Article JGRAPH_7, MML version 4.99.1005

:: JGRAPH_7:th 1
theorem
for b1, b2, b3 being real set
for b4 being Element of the carrier of TOP-REAL 2
      st b1 < b2 & b4 `2 = b3 & b1 <= b4 `1 & b4 `1 <= b2
   holds b4 in LSeg(|[b1,b3]|,|[b2,b3]|);

:: JGRAPH_7:th 2
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 Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL b1 st
      b5 is continuous(I[01], TOP-REAL b1) & b5 is one-to-one & rng b5 = b2 & b5 . 0 = b3 & b5 . 1 = b4;

:: JGRAPH_7:th 3
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4, b5 being real set
      st b1 `1 < b3 & b1 `1 = b2 `1 & b4 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b5
   holds LE b1,b2,[.b1 `1,b3,b4,b5.];

:: JGRAPH_7:th 4
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4 being real set
      st b1 `1 < b3 & b4 < b2 `2 & b4 <= b1 `2 & b1 `2 <= b2 `2 & b1 `1 <= b2 `1 & b2 `1 <= b3
   holds LE b1,b2,[.b1 `1,b3,b4,b2 `2.];

:: JGRAPH_7:th 5
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4 being real set
      st b1 `1 < b2 `1 & b3 < b4 & b3 <= b1 `2 & b1 `2 <= b4 & b3 <= b2 `2 & b2 `2 <= b4
   holds LE b1,b2,[.b1 `1,b2 `1,b3,b4.];

:: JGRAPH_7:th 6
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4 being real set
      st b2 `2 < b4 & b2 `2 <= b1 `2 & b1 `2 <= b4 & b1 `1 < b2 `1 & b2 `1 <= b3
   holds LE b1,b2,[.b1 `1,b3,b2 `2,b4.];

:: JGRAPH_7:th 7
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4, b5, b6 being real set
      st b3 < b4 & b5 < b6 & b1 `2 = b6 & b2 `2 = b6 & b3 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 <= b4
   holds LE b1,b2,[.b3,b4,b5,b6.];

:: JGRAPH_7:th 8
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4, b5, b6 being real set
      st b3 < b4 & b5 < b6 & b1 `2 = b6 & b2 `1 = b4 & b3 <= b1 `1 & b1 `1 <= b4 & b5 <= b2 `2 & b2 `2 <= b6
   holds LE b1,b2,[.b3,b4,b5,b6.];

:: JGRAPH_7:th 9
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4, b5, b6 being real set
      st b3 < b4 & b5 < b6 & b1 `2 = b6 & b2 `2 = b5 & b3 <= b1 `1 & b1 `1 <= b4 & b3 < b2 `1 & b2 `1 <= b4
   holds LE b1,b2,[.b3,b4,b5,b6.];

:: JGRAPH_7:th 10
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4, b5, b6 being real set
      st b3 < b4 & b5 < b6 & b1 `1 = b4 & b2 `1 = b4 & b5 <= b2 `2 & b2 `2 < b1 `2 & b1 `2 <= b6
   holds LE b1,b2,[.b3,b4,b5,b6.];

:: JGRAPH_7:th 11
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4, b5, b6 being real set
      st b3 < b4 & b5 < b6 & b1 `1 = b4 & b2 `2 = b5 & b5 <= b1 `2 & b1 `2 <= b6 & b3 < b2 `1 & b2 `1 <= b4
   holds LE b1,b2,[.b3,b4,b5,b6.];

:: JGRAPH_7:th 12
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4, b5, b6 being real set
      st b3 < b4 & b5 < b6 & b1 `2 = b5 & b2 `2 = b5 & b3 < b2 `1 & b2 `1 < b1 `1 & b1 `1 <= b4
   holds LE b1,b2,[.b3,b4,b5,b6.];

:: JGRAPH_7:th 13
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4, b5, b6 being real set
      st b3 < b4 & b5 < b6 & b1 `2 = b6 & b2 `1 = b4 & b3 <= b1 `1 & b1 `1 <= b4 & b5 <= b2 `2 & b2 `2 <= b6
   holds LE b1,b2,[.b3,b4,b5,b6.];

:: JGRAPH_7:th 14
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `1 = b5 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 < b4 `2 & b4 `2 <= b8
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 15
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `2 = b8 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 <= b8 & b5 <= b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 16
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 <= b8 & b7 <= b4 `2 & b4 `2 <= b8
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 17
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 18
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b8 & b4 `2 = b8 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 <= b3 `1 & b3 `1 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 19
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b8 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 <= b3 `1 & b3 `1 <= b6 & b7 <= b4 `2 & b4 `2 <= b8
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 20
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b8 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 <= b3 `1 & b3 `1 <= b6 & b5 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 21
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b6 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 <= b8
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 22
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b6 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b7 <= b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 23
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 24
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b8 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 < b3 `1 & b3 `1 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 25
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b8 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b7 <= b4 `2 & b4 `2 <= b8
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 26
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b5 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 27
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `1 = b6 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 <= b6 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 <= b8
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 28
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 `1 <> b3 `1 & b4 `2 <> b2 `2 & b4 `2 <= b1 `2 & b1 `2 <= b2 `2 & b1 `1 <= b2 `1 & b2 `1 <= b3 `1 & b4 `2 <= b3 `2 & b3 `2 <= b2 `2 & b1 `1 < b4 `1 & b4 `1 <= b3 `1
   holds b1,b2,b3,b4 are_in_this_order_on [.b1 `1,b3 `1,b4 `2,b2 `2.];

:: JGRAPH_7:th 29
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 <= b6 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 30
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b6 & b3 `1 = b6 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 <= b8 & b2 `2 <= b8 & b3 `2 < b2 `2 & b4 `2 < b3 `2 & b7 <= b4 `2
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 31
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b6 & b3 `1 = b6 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b2 `2 <= b8 & b3 `2 < b2 `2 & b7 <= b3 `2 & b5 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 32
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b6 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b7 <= b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 33
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 < b2 `1 & b2 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 34
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b8 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 < b3 `1 & b3 `1 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 35
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b8 & b4 `1 = b6 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b7 <= b4 `2 & b4 `2 <= b8
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 36
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b5 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 37
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `1 = b6 & b4 `1 = b6 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 <= b6 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 <= b8
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 38
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `1 = b6 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 <= b6 & b7 <= b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 39
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b7 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 <= b6 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 40
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `1 = b6 & b3 `1 = b6 & b4 `1 = b6 & b5 <= b1 `1 & b1 `1 <= b6 & b2 `2 <= b8 & b3 `2 < b2 `2 & b4 `2 < b3 `2 & b7 <= b4 `2
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 41
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `1 = b6 & b3 `1 = b6 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 <= b6 & b2 `2 <= b8 & b3 `2 < b2 `2 & b7 <= b3 `2 & b5 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 42
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `1 = b6 & b3 `2 = b7 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 <= b6 & b7 <= b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 43
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 <= b6 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 < b2 `1 & b2 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 44
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `1 = b6 & b3 `1 = b6 & b4 `1 = b6 & b1 `2 <= b8 & b2 `2 < b1 `2 & b3 `2 < b2 `2 & b4 `2 < b3 `2 & b7 <= b4 `2
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 45
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `1 = b6 & b3 `1 = b6 & b4 `2 = b7 & b1 `2 <= b8 & b2 `2 < b1 `2 & b3 `2 < b2 `2 & b7 <= b3 `2 & b5 < b4 `1 & b4 `1 <= b6
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 46
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `1 = b6 & b3 `2 = b7 & b4 `2 = b7 & b1 `2 <= b8 & b2 `2 < b1 `2 & b7 <= b2 `2 & b3 `1 <= b6 & b4 `1 < b3 `1 & b5 < b4 `1
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 47
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b2 `1 <= b6 & b3 `1 < b2 `1 & b4 `1 < b3 `1 & b5 < b4 `1
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 48
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
      st b5 < b6 & b7 < b8 & b1 `2 = b7 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b1 `1 <= b6 & b2 `1 < b1 `1 & b3 `1 < b2 `1 & b4 `1 < b3 `1 & b5 < b4 `1
   holds b1,b2,b3,b4 are_in_this_order_on [.b5,b6,b7,b8.];

:: JGRAPH_7:th 49
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
      st 0 < b1 &
         0 < b3 &
         b5 = AffineMap(b1,b2,b3,b4) &
         b6 = AffineMap(1 / b1,- (b2 / b1),1 / b3,- (b4 / b3))
   holds b6 = b5 " & b5 = b6 ";

:: JGRAPH_7:th 50
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
      st 0 < b1 & 0 < b3 & b5 = AffineMap(b1,b2,b3,b4)
   holds b5 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
    (for b6, b7 being Element of the carrier of TOP-REAL 2
          st b6 `1 < b7 `1
       holds (b5 . b6) `1 < (b5 . b7) `1);

:: JGRAPH_7:th 51
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
      st 0 < b1 & 0 < b3 & b5 = AffineMap(b1,b2,b3,b4)
   holds b5 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
    (for b6, b7 being Element of the carrier of TOP-REAL 2
          st b6 `2 < b7 `2
       holds (b5 . b6) `2 < (b5 . b7) `2);

:: JGRAPH_7:th 52
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         rng b6 c= closed_inside_of_rectangle(b1,b2,b3,b4)
   holds rng (b5 * b6) c= closed_inside_of_rectangle(- 1,1,- 1,1);

:: JGRAPH_7:th 53
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         b6 is continuous(I[01], TOP-REAL 2) &
         b6 is one-to-one
   holds b5 * b6 is continuous(I[01], TOP-REAL 2) & b5 * b6 is one-to-one;

:: JGRAPH_7:th 54
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         (b6 . b7) `1 = b1
   holds ((b5 * b6) . b7) `1 = - 1;

:: JGRAPH_7:th 55
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         (b6 . b7) `2 = b4
   holds ((b5 * b6) . b7) `2 = 1;

:: JGRAPH_7:th 56
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         (b6 . b7) `1 = b2
   holds ((b5 * b6) . b7) `1 = 1;

:: JGRAPH_7:th 57
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         (b6 . b7) `2 = b3
   holds ((b5 * b6) . b7) `2 = - 1;

:: JGRAPH_7:th 58
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         b3 <= (b6 . b7) `2 &
         (b6 . b7) `2 < (b6 . b8) `2 &
         (b6 . b8) `2 <= b4
   holds - 1 <= ((b5 * b6) . b7) `2 &
    ((b5 * b6) . b7) `2 < ((b5 * b6) . b8) `2 &
    ((b5 * b6) . b8) `2 <= 1;

:: JGRAPH_7:th 59
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         b3 <= (b6 . b7) `2 &
         (b6 . b7) `2 <= b4 &
         b1 <= (b6 . b8) `1 &
         (b6 . b8) `1 <= b2
   holds - 1 <= ((b5 * b6) . b7) `2 &
    ((b5 * b6) . b7) `2 <= 1 &
    - 1 <= ((b5 * b6) . b8) `1 &
    ((b5 * b6) . b8) `1 <= 1;

:: JGRAPH_7:th 60
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         b3 <= (b6 . b7) `2 &
         (b6 . b7) `2 <= b4 &
         b3 <= (b6 . b8) `2 &
         (b6 . b8) `2 <= b4
   holds - 1 <= ((b5 * b6) . b7) `2 &
    ((b5 * b6) . b7) `2 <= 1 &
    - 1 <= ((b5 * b6) . b8) `2 &
    ((b5 * b6) . b8) `2 <= 1;

:: JGRAPH_7:th 61
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         b3 <= (b6 . b7) `2 &
         (b6 . b7) `2 <= b4 &
         b1 < (b6 . b8) `1 &
         (b6 . b8) `1 <= b2
   holds - 1 <= ((b5 * b6) . b7) `2 &
    ((b5 * b6) . b7) `2 <= 1 &
    - 1 < ((b5 * b6) . b8) `1 &
    ((b5 * b6) . b8) `1 <= 1;

:: JGRAPH_7:th 62
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         b1 <= (b6 . b7) `1 &
         (b6 . b7) `1 < (b6 . b8) `1 &
         (b6 . b8) `1 <= b2
   holds - 1 <= ((b5 * b6) . b7) `1 &
    ((b5 * b6) . b7) `1 < ((b5 * b6) . b8) `1 &
    ((b5 * b6) . b8) `1 <= 1;

:: JGRAPH_7:th 63
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         b1 <= (b6 . b7) `1 &
         (b6 . b7) `1 <= b2 &
         b3 <= (b6 . b8) `2 &
         (b6 . b8) `2 <= b4
   holds - 1 <= ((b5 * b6) . b7) `1 &
    ((b5 * b6) . b7) `1 <= 1 &
    - 1 <= ((b5 * b6) . b8) `2 &
    ((b5 * b6) . b8) `2 <= 1;

:: JGRAPH_7:th 64
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         b1 <= (b6 . b7) `1 &
         (b6 . b7) `1 <= b2 &
         b1 < (b6 . b8) `1 &
         (b6 . b8) `1 <= b2
   holds - 1 <= ((b5 * b6) . b7) `1 &
    ((b5 * b6) . b7) `1 <= 1 &
    - 1 < ((b5 * b6) . b8) `1 &
    ((b5 * b6) . b8) `1 <= 1;

:: JGRAPH_7:th 65
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         (b6 . b7) `2 <= b4 &
         (b6 . b8) `2 < (b6 . b7) `2 &
         b3 <= (b6 . b8) `2
   holds ((b5 * b6) . b7) `2 <= 1 &
    ((b5 * b6) . b8) `2 < ((b5 * b6) . b7) `2 &
    - 1 <= ((b5 * b6) . b8) `2;

:: JGRAPH_7:th 66
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         b3 <= (b6 . b7) `2 &
         (b6 . b7) `2 <= b4 &
         b1 < (b6 . b8) `1 &
         (b6 . b8) `1 <= b2
   holds - 1 <= ((b5 * b6) . b7) `2 &
    ((b5 * b6) . b7) `2 <= 1 &
    - 1 < ((b5 * b6) . b8) `1 &
    ((b5 * b6) . b8) `1 <= 1;

:: JGRAPH_7:th 67
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
      st b1 < b2 &
         b3 < b4 &
         b5 = AffineMap(2 / (b2 - b1),- ((b2 + b1) / (b2 - b1)),2 / (b4 - b3),- ((b4 + b3) / (b4 - b3))) &
         b1 < (b6 . b8) `1 &
         (b6 . b8) `1 < (b6 . b7) `1 &
         (b6 . b7) `1 <= b2
   holds - 1 < ((b5 * b6) . b8) `1 &
    ((b5 * b6) . b8) `1 < ((b5 * b6) . b7) `1 &
    ((b5 * b6) . b7) `1 <= 1;

:: JGRAPH_7:th 68
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `1 = b5 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 < b4 `2 & b4 `2 <= b8 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 69
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `1 = b5 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 < b4 `2 & b4 `2 <= b8 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 70
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `2 = b8 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 <= b8 & b5 <= b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 71
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `2 = b8 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 <= b8 & b5 <= b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 72
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 <= b8 & b7 <= b4 `2 & b4 `2 <= b8 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 73
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 <= b8 & b7 <= b4 `2 & b4 `2 <= b8 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 74
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 75
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b5 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 < b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 76
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b8 & b4 `2 = b8 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 <= b3 `1 & b3 `1 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 77
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b8 & b4 `2 = b8 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 <= b3 `1 & b3 `1 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 78
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b8 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 <= b3 `1 & b3 `1 <= b6 & b7 <= b4 `2 & b4 `2 <= b8 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 79
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b8 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 <= b3 `1 & b3 `1 <= b6 & b7 <= b4 `2 & b4 `2 <= b8 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 80
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b8 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 <= b3 `1 & b3 `1 <= b6 & b5 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 81
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b8 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 <= b3 `1 & b3 `1 <= b6 & b5 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 82
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b6 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 <= b8 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 83
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b6 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 <= b8 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 84
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b6 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b7 <= b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 85
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `1 = b6 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b7 <= b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 86
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 87
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b5 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 < b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 88
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b8 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 < b3 `1 & b3 `1 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 89
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b8 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 < b3 `1 & b3 `1 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 90
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b8 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b7 <= b4 `2 & b4 `2 <= b8 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 91
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b8 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b7 <= b4 `2 & b4 `2 <= b8 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 92
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b5 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 93
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b5 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 94
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `1 = b6 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 <= b6 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 <= b8 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 95
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `1 = b6 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 <= b6 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 <= b8 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 96
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `1 = b6 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 <= b6 & b7 <= b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 97
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `1 = b6 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 <= b6 & b7 <= b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 98
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 <= b6 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 99
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b8 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 <= b2 `1 & b2 `1 <= b6 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 100
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b6 & b3 `1 = b6 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 <= b8 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 < b2 `2 & b2 `2 <= b8 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 101
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b6 & b3 `1 = b6 & b4 `1 = b6 & b7 <= b1 `2 & b1 `2 <= b8 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 < b2 `2 & b2 `2 <= b8 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 102
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b6 & b3 `1 = b6 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b7 <= b3 `2 & b3 `2 < b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 103
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b6 & b3 `1 = b6 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b7 <= b3 `2 & b3 `2 < b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 104
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b6 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b7 <= b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 105
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `1 = b6 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b7 <= b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 106
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 < b2 `1 & b2 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 107
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b5 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 < b2 `1 & b2 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 108
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b8 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 < b3 `1 & b3 `1 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 109
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b8 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 < b3 `1 & b3 `1 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 110
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b8 & b4 `1 = b6 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b7 <= b4 `2 & b4 `2 <= b8 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 111
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b8 & b4 `1 = b6 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b7 <= b4 `2 & b4 `2 <= b8 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 112
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b5 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 113
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b8 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 < b3 `1 & b3 `1 <= b6 & b5 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 114
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `1 = b6 & b4 `1 = b6 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 <= b6 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 <= b8 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 115
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `1 = b6 & b4 `1 = b6 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 <= b6 & b7 <= b4 `2 & b4 `2 < b3 `2 & b3 `2 <= b8 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 116
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `1 = b6 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 <= b6 & b7 <= b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 117
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `1 = b6 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 <= b6 & b7 <= b3 `2 & b3 `2 <= b8 & b5 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 118
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b7 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 <= b6 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 119
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b8 & b3 `2 = b7 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 < b2 `1 & b2 `1 <= b6 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 120
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `1 = b6 & b3 `1 = b6 & b4 `1 = b6 & b5 <= b1 `1 & b1 `1 <= b6 & b2 `2 <= b8 & b3 `2 < b2 `2 & b4 `2 < b3 `2 & b7 <= b4 `2 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 121
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `1 = b6 & b3 `1 = b6 & b4 `1 = b6 & b5 <= b1 `1 & b1 `1 <= b6 & b2 `2 <= b8 & b3 `2 < b2 `2 & b4 `2 < b3 `2 & b7 <= b4 `2 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 122
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `1 = b6 & b3 `1 = b6 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 <= b6 & b2 `2 <= b8 & b3 `2 < b2 `2 & b7 <= b3 `2 & b5 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 123
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `1 = b6 & b3 `1 = b6 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 <= b6 & b2 `2 <= b8 & b3 `2 < b2 `2 & b7 <= b3 `2 & b5 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 124
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `1 = b6 & b3 `2 = b7 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 <= b6 & b7 <= b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 125
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `1 = b6 & b3 `2 = b7 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 <= b6 & b7 <= b2 `2 & b2 `2 <= b8 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 126
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 <= b6 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 < b2 `1 & b2 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 127
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b8 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b5 <= b1 `1 & b1 `1 <= b6 & b5 < b4 `1 & b4 `1 < b3 `1 & b3 `1 < b2 `1 & b2 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 128
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `1 = b6 & b3 `1 = b6 & b4 `1 = b6 & b1 `2 <= b8 & b2 `2 < b1 `2 & b3 `2 < b2 `2 & b4 `2 < b3 `2 & b7 <= b4 `2 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 129
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `1 = b6 & b3 `1 = b6 & b4 `1 = b6 & b1 `2 <= b8 & b2 `2 < b1 `2 & b3 `2 < b2 `2 & b4 `2 < b3 `2 & b7 <= b4 `2 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 130
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `1 = b6 & b3 `1 = b6 & b4 `2 = b7 & b1 `2 <= b8 & b2 `2 < b1 `2 & b3 `2 < b2 `2 & b7 <= b3 `2 & b5 < b4 `1 & b4 `1 <= b6 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 131
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `1 = b6 & b3 `1 = b6 & b4 `2 = b7 & b1 `2 <= b8 & b2 `2 < b1 `2 & b3 `2 < b2 `2 & b7 <= b3 `2 & b5 < b4 `1 & b4 `1 <= b6 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 132
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `1 = b6 & b3 `2 = b7 & b4 `2 = b7 & b1 `2 <= b8 & b2 `2 < b1 `2 & b7 <= b2 `2 & b3 `1 <= b6 & b4 `1 < b3 `1 & b5 < b4 `1 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 133
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `1 = b6 & b3 `2 = b7 & b4 `2 = b7 & b1 `2 <= b8 & b2 `2 < b1 `2 & b7 <= b2 `2 & b3 `1 <= b6 & b4 `1 < b3 `1 & b5 < b4 `1 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 134
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b2 `1 <= b6 & b3 `1 < b2 `1 & b4 `1 < b3 `1 & b5 < b4 `1 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 135
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `1 = b6 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b7 <= b1 `2 & b1 `2 <= b8 & b2 `1 <= b6 & b3 `1 < b2 `1 & b4 `1 < b3 `1 & b5 < b4 `1 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;

:: JGRAPH_7:th 136
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b7 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b1 `1 <= b6 & b2 `1 < b1 `1 & b3 `1 < b2 `1 & b4 `1 < b3 `1 & b5 < b4 `1 & b9 . 0 = b1 & b9 . 1 = b3 & b10 . 0 = b2 & b10 . 1 = b4 & b9 is continuous(I[01], TOP-REAL 2) & b9 is one-to-one & b10 is continuous(I[01], TOP-REAL 2) & b10 is one-to-one & rng b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & rng b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds rng b9 meets rng b10;

:: JGRAPH_7:th 137
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5, b6, b7, b8 being real set
for b9, b10 being Element of bool the carrier of TOP-REAL 2
      st b5 < b6 & b7 < b8 & b1 `2 = b7 & b2 `2 = b7 & b3 `2 = b7 & b4 `2 = b7 & b1 `1 <= b6 & b2 `1 < b1 `1 & b3 `1 < b2 `1 & b4 `1 < b3 `1 & b5 < b4 `1 & b9 is_an_arc_of b1,b3 & b10 is_an_arc_of b2,b4 & b9 c= closed_inside_of_rectangle(b5,b6,b7,b8) & b10 c= closed_inside_of_rectangle(b5,b6,b7,b8)
   holds b9 meets b10;