Article JORDAN1, MML version 4.99.1005

:: JORDAN1:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2, b3 being Element of the carrier of b1 holds
        ex b4 being non empty TopSpace-like TopStruct st
           b4 is connected &
            (ex b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b1 st
               b5 is continuous(b4, b1) & b2 in rng b5 & b3 in rng b5)
   holds b1 is connected;

:: JORDAN1:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2, b3 being Element of the carrier of b1 holds
        ex b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 st
           b4 is continuous(I[01], b1) & b2 = b4 . 0 & b3 = b4 . 1
   holds b1 is connected;

:: JORDAN1:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st for b3, b4 being Element of the carrier of b1
              st b3 in b2 & b4 in b2 & b3 <> b4
           holds ex b5 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 | b2 st
              b5 is continuous(I[01], b1 | b2) & b3 = b5 . 0 & b4 = b5 . 1
   holds b2 is connected(b1);

:: JORDAN1:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is connected(b1) & b3 is connected(b1) & b2 meets b3
   holds b2 \/ b3 is connected(b1);

:: JORDAN1:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2 is connected(b1) & b3 is connected(b1) & b4 is connected(b1) & b2 meets b3 & b3 meets b4
   holds (b2 \/ b3) \/ b4 is connected(b1);

:: JORDAN1:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b2 is connected(b1) & b3 is connected(b1) & b4 is connected(b1) & b5 is connected(b1) & b2 meets b3 & b3 meets b4 & b4 meets b5
   holds ((b2 \/ b3) \/ b4) \/ b5 is connected(b1);

:: JORDAN1:attrnot 1 => JORDAN1:attr 1
definition
  let a1 be Element of NAT;
  let a2 be Element of bool the carrier of TOP-REAL a1;
  attr a2 is convex means
    for b1, b2 being Element of the carrier of TOP-REAL a1
          st b1 in a2 & b2 in a2
       holds LSeg(b1,b2) c= a2;
end;

:: JORDAN1:dfs 1
definiens
  let a1 be Element of NAT;
  let a2 be Element of bool the carrier of TOP-REAL a1;
To prove
     a2 is convex
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of TOP-REAL a1
          st b1 in a2 & b2 in a2
       holds LSeg(b1,b2) c= a2;

:: JORDAN1:def 1
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
      b2 is convex(b1)
   iff
      for b3, b4 being Element of the carrier of TOP-REAL b1
            st b3 in b2 & b4 in b2
         holds LSeg(b3,b4) c= b2;

:: JORDAN1:th 9
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
      st b2 is convex(b1)
   holds b2 is connected(TOP-REAL b1);

:: JORDAN1:condreg 1
registration
  let a1 be Element of NAT;
  cluster convex -> connected (Element of bool the carrier of TOP-REAL a1);
end;

:: JORDAN1:th 12
theorem
for b1, b2, b3, b4 being Element of REAL holds
{|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: b1 < b5 & b5 < b2 & b3 < b6 & b6 < b4} = (({|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: b1 < b5} /\ {|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: b5 < b2}) /\ {|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: b3 < b6}) /\ {|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: b6 < b4};

:: JORDAN1:th 13
theorem
for b1, b2, b3, b4 being Element of REAL holds
{|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: (b1 <= b5 & b5 <= b2 & b3 <= b6 implies b4 < b6)} = (({|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: b5 < b1} \/ {|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: b6 < b3}) \/ {|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: b2 < b5}) \/ {|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: b4 < b6};

:: JORDAN1:th 14
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of bool the carrier of TOP-REAL 2
      st b5 = {|[b6,b7]| where b6 is Element of REAL, b7 is Element of REAL: b1 < b6 & b6 < b3 & b2 < b7 & b7 < b4}
   holds b5 is convex(2);

:: JORDAN1:th 15
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of bool the carrier of TOP-REAL 2
      st b5 = {|[b6,b7]| where b6 is Element of REAL, b7 is Element of REAL: b1 < b6 & b6 < b3 & b2 < b7 & b7 < b4}
   holds b5 is connected(TOP-REAL 2);

:: JORDAN1:th 16
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b1 < b3}
   holds b2 is convex(2);

:: JORDAN1:th 17
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b1 < b3}
   holds b2 is connected(TOP-REAL 2);

:: JORDAN1:th 18
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b3 < b1}
   holds b2 is convex(2);

:: JORDAN1:th 19
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b3 < b1}
   holds b2 is connected(TOP-REAL 2);

:: JORDAN1:th 20
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b1 < b4}
   holds b2 is convex(2);

:: JORDAN1:th 21
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b1 < b4}
   holds b2 is connected(TOP-REAL 2);

:: JORDAN1:th 22
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b4 < b1}
   holds b2 is convex(2);

:: JORDAN1:th 23
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b4 < b1}
   holds b2 is connected(TOP-REAL 2);

:: JORDAN1:th 24
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of bool the carrier of TOP-REAL 2
      st b5 = {|[b6,b7]| where b6 is Element of REAL, b7 is Element of REAL: (b1 <= b6 & b6 <= b3 & b2 <= b7 implies b4 < b7)}
   holds b5 is connected(TOP-REAL 2);

:: JORDAN1:th 25
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b1 < b3}
   holds b2 is open(TOP-REAL 2);

:: JORDAN1:th 26
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b3 < b1}
   holds b2 is open(TOP-REAL 2);

:: JORDAN1:th 27
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b1 < b4}
   holds b2 is open(TOP-REAL 2);

:: JORDAN1:th 28
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b4 < b1}
   holds b2 is open(TOP-REAL 2);

:: JORDAN1:th 29
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of bool the carrier of TOP-REAL 2
      st b5 = {|[b6,b7]| where b6 is Element of REAL, b7 is Element of REAL: b1 < b6 & b6 < b3 & b2 < b7 & b7 < b4}
   holds b5 is open(TOP-REAL 2);

:: JORDAN1:th 30
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of bool the carrier of TOP-REAL 2
      st b5 = {|[b6,b7]| where b6 is Element of REAL, b7 is Element of REAL: (b1 <= b6 & b6 <= b3 & b2 <= b7 implies b4 < b7)}
   holds b5 is open(TOP-REAL 2);

:: JORDAN1:th 31
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6 being Element of bool the carrier of TOP-REAL 2
      st b5 = {|[b7,b8]| where b7 is Element of REAL, b8 is Element of REAL: b1 < b7 & b7 < b3 & b2 < b8 & b8 < b4} &
         b6 = {|[b7,b8]| where b7 is Element of REAL, b8 is Element of REAL: (b1 <= b7 & b7 <= b3 & b2 <= b8 implies b4 < b8)}
   holds b5 misses b6;

:: JORDAN1:th 32
theorem
for b1, b2, b3, b4 being Element of REAL holds
{b5 where b5 is Element of the carrier of TOP-REAL 2: b1 < b5 `1 & b5 `1 < b2 & b3 < b5 `2 & b5 `2 < b4} = {|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: b1 < b5 & b5 < b2 & b3 < b6 & b6 < b4};

:: JORDAN1:th 33
theorem
for b1, b2, b3, b4 being Element of REAL holds
{b5 where b5 is Element of the carrier of TOP-REAL 2: (b1 <= b5 `1 & b5 `1 <= b2 & b3 <= b5 `2 implies b4 < b5 `2)} = {|[b5,b6]| where b5 is Element of REAL, b6 is Element of REAL: (b1 <= b5 & b5 <= b2 & b3 <= b6 implies b4 < b6)};

:: JORDAN1:th 34
theorem
for b1, b2, b3, b4 being Element of REAL holds
{b5 where b5 is Element of the carrier of TOP-REAL 2: b1 < b5 `1 & b5 `1 < b2 & b3 < b5 `2 & b5 `2 < b4} is Element of bool the carrier of TOP-REAL 2;

:: JORDAN1:th 35
theorem
for b1, b2, b3, b4 being Element of REAL holds
{b5 where b5 is Element of the carrier of TOP-REAL 2: (b1 <= b5 `1 & b5 `1 <= b2 & b3 <= b5 `2 implies b4 < b5 `2)} is Element of bool the carrier of TOP-REAL 2;

:: JORDAN1:th 36
theorem
for b1, b2, b3, b4 being Element of REAL
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: b1 < b6 `1 & b6 `1 < b3 & b2 < b6 `2 & b6 `2 < b4}
   holds b5 is connected(TOP-REAL 2);

:: JORDAN1:th 37
theorem
for b1, b2, b3, b4 being Element of REAL
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: (b1 <= b6 `1 & b6 `1 <= b3 & b2 <= b6 `2 implies b4 < b6 `2)}
   holds b5 is connected(TOP-REAL 2);

:: JORDAN1:th 38
theorem
for b1, b2, b3, b4 being Element of REAL
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: b1 < b6 `1 & b6 `1 < b3 & b2 < b6 `2 & b6 `2 < b4}
   holds b5 is open(TOP-REAL 2);

:: JORDAN1:th 39
theorem
for b1, b2, b3, b4 being Element of REAL
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: (b1 <= b6 `1 & b6 `1 <= b3 & b2 <= b6 `2 implies b4 < b6 `2)}
   holds b5 is open(TOP-REAL 2);

:: JORDAN1:th 40
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6 being Element of bool the carrier of TOP-REAL 2
      st b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: b1 < b7 `1 & b7 `1 < b3 & b2 < b7 `2 & b7 `2 < b4} &
         b6 = {b7 where b7 is Element of the carrier of TOP-REAL 2: (b1 <= b7 `1 & b7 `1 <= b3 & b2 <= b7 `2 implies b4 < b7 `2)}
   holds b5 misses b6;

:: JORDAN1:th 41
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6, b7 being Element of bool the carrier of TOP-REAL 2
      st b1 < b3 &
         b2 < b4 &
         b5 = {b8 where b8 is Element of the carrier of TOP-REAL 2: ((b8 `1 = b1 & b8 `2 <= b4 implies b8 `2 < b2) &
          (b8 `1 <= b3 & b1 <= b8 `1 implies b8 `2 <> b4) &
          (b8 `1 <= b3 & b1 <= b8 `1 implies b8 `2 <> b2) implies b8 `1 = b3 & b8 `2 <= b4 & b2 <= b8 `2)} &
         b6 = {b8 where b8 is Element of the carrier of TOP-REAL 2: b1 < b8 `1 & b8 `1 < b3 & b2 < b8 `2 & b8 `2 < b4} &
         b7 = {b8 where b8 is Element of the carrier of TOP-REAL 2: (b1 <= b8 `1 & b8 `1 <= b3 & b2 <= b8 `2 implies b4 < b8 `2)}
   holds b5 ` = b6 \/ b7 &
    b5 ` <> {} &
    b6 misses b7 &
    (for b8, b9 being Element of bool the carrier of (TOP-REAL 2) | (b5 `)
          st b8 = b6 & b9 = b7
       holds b8 is_a_component_of (TOP-REAL 2) | (b5 `) & b9 is_a_component_of (TOP-REAL 2) | (b5 `));

:: JORDAN1:th 42
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6, b7 being Element of bool the carrier of TOP-REAL 2
      st b1 < b3 &
         b2 < b4 &
         b5 = {b8 where b8 is Element of the carrier of TOP-REAL 2: ((b8 `1 = b1 & b8 `2 <= b4 implies b8 `2 < b2) &
          (b8 `1 <= b3 & b1 <= b8 `1 implies b8 `2 <> b4) &
          (b8 `1 <= b3 & b1 <= b8 `1 implies b8 `2 <> b2) implies b8 `1 = b3 & b8 `2 <= b4 & b2 <= b8 `2)} &
         b6 = {b8 where b8 is Element of the carrier of TOP-REAL 2: b1 < b8 `1 & b8 `1 < b3 & b2 < b8 `2 & b8 `2 < b4} &
         b7 = {b8 where b8 is Element of the carrier of TOP-REAL 2: (b1 <= b8 `1 & b8 `1 <= b3 & b2 <= b8 `2 implies b4 < b8 `2)}
   holds b5 = (Cl b6) \ b6 & b5 = (Cl b7) \ b7;

:: JORDAN1:th 43
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6 being Element of bool the carrier of TOP-REAL 2
      st b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: ((b7 `1 = b1 & b7 `2 <= b4 implies b7 `2 < b3) &
          (b7 `1 <= b2 & b1 <= b7 `1 implies b7 `2 <> b4) &
          (b7 `1 <= b2 & b1 <= b7 `1 implies b7 `2 <> b3) implies b7 `1 = b2 & b7 `2 <= b4 & b3 <= b7 `2)} &
         b6 = {b7 where b7 is Element of the carrier of TOP-REAL 2: b1 < b7 `1 & b7 `1 < b2 & b3 < b7 `2 & b7 `2 < b4}
   holds b6 c= [#] ((TOP-REAL 2) | (b5 `));

:: JORDAN1:th 44
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6 being Element of bool the carrier of TOP-REAL 2
      st b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: ((b7 `1 = b1 & b7 `2 <= b4 implies b7 `2 < b3) &
          (b7 `1 <= b2 & b1 <= b7 `1 implies b7 `2 <> b4) &
          (b7 `1 <= b2 & b1 <= b7 `1 implies b7 `2 <> b3) implies b7 `1 = b2 & b7 `2 <= b4 & b3 <= b7 `2)} &
         b6 = {b7 where b7 is Element of the carrier of TOP-REAL 2: b1 < b7 `1 & b7 `1 < b2 & b3 < b7 `2 & b7 `2 < b4}
   holds b6 is Element of bool the carrier of (TOP-REAL 2) | (b5 `);

:: JORDAN1:th 45
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6 being Element of bool the carrier of TOP-REAL 2
      st b1 < b2 &
         b3 < b4 &
         b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: ((b7 `1 = b1 & b7 `2 <= b4 implies b7 `2 < b3) &
          (b7 `1 <= b2 & b1 <= b7 `1 implies b7 `2 <> b4) &
          (b7 `1 <= b2 & b1 <= b7 `1 implies b7 `2 <> b3) implies b7 `1 = b2 & b7 `2 <= b4 & b3 <= b7 `2)} &
         b6 = {b7 where b7 is Element of the carrier of TOP-REAL 2: (b1 <= b7 `1 & b7 `1 <= b2 & b3 <= b7 `2 implies b4 < b7 `2)}
   holds b6 c= [#] ((TOP-REAL 2) | (b5 `));

:: JORDAN1:th 46
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6 being Element of bool the carrier of TOP-REAL 2
      st b1 < b2 &
         b3 < b4 &
         b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: ((b7 `1 = b1 & b7 `2 <= b4 implies b7 `2 < b3) &
          (b7 `1 <= b2 & b1 <= b7 `1 implies b7 `2 <> b4) &
          (b7 `1 <= b2 & b1 <= b7 `1 implies b7 `2 <> b3) implies b7 `1 = b2 & b7 `2 <= b4 & b3 <= b7 `2)} &
         b6 = {b7 where b7 is Element of the carrier of TOP-REAL 2: (b1 <= b7 `1 & b7 `1 <= b2 & b3 <= b7 `2 implies b4 < b7 `2)}
   holds b6 is Element of bool the carrier of (TOP-REAL 2) | (b5 `);

:: JORDAN1:attrnot 2 => JORDAN1:attr 2
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  attr a1 is Jordan means
    a1 ` <> {} &
     (ex b1, b2 being Element of bool the carrier of TOP-REAL 2 st
        a1 ` = b1 \/ b2 &
         b1 misses b2 &
         (Cl b1) \ b1 = (Cl b2) \ b2 &
         (for b3, b4 being Element of bool the carrier of (TOP-REAL 2) | (a1 `)
               st b3 = b1 & b4 = b2
            holds b3 is_a_component_of (TOP-REAL 2) | (a1 `) & b4 is_a_component_of (TOP-REAL 2) | (a1 `)));
end;

:: JORDAN1:dfs 2
definiens
  let a1 be Element of bool the carrier of TOP-REAL 2;
To prove
     a1 is Jordan
it is sufficient to prove
  thus a1 ` <> {} &
     (ex b1, b2 being Element of bool the carrier of TOP-REAL 2 st
        a1 ` = b1 \/ b2 &
         b1 misses b2 &
         (Cl b1) \ b1 = (Cl b2) \ b2 &
         (for b3, b4 being Element of bool the carrier of (TOP-REAL 2) | (a1 `)
               st b3 = b1 & b4 = b2
            holds b3 is_a_component_of (TOP-REAL 2) | (a1 `) & b4 is_a_component_of (TOP-REAL 2) | (a1 `)));

:: JORDAN1:def 2
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
      b1 is Jordan
   iff
      b1 ` <> {} &
       (ex b2, b3 being Element of bool the carrier of TOP-REAL 2 st
          b1 ` = b2 \/ b3 &
           b2 misses b3 &
           (Cl b2) \ b2 = (Cl b3) \ b3 &
           (for b4, b5 being Element of bool the carrier of (TOP-REAL 2) | (b1 `)
                 st b4 = b2 & b5 = b3
              holds b4 is_a_component_of (TOP-REAL 2) | (b1 `) & b5 is_a_component_of (TOP-REAL 2) | (b1 `)));

:: JORDAN1:prednot 1 => JORDAN1:attr 2
notation
  let a1 be Element of bool the carrier of TOP-REAL 2;
  synonym a1 has_property_J for Jordan;
end;

:: JORDAN1:th 47
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is Jordan
   holds b1 ` <> {} &
    (ex b2, b3 being Element of bool the carrier of TOP-REAL 2 st
       ex b4, b5 being Element of bool the carrier of (TOP-REAL 2) | (b1 `) st
          b1 ` = b2 \/ b3 &
           b2 misses b3 &
           (Cl b2) \ b2 = (Cl b3) \ b3 &
           b4 = b2 &
           b5 = b3 &
           b4 is_a_component_of (TOP-REAL 2) | (b1 `) &
           b5 is_a_component_of (TOP-REAL 2) | (b1 `) &
           (for b6 being Element of bool the carrier of (TOP-REAL 2) | (b1 `)
                 st b6 is_a_component_of (TOP-REAL 2) | (b1 `) & b6 <> b4
              holds b6 = b5));

:: JORDAN1:th 48
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of bool the carrier of TOP-REAL 2
      st b1 < b2 &
         b3 < b4 &
         b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: ((b6 `1 = b1 & b6 `2 <= b4 implies b6 `2 < b3) &
          (b6 `1 <= b2 & b1 <= b6 `1 implies b6 `2 <> b4) &
          (b6 `1 <= b2 & b1 <= b6 `1 implies b6 `2 <> b3) implies b6 `1 = b2 & b6 `2 <= b4 & b3 <= b6 `2)}
   holds b5 is Jordan;

:: JORDAN1:th 49
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6 being Element of bool the carrier of TOP-REAL 2
      st b1 < b3 &
         b2 < b4 &
         b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: ((b7 `1 = b1 & b7 `2 <= b4 implies b7 `2 < b2) &
          (b7 `1 <= b3 & b1 <= b7 `1 implies b7 `2 <> b4) &
          (b7 `1 <= b3 & b1 <= b7 `1 implies b7 `2 <> b2) implies b7 `1 = b3 & b7 `2 <= b4 & b2 <= b7 `2)} &
         b6 = {b7 where b7 is Element of the carrier of TOP-REAL 2: (b1 <= b7 `1 & b7 `1 <= b3 & b2 <= b7 `2 implies b4 < b7 `2)}
   holds Cl b6 = b5 \/ b6;

:: JORDAN1:th 50
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6 being Element of bool the carrier of TOP-REAL 2
      st b1 < b3 &
         b2 < b4 &
         b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: ((b7 `1 = b1 & b7 `2 <= b4 implies b7 `2 < b2) &
          (b7 `1 <= b3 & b1 <= b7 `1 implies b7 `2 <> b4) &
          (b7 `1 <= b3 & b1 <= b7 `1 implies b7 `2 <> b2) implies b7 `1 = b3 & b7 `2 <= b4 & b2 <= b7 `2)} &
         b6 = {b7 where b7 is Element of the carrier of TOP-REAL 2: b1 < b7 `1 & b7 `1 < b3 & b2 < b7 `2 & b7 `2 < b4}
   holds Cl b6 = b5 \/ b6;

:: JORDAN1:th 51
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
(LSeg(b1,b2)) \ {b1,b2} is convex(2);

:: JORDAN1:th 52
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
(LSeg(b1,b2)) \ {b1,b2} is connected(TOP-REAL 2);

:: JORDAN1:funcreg 1
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster north_halfline a1 -> convex;
end;

:: JORDAN1:funcreg 2
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster east_halfline a1 -> convex;
end;

:: JORDAN1:funcreg 3
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster south_halfline a1 -> convex;
end;

:: JORDAN1:funcreg 4
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster west_halfline a1 -> convex;
end;