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2 holds
R_Segment(b1,b2,b3,b4) c= b1;

:: JORDAN6:th 22
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3
   holds L_Segment(b1,b2,b3,b2) = {b2};

:: JORDAN6:th 25
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3
   holds L_Segment(b1,b2,b3,b3) = b1;

:: JORDAN6:th 26
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3
   holds R_Segment(b1,b2,b3,b3) = {b3};

:: JORDAN6:th 27
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3
   holds R_Segment(b1,b2,b3,b2) = b1;

:: JORDAN6:th 28
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & b4 in b1
   holds R_Segment(b1,b2,b3,b4) = L_Segment(b1,b3,b2,b4);

:: JORDAN6:funcnot 5 => JORDAN6:func 5
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3, a4, a5 be Element of the carrier of TOP-REAL 2;
  func Segment(A1,A2,A3,A4,A5) -> Element of bool the carrier of TOP-REAL 2 equals
    (R_Segment(a1,a2,a3,a4)) /\ L_Segment(a1,a2,a3,a5);
end;

:: JORDAN6:def 5
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2 holds
Segment(b1,b2,b3,b4,b5) = (R_Segment(b1,b2,b3,b4)) /\ L_Segment(b1,b2,b3,b5);

:: JORDAN6:th 29
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2 holds
Segment(b1,b2,b3,b4,b5) = {b6 where b6 is Element of the carrier of TOP-REAL 2: LE b4,b6,b1,b2,b3 & LE b6,b5,b1,b2,b3};

:: JORDAN6:th 30
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & LE b5,b4,b1,b3,b2
   holds LE b4,b5,b1,b2,b3;

:: JORDAN6:th 31
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & b4 in b1
   holds L_Segment(b1,b2,b3,b4) = R_Segment(b1,b3,b2,b4);

:: JORDAN6:th 32
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & b4 in b1 & b5 in b1
   holds Segment(b1,b2,b3,b4,b5) = Segment(b1,b3,b2,b5,b4);

:: JORDAN6:funcnot 6 => JORDAN6:func 6
definition
  let a1 be real set;
  func Vertical_Line A1 -> Element of bool the carrier of TOP-REAL 2 equals
    {b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `1 = a1};
end;

:: JORDAN6:def 6
theorem
for b1 being real set holds
   Vertical_Line b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `1 = b1};

:: JORDAN6:funcnot 7 => JORDAN6:func 7
definition
  let a1 be real set;
  func Horizontal_Line A1 -> Element of bool the carrier of TOP-REAL 2 equals
    {b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `2 = a1};
end;

:: JORDAN6:def 7
theorem
for b1 being real set holds
   Horizontal_Line b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `2 = b1};

:: JORDAN6:th 33
theorem
for b1 being real set holds
   Vertical_Line b1 is closed(TOP-REAL 2) & Horizontal_Line b1 is closed(TOP-REAL 2);

:: JORDAN6:th 34
theorem
for b1 being real set
for b2 being Element of the carrier of TOP-REAL 2 holds
      b2 in Vertical_Line b1
   iff
      b2 `1 = b1;

:: JORDAN6:th 35
theorem
for b1 being real set
for b2 being Element of the carrier of TOP-REAL 2 holds
      b2 in Horizontal_Line b1
   iff
      b2 `2 = b1;

:: JORDAN6:th 40
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve
   holds ex b2, b3 being non empty Element of bool the carrier of TOP-REAL 2 st
      b2 is_an_arc_of W-min b1,E-max b1 &
       b3 is_an_arc_of E-max b1,W-min b1 &
       b2 /\ b3 = {W-min b1,E-max b1} &
       b2 \/ b3 = b1 &
       (Last_Point(b3,E-max b1,W-min b1,Vertical_Line (((W-bound b1) + E-bound b1) / 2))) `2 < (First_Point(b2,W-min b1,E-max b1,Vertical_Line (((W-bound b1) + E-bound b1) / 2))) `2;

:: JORDAN6:th 41
theorem
for b1 being Element of bool the carrier of I[01]
      st b1 = (the carrier of I[01]) \ {0,1}
   holds b1 is open(I[01]);

:: JORDAN6:th 46
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
      st b1 = ].b2,b3.[
   holds b1 is open(R^1);

:: JORDAN6:th 48
theorem
for b1 being Element of bool the carrier of I[01]
      st b1 = (the carrier of I[01]) \ {0,1}
   holds b1 <> {} & b1 is connected(I[01]);

:: JORDAN6:th 49
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b3,b4
   holds b3 <> b4;

:: JORDAN6:th 50
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of (TOP-REAL b1) | b2
for b4, b5 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b4,b5 & b3 = b2 \ {b4,b5}
   holds b3 is open((TOP-REAL b1) | b2);

:: JORDAN6:th 52
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
for b5 being Element of bool the carrier of (TOP-REAL b1) | b2
for b6, b7 being Element of the carrier of TOP-REAL b1
      st b6 in b2 & b7 in b2 & b3 is_an_arc_of b6,b7 & b4 is_an_arc_of b6,b7 & b3 \/ b4 = b2 & b3 /\ b4 = {b6,b7} & b5 = b3 \ {b6,b7}
   holds b5 is open((TOP-REAL b1) | b2);

:: JORDAN6:th 53
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of (TOP-REAL b1) | b2
for b4, b5 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b4,b5 & b3 = b2 \ {b4,b5}
   holds b3 is connected((TOP-REAL b1) | b2);

:: JORDAN6:th 54
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b1 | b2
for b5 being Element of bool the carrier of b1 | b3
      st b2 c= b3 & b5 = b4 & b4 is connected(b1 | b2)
   holds b5 is connected(b1 | b3);

:: JORDAN6:th 55
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b3,b4
   holds ex b5 being Element of the carrier of TOP-REAL b1 st
      b5 in b2 & b5 <> b3 & b5 <> b4;

:: JORDAN6:th 56
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b3,b4
   holds b2 \ {b3,b4} <> {};

:: JORDAN6:th 57
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of bool the carrier of (TOP-REAL b1) | b3
for b5, b6 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b5,b6 & b2 c= b3 & b4 = b2 \ {b5,b6}
   holds b4 is connected((TOP-REAL b1) | b3);

:: JORDAN6:th 58
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being non empty Element of bool the carrier of b2
for b5 being Element of bool the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 | b4
for b7 being Function-like quasi_total Relation of the carrier of b2 | b5,the carrier of b3
      st b4 c= b5 & b6 is continuous(b1, b2 | b4) & b7 is continuous(b2 | b5, b3)
   holds ex b8 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3 st
      b8 = b7 * b6 & b8 is continuous(b1, b3);

:: JORDAN6:th 59
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4, b5 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b4,b5 & b2 c= b3
   holds b2 = b3;

:: JORDAN6:th 60
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & b3 in b1 & b4 in b1 & b3 <> b4 & b2 = b1 \ {b3,b4}
   holds b2 is connected(not TOP-REAL 2 | b1);

:: JORDAN6:th 61
theorem
for b1, b2, b3, b4, b5 being Element of bool the carrier of TOP-REAL 2
for b6, b7 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & b2 is_an_arc_of b6,b7 & b3 is_an_arc_of b6,b7 & b2 \/ b3 = b1 & b4 is_an_arc_of b6,b7 & b5 is_an_arc_of b6,b7 & b4 \/ b5 = b1 & (b2 = b4 implies b3 <> b5)
   holds b2 = b5 & b3 = b4;

:: JORDAN6:condreg 1
registration
  cluster -> real (Element of the carrier of R^1);
end;

:: JORDAN6:th 64
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being real set
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 `1 <= b2 & b2 <= b4 `1 & b1 is_an_arc_of b3,b4
   holds b1 meets Vertical_Line b2 & b1 /\ Vertical_Line b2 is closed(TOP-REAL 2);

:: JORDAN6:funcnot 8 => JORDAN6:func 8
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  assume a1 is being_simple_closed_curve;
  func Upper_Arc A1 -> non empty Element of bool the carrier of TOP-REAL 2 means
    it is_an_arc_of W-min a1,E-max a1 &
     (ex b1 being non empty Element of bool the carrier of TOP-REAL 2 st
        b1 is_an_arc_of E-max a1,W-min a1 &
         it /\ b1 = {W-min a1,E-max a1} &
         it \/ b1 = a1 &
         (Last_Point(b1,E-max a1,W-min a1,Vertical_Line (((W-bound a1) + E-bound a1) / 2))) `2 < (First_Point(it,W-min a1,E-max a1,Vertical_Line (((W-bound a1) + E-bound a1) / 2))) `2);
end;

:: JORDAN6:def 8
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
   st b1 is being_simple_closed_curve
for b2 being non empty Element of bool the carrier of TOP-REAL 2 holds
      b2 = Upper_Arc b1
   iff
      b2 is_an_arc_of W-min b1,E-max b1 &
       (ex b3 being non empty Element of bool the carrier of TOP-REAL 2 st
          b3 is_an_arc_of E-max b1,W-min b1 &
           b2 /\ b3 = {W-min b1,E-max b1} &
           b2 \/ b3 = b1 &
           (Last_Point(b3,E-max b1,W-min b1,Vertical_Line (((W-bound b1) + E-bound b1) / 2))) `2 < (First_Point(b2,W-min b1,E-max b1,Vertical_Line (((W-bound b1) + E-bound b1) / 2))) `2);

:: JORDAN6:funcnot 9 => JORDAN6:func 9
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  assume a1 is being_simple_closed_curve;
  func Lower_Arc A1 -> non empty Element of bool the carrier of TOP-REAL 2 means
    it is_an_arc_of E-max a1,W-min a1 &
     (Upper_Arc a1) /\ it = {W-min a1,E-max a1} &
     (Upper_Arc a1) \/ it = a1 &
     (Last_Point(it,E-max a1,W-min a1,Vertical_Line (((W-bound a1) + E-bound a1) / 2))) `2 < (First_Point(Upper_Arc a1,W-min a1,E-max a1,Vertical_Line (((W-bound a1) + E-bound a1) / 2))) `2;
end;

:: JORDAN6:def 9
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
   st b1 is being_simple_closed_curve
for b2 being non empty Element of bool the carrier of TOP-REAL 2 holds
      b2 = Lower_Arc b1
   iff
      b2 is_an_arc_of E-max b1,W-min b1 &
       (Upper_Arc b1) /\ b2 = {W-min b1,E-max b1} &
       (Upper_Arc b1) \/ b2 = b1 &
       (Last_Point(b2,E-max b1,W-min b1,Vertical_Line (((W-bound b1) + E-bound b1) / 2))) `2 < (First_Point(Upper_Arc b1,W-min b1,E-max b1,Vertical_Line (((W-bound b1) + E-bound b1) / 2))) `2;

:: JORDAN6:th 65
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve
   holds Upper_Arc b1 is_an_arc_of W-min b1,E-max b1 &
    Upper_Arc b1 is_an_arc_of E-max b1,W-min b1 &
    Lower_Arc b1 is_an_arc_of E-max b1,W-min b1 &
    Lower_Arc b1 is_an_arc_of W-min b1,E-max b1 &
    (Upper_Arc b1) /\ Lower_Arc b1 = {W-min b1,E-max b1} &
    (Upper_Arc b1) \/ Lower_Arc b1 = b1 &
    (Last_Point(Lower_Arc b1,E-max b1,W-min b1,Vertical_Line (((W-bound b1) + E-bound b1) / 2))) `2 < (First_Point(Upper_Arc b1,W-min b1,E-max b1,Vertical_Line (((W-bound b1) + E-bound b1) / 2))) `2;

:: JORDAN6:th 66
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve
   holds Lower_Arc b1 = (b1 \ Upper_Arc b1) \/ {W-min b1,E-max b1} &
    Upper_Arc b1 = (b1 \ Lower_Arc b1) \/ {W-min b1,E-max b1};

:: JORDAN6:th 67
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
      st b1 is being_simple_closed_curve &
         (Upper_Arc b1) /\ b2 = {W-min b1,E-max b1} &
         (Upper_Arc b1) \/ b2 = b1
   holds b2 = Lower_Arc b1;

:: JORDAN6:th 68
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
      st b1 is being_simple_closed_curve &
         b2 /\ Lower_Arc b1 = {W-min b1,E-max b1} &
         b2 \/ Lower_Arc b1 = b1
   holds b2 = Upper_Arc b1;

:: JORDAN6:th 69
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & LE b4,b2,b1,b2,b3
   holds b4 = b2;

:: JORDAN6:th 70
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & LE b3,b4,b1,b2,b3
   holds b4 = b3;

:: JORDAN6:prednot 1 => JORDAN6:pred 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3 be Element of the carrier of TOP-REAL 2;
  pred LE A2,A3,A1 means
    ((a2 in Upper_Arc a1 & a3 in Lower_Arc a1 implies a3 = W-min a1) &
     (a2 in Upper_Arc a1 & a3 in Upper_Arc a1 implies not LE a2,a3,Upper_Arc a1,W-min a1,E-max a1)) implies a2 in Lower_Arc a1 & a3 in Lower_Arc a1 & a3 <> W-min a1 & LE a2,a3,Lower_Arc a1,E-max a1,W-min a1;
end;

:: JORDAN6:dfs 10
definiens
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3 be Element of the carrier of TOP-REAL 2;
To prove
     LE a2,a3,a1
it is sufficient to prove
  thus ((a2 in Upper_Arc a1 & a3 in Lower_Arc a1 implies a3 = W-min a1) &
     (a2 in Upper_Arc a1 & a3 in Upper_Arc a1 implies not LE a2,a3,Upper_Arc a1,W-min a1,E-max a1)) implies a2 in Lower_Arc a1 & a3 in Lower_Arc a1 & a3 <> W-min a1 & LE a2,a3,Lower_Arc a1,E-max a1,W-min a1;

:: JORDAN6:def 10
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2 holds
   LE b2,b3,b1
iff
   ((b2 in Upper_Arc b1 & b3 in Lower_Arc b1 implies b3 = W-min b1) &
    (b2 in Upper_Arc b1 & b3 in Upper_Arc b1 implies not LE b2,b3,Upper_Arc b1,W-min b1,E-max b1) implies b2 in Lower_Arc b1 & b3 in Lower_Arc b1 & b3 <> W-min b1 & LE b2,b3,Lower_Arc b1,E-max b1,W-min b1);

:: JORDAN6:th 71
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 b1 is being_simple_closed_curve & b2 in b1
   holds LE b2,b2,b1;

:: JORDAN6:th 72
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & LE b2,b3,b1 & LE b3,b2,b1
   holds b2 = b3;

:: JORDAN6:th 73
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & LE b2,b3,b1 & LE b3,b4,b1
   holds LE b2,b4,b1;

:: JORDAN6:th 74
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & b4 <> b3
   holds not b3 in L_Segment(b1,b2,b3,b4);

:: JORDAN6:th 75
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & b4 <> b2
   holds not b2 in R_Segment(b1,b2,b3,b4);

:: JORDAN6:funcreg 1
registration
  let a1 be non empty being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  cluster Lower_Arc a1 -> non empty compact;
end;

:: JORDAN6:funcreg 2
registration
  let a1 be non empty being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  cluster Upper_Arc a1 -> non empty compact;
end;

:: JORDAN6:th 76
theorem
for b1 being non empty being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   Lower_Arc b1 c= b1 & Upper_Arc b1 c= b1;

:: JORDAN6:funcnot 10 => JORDAN6:func 10
definition
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  func Lower_Middle_Point A1 -> Element of the carrier of TOP-REAL 2 equals
    First_Point(Lower_Arc a1,W-min a1,E-max a1,Vertical_Line (((W-bound a1) + E-bound a1) / 2));
end;

:: JORDAN6:def 11
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   Lower_Middle_Point b1 = First_Point(Lower_Arc b1,W-min b1,E-max b1,Vertical_Line (((W-bound b1) + E-bound b1) / 2));

:: JORDAN6:funcnot 11 => JORDAN6:func 11
definition
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  func Upper_Middle_Point A1 -> Element of the carrier of TOP-REAL 2 equals
    First_Point(Upper_Arc a1,W-min a1,E-max a1,Vertical_Line (((W-bound a1) + E-bound a1) / 2));
end;

:: JORDAN6:def 12
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   Upper_Middle_Point b1 = First_Point(Upper_Arc b1,W-min b1,E-max b1,Vertical_Line (((W-bound b1) + E-bound b1) / 2));

:: JORDAN6:th 77
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   Lower_Arc b1 meets Vertical_Line (((W-bound b1) + E-bound b1) / 2);

:: JORDAN6:th 78
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   Upper_Arc b1 meets Vertical_Line (((W-bound b1) + E-bound b1) / 2);

:: JORDAN6:th 79
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   (Lower_Middle_Point b1) `1 = ((W-bound b1) + E-bound b1) / 2;

:: JORDAN6:th 80
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   (Upper_Middle_Point b1) `1 = ((W-bound b1) + E-bound b1) / 2;

:: JORDAN6:th 81
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   Lower_Middle_Point b1 in Lower_Arc b1;

:: JORDAN6:th 82
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   Upper_Middle_Point b1 in Upper_Arc b1;

:: JORDAN6:th 83
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   Upper_Middle_Point b1 in b1;

:: JORDAN6:th 84
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being real set
      st W-bound b1 <= b2 & b2 <= E-bound b1
   holds LSeg(|[b2,S-bound b1]|,|[b2,N-bound b1]|) meets Upper_Arc b1;

:: JORDAN6:th 85
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being real set
      st W-bound b1 <= b2 & b2 <= E-bound b1
   holds LSeg(|[b2,S-bound b1]|,|[b2,N-bound b1]|) meets Lower_Arc b1;