Article JGRAPH_2, MML version 4.99.1005

:: JGRAPH_2:th 9
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b7, b8 being Element of bool the carrier of b4
      st b1 is SubSpace of b4 &
         b3 is SubSpace of b4 &
         b7 = [#] b1 &
         b8 = [#] b3 &
         ([#] b1) \/ [#] b3 = [#] b4 &
         b7 is closed(b4) &
         b8 is closed(b4) &
         b5 is continuous(b1, b2) &
         b6 is continuous(b3, b2) &
         (for b9 being set
               st b9 in ([#] b1) /\ [#] b3
            holds b5 . b9 = b6 . b9)
   holds ex b9 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2 st
      b9 = b5 +* b6 & b9 is continuous(b4, b2);

:: JGRAPH_2:th 10
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of Euclid b1
for b3 being Element of the carrier of TOP-REAL b1
for b4 being real set
      st b3 = b2
   holds Ball(b2,b4) = {b5 where b5 is Element of the carrier of TOP-REAL b1: |.b3 - b5.| < b4};

:: JGRAPH_2:th 11
theorem
(0.REAL 2) `1 = 0 & (0.REAL 2) `2 = 0;

:: JGRAPH_2:th 12
theorem
1.REAL 2 = <*1,1*>;

:: JGRAPH_2:th 13
theorem
(1.REAL 2) `1 = 1 & (1.REAL 2) `2 = 1;

:: JGRAPH_2:th 14
theorem
dom proj1 = the carrier of TOP-REAL 2 & dom proj1 = REAL 2;

:: JGRAPH_2:th 15
theorem
dom proj2 = the carrier of TOP-REAL 2 & dom proj2 = REAL 2;

:: JGRAPH_2:th 18
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   b1 = |[proj1 . b1,proj2 . b1]|;

:: JGRAPH_2:th 19
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 = {0.REAL 2}
   holds b1 ` <> {} &
    (the carrier of TOP-REAL 2) \ b1 <> {};

:: JGRAPH_2:th 20
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      for b4 being Element of the carrier of b1
      for b5 being Element of bool the carrier of b2
            st b3 . b4 in b5 & b5 is open(b2)
         holds ex b6 being Element of bool the carrier of b1 st
            b4 in b6 & b6 is open(b1) & b3 .: b6 c= b5;

:: JGRAPH_2:th 21
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 is open(TOP-REAL 2) & b1 in b2
   holds ex b3 being real set st
      0 < b3 &
       {b4 where b4 is Element of the carrier of TOP-REAL 2: b1 `1 - b3 < b4 `1 & b4 `1 < b1 `1 + b3 & b1 `2 - b3 < b4 `2 & b4 `2 < b1 `2 + b3} c= b2;

:: JGRAPH_2:th 22
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Element of bool the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b7 being Function-like quasi_total Relation of the carrier of b2 | b4,the carrier of b3 | b5
      st b6 is continuous(b1, b2) & b7 is continuous(b2 | b4, b3 | b5) & rng b6 c= b4 & b4 <> {} & b5 <> {}
   holds ex b8 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3 st
      b8 is continuous(b1, b3) & b8 = b7 * b6;

:: JGRAPH_2:funcnot 1 => JGRAPH_2:func 1
definition
  func Out_In_Sq -> Function-like quasi_total Relation of (the carrier of TOP-REAL 2) \ {0.REAL 2},(the carrier of TOP-REAL 2) \ {0.REAL 2} means
    for b1 being Element of the carrier of TOP-REAL 2
          st b1 <> 0.REAL 2
       holds ((b1 `2 <= b1 `1 implies b1 `2 < - (b1 `1)) &
         (b1 `1 <= b1 `2 implies - (b1 `1) < b1 `2) or it . b1 = |[1 / (b1 `1),(b1 `2 / (b1 `1)) / (b1 `1)]|) &
        ((b1 `2 <= b1 `1 implies b1 `2 < - (b1 `1)) &
         (b1 `1 <= b1 `2 implies - (b1 `1) < b1 `2) implies it . b1 = |[(b1 `1 / (b1 `2)) / (b1 `2),1 / (b1 `2)]|);
end;

:: JGRAPH_2:def 1
theorem
for b1 being Function-like quasi_total Relation of (the carrier of TOP-REAL 2) \ {0.REAL 2},(the carrier of TOP-REAL 2) \ {0.REAL 2} holds
      b1 = Out_In_Sq
   iff
      for b2 being Element of the carrier of TOP-REAL 2
            st b2 <> 0.REAL 2
         holds ((b2 `2 <= b2 `1 implies b2 `2 < - (b2 `1)) &
           (b2 `1 <= b2 `2 implies - (b2 `1) < b2 `2) or b1 . b2 = |[1 / (b2 `1),(b2 `2 / (b2 `1)) / (b2 `1)]|) &
          ((b2 `2 <= b2 `1 implies b2 `2 < - (b2 `1)) &
           (b2 `1 <= b2 `2 implies - (b2 `1) < b2 `2) implies b1 . b2 = |[(b2 `1 / (b2 `2)) / (b2 `2),1 / (b2 `2)]|);

:: JGRAPH_2:th 23
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st (b1 `2 <= b1 `1 implies b1 `2 < - (b1 `1)) &
         (b1 `1 <= b1 `2 implies - (b1 `1) < b1 `2) &
         (b1 `1 <= b1 `2 implies b1 `1 < - (b1 `2))
   holds b1 `2 <= b1 `1 & b1 `1 <= - (b1 `2);

:: JGRAPH_2:th 24
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st b1 <> 0.REAL 2
   holds ((b1 `1 <= b1 `2 implies b1 `1 < - (b1 `2)) &
     (b1 `2 <= b1 `1 implies - (b1 `2) < b1 `1) or Out_In_Sq . b1 = |[(b1 `1 / (b1 `2)) / (b1 `2),1 / (b1 `2)]|) &
    ((b1 `1 <= b1 `2 implies b1 `1 < - (b1 `2)) &
     (b1 `2 <= b1 `1 implies - (b1 `2) < b1 `1) implies Out_In_Sq . b1 = |[1 / (b1 `1),(b1 `2 / (b1 `1)) / (b1 `1)]|);

:: JGRAPH_2:th 25
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 b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: (b3 `2 <= b3 `1 & - (b3 `1) <= b3 `2 or b3 `1 <= b3 `2 & b3 `2 <= - (b3 `1)) &
         b3 <> 0.REAL 2}
   holds rng (Out_In_Sq | b2) c= the carrier of ((TOP-REAL 2) | b1) | b2;

:: JGRAPH_2:th 26
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 b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: (b3 `1 <= b3 `2 & - (b3 `2) <= b3 `1 or b3 `2 <= b3 `1 & b3 `1 <= - (b3 `2)) &
         b3 <> 0.REAL 2}
   holds rng (Out_In_Sq | b2) c= the carrier of ((TOP-REAL 2) | b1) | b2;

:: JGRAPH_2:th 27
theorem
for b1 being set
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 = {b3 where b3 is Element of the carrier of TOP-REAL 2: (b3 `2 <= b3 `1 & - (b3 `1) <= b3 `2 or b3 `1 <= b3 `2 & b3 `2 <= - (b3 `1)) &
          b3 <> 0.REAL 2} &
         b2 ` = {0.REAL 2}
   holds b1 is non empty Element of bool the carrier of (TOP-REAL 2) | b2 &
    b1 is non empty Element of bool the carrier of TOP-REAL 2;

:: JGRAPH_2:th 28
theorem
for b1 being set
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 = {b3 where b3 is Element of the carrier of TOP-REAL 2: (b3 `1 <= b3 `2 & - (b3 `2) <= b3 `1 or b3 `2 <= b3 `1 & b3 `1 <= - (b3 `2)) &
          b3 <> 0.REAL 2} &
         b2 ` = {0.REAL 2}
   holds b1 is non empty Element of bool the carrier of (TOP-REAL 2) | b2 &
    b1 is non empty Element of bool the carrier of TOP-REAL 2;

:: JGRAPH_2:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
      st b2 is continuous(b1, R^1) & b3 is continuous(b1, R^1)
   holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b5 being Element of the carrier of b1
       for b6, b7 being real set
             st b2 . b5 = b6 & b3 . b5 = b7
          holds b4 . b5 = b6 + b7) &
       b4 is continuous(b1, R^1);

:: JGRAPH_2:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being real set holds
   ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b4 being Element of the carrier of b1 holds
          b3 . b4 = b2) &
       b3 is continuous(b1, R^1);

:: JGRAPH_2:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
      st b2 is continuous(b1, R^1) & b3 is continuous(b1, R^1)
   holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b5 being Element of the carrier of b1
       for b6, b7 being real set
             st b2 . b5 = b6 & b3 . b5 = b7
          holds b4 . b5 = b6 - b7) &
       b4 is continuous(b1, R^1);

:: JGRAPH_2:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
      st b2 is continuous(b1, R^1)
   holds ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b4 being Element of the carrier of b1
       for b5 being real set
             st b2 . b4 = b5
          holds b3 . b4 = b5 * b5) &
       b3 is continuous(b1, R^1);

:: JGRAPH_2:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
for b3 being real set
      st b2 is continuous(b1, R^1)
   holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b5 being Element of the carrier of b1
       for b6 being real set
             st b2 . b5 = b6
          holds b4 . b5 = b3 * b6) &
       b4 is continuous(b1, R^1);

:: JGRAPH_2:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
for b3 being real set
      st b2 is continuous(b1, R^1)
   holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b5 being Element of the carrier of b1
       for b6 being real set
             st b2 . b5 = b6
          holds b4 . b5 = b6 + b3) &
       b4 is continuous(b1, R^1);

:: JGRAPH_2:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
      st b2 is continuous(b1, R^1) & b3 is continuous(b1, R^1)
   holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b5 being Element of the carrier of b1
       for b6, b7 being real set
             st b2 . b5 = b6 & b3 . b5 = b7
          holds b4 . b5 = b6 * b7) &
       b4 is continuous(b1, R^1);

:: JGRAPH_2:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
      st b2 is continuous(b1, R^1) &
         (for b3 being Element of the carrier of b1 holds
            b2 . b3 <> 0)
   holds ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b4 being Element of the carrier of b1
       for b5 being real set
             st b2 . b4 = b5
          holds b3 . b4 = 1 / b5) &
       b3 is continuous(b1, R^1);

:: JGRAPH_2:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
      st b2 is continuous(b1, R^1) &
         b3 is continuous(b1, R^1) &
         (for b4 being Element of the carrier of b1 holds
            b3 . b4 <> 0)
   holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b5 being Element of the carrier of b1
       for b6, b7 being real set
             st b2 . b5 = b6 & b3 . b5 = b7
          holds b4 . b5 = b6 / b7) &
       b4 is continuous(b1, R^1);

:: JGRAPH_2:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
      st b2 is continuous(b1, R^1) &
         b3 is continuous(b1, R^1) &
         (for b4 being Element of the carrier of b1 holds
            b3 . b4 <> 0)
   holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b5 being Element of the carrier of b1
       for b6, b7 being real set
             st b2 . b5 = b6 & b3 . b5 = b7
          holds b4 . b5 = (b6 / b7) / b7) &
       b4 is continuous(b1, R^1);

:: JGRAPH_2:th 39
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
      st for b3 being Element of the carrier of (TOP-REAL 2) | b1 holds
           b2 . b3 = proj1 . b3
   holds b2 is continuous((TOP-REAL 2) | b1, R^1);

:: JGRAPH_2:th 40
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
      st for b3 being Element of the carrier of (TOP-REAL 2) | b1 holds
           b2 . b3 = proj2 . b3
   holds b2 is continuous((TOP-REAL 2) | b1, R^1);

:: JGRAPH_2:th 41
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
      st (for b3 being Element of the carrier of TOP-REAL 2
               st b3 in the carrier of (TOP-REAL 2) | b1
            holds b2 . b3 = 1 / (b3 `1)) &
         (for b3 being Element of the carrier of TOP-REAL 2
               st b3 in the carrier of (TOP-REAL 2) | b1
            holds b3 `1 <> 0)
   holds b2 is continuous((TOP-REAL 2) | b1, R^1);

:: JGRAPH_2:th 42
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
      st (for b3 being Element of the carrier of TOP-REAL 2
               st b3 in the carrier of (TOP-REAL 2) | b1
            holds b2 . b3 = 1 / (b3 `2)) &
         (for b3 being Element of the carrier of TOP-REAL 2
               st b3 in the carrier of (TOP-REAL 2) | b1
            holds b3 `2 <> 0)
   holds b2 is continuous((TOP-REAL 2) | b1, R^1);

:: JGRAPH_2:th 43
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
      st (for b3 being Element of the carrier of TOP-REAL 2
               st b3 in the carrier of (TOP-REAL 2) | b1
            holds b2 . b3 = (b3 `2 / (b3 `1)) / (b3 `1)) &
         (for b3 being Element of the carrier of TOP-REAL 2
               st b3 in the carrier of (TOP-REAL 2) | b1
            holds b3 `1 <> 0)
   holds b2 is continuous((TOP-REAL 2) | b1, R^1);

:: JGRAPH_2:th 44
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
      st (for b3 being Element of the carrier of TOP-REAL 2
               st b3 in the carrier of (TOP-REAL 2) | b1
            holds b2 . b3 = (b3 `1 / (b3 `2)) / (b3 `2)) &
         (for b3 being Element of the carrier of TOP-REAL 2
               st b3 in the carrier of (TOP-REAL 2) | b1
            holds b3 `2 <> 0)
   holds b2 is continuous((TOP-REAL 2) | b1, R^1);

:: JGRAPH_2:th 45
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b2
for b4, b5 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
      st b4 is continuous((TOP-REAL 2) | b1, R^1) &
         b5 is continuous((TOP-REAL 2) | b1, R^1) &
         b1 <> {} &
         b2 <> {} &
         (for b6, b7, b8, b9 being real set
               st |[b6,b7]| in b1 & b8 = b4 . |[b6,b7]| & b9 = b5 . |[b6,b7]|
            holds b3 . |[b6,b7]| = |[b8,b9]|)
   holds b3 is continuous((TOP-REAL 2) | b1, (TOP-REAL 2) | b2);

:: JGRAPH_2:th 46
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b2
      st b3 = Out_In_Sq | b1 &
         b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b1 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `2 <= b4 `1 & - (b4 `1) <= b4 `2 or b4 `1 <= b4 `2 & b4 `2 <= - (b4 `1)) &
          b4 <> 0.REAL 2}
   holds b3 is continuous((TOP-REAL 2) | b1, (TOP-REAL 2) | b2);

:: JGRAPH_2:th 47
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b2
      st b3 = Out_In_Sq | b1 &
         b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b1 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `1 <= b4 `2 & - (b4 `2) <= b4 `1 or b4 `2 <= b4 `1 & b4 `1 <= - (b4 `2)) &
          b4 <> 0.REAL 2}
   holds b3 is continuous((TOP-REAL 2) | b1, (TOP-REAL 2) | b2);

:: JGRAPH_2:sch 1
scheme JGRAPH_2:sch 1
{b1 where b1 is Element of the carrier of TOP-REAL 2: P1[b1]} is Element of bool the carrier of TOP-REAL 2


:: JGRAPH_2:sch 2
scheme JGRAPH_2:sch 2
{F1 -> Element of bool the carrier of TOP-REAL 2}:
F1() ` = {b1 where b1 is Element of the carrier of TOP-REAL 2: not (P1[b1])}
provided
   F1() = {b1 where b1 is Element of the carrier of TOP-REAL 2: P1[b1]};


:: JGRAPH_2:sch 3
scheme JGRAPH_2:sch 3
{F1 -> real set,
  F2 -> real set}:
{b1 where b1 is Element of the carrier of TOP-REAL 2: F1(b1) <= F2(b1)} is closed Element of bool the carrier of TOP-REAL 2
provided
   for b1, b2 being Element of the carrier of TOP-REAL 2 holds
   F1(b1 - b2) = F1(b1) - F1(b2) & F2(b1 - b2) = F2(b1) - F2(b2)
and
   for b1, b2 being Element of the carrier of TOP-REAL 2 holds
   |.b1 - b2.| ^2 = F1(b1 - b2) ^2 + (F2(b1 - b2) ^2);


:: JGRAPH_2:th 48
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 being Function-like quasi_total Relation of the carrier of ((TOP-REAL 2) | b1) | b2,the carrier of (TOP-REAL 2) | b1
      st b3 = Out_In_Sq | b2 &
         b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `2 <= b4 `1 & - (b4 `1) <= b4 `2 or b4 `1 <= b4 `2 & b4 `2 <= - (b4 `1)) &
          b4 <> 0.REAL 2}
   holds b3 is continuous(((TOP-REAL 2) | b1) | b2, (TOP-REAL 2) | b1) &
    b2 is closed((TOP-REAL 2) | b1);

:: JGRAPH_2:th 49
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 being Function-like quasi_total Relation of the carrier of ((TOP-REAL 2) | b1) | b2,the carrier of (TOP-REAL 2) | b1
      st b3 = Out_In_Sq | b2 &
         b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `1 <= b4 `2 & - (b4 `2) <= b4 `1 or b4 `2 <= b4 `1 & b4 `1 <= - (b4 `2)) &
          b4 <> 0.REAL 2}
   holds b3 is continuous(((TOP-REAL 2) | b1) | b2, (TOP-REAL 2) | b1) &
    b2 is closed((TOP-REAL 2) | b1);

:: JGRAPH_2:th 50
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 ` = {0.REAL 2}
   holds ex b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b1 st
      b2 = Out_In_Sq &
       b2 is continuous((TOP-REAL 2) | b1, (TOP-REAL 2) | b1);

:: JGRAPH_2:th 51
theorem
for b1, b2, b3 being Element of bool the carrier of TOP-REAL 2
      st b1 = {0.REAL 2} &
         b2 = {b4 where b4 is Element of the carrier of TOP-REAL 2: - 1 < b4 `1 & b4 `1 < 1 & - 1 < b4 `2 & b4 `2 < 1} &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: ((- 1 = b4 `1 & - 1 <= b4 `2 implies 1 < b4 `2) &
          (b4 `1 = 1 & - 1 <= b4 `2 implies 1 < b4 `2) &
          (- 1 = b4 `2 & - 1 <= b4 `1 implies 1 < b4 `1) implies 1 = b4 `2 & - 1 <= b4 `1 & b4 `1 <= 1)}
   holds ex b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | (b1 `),the carrier of (TOP-REAL 2) | (b1 `) st
      b4 is continuous((TOP-REAL 2) | (b1 `), (TOP-REAL 2) | (b1 `)) &
       b4 is one-to-one &
       (for b5 being Element of the carrier of TOP-REAL 2
             st b5 in b2 & b5 <> 0.REAL 2
          holds not b4 . b5 in b2 \/ b3) &
       (for b5 being Element of the carrier of TOP-REAL 2
             st not b5 in b2 \/ b3
          holds b4 . b5 in b2) &
       (for b5 being Element of the carrier of TOP-REAL 2
             st b5 in b3
          holds b4 . b5 = b5);

:: JGRAPH_2:th 52
theorem
for b1, b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of TOP-REAL 2
for b4, b5 being Element of the carrier of I[01]
      st b4 = 0 &
         b5 = 1 &
         b1 is continuous(I[01], TOP-REAL 2) &
         b1 is one-to-one &
         b2 is continuous(I[01], TOP-REAL 2) &
         b2 is one-to-one &
         b3 = {b6 where b6 is Element of the carrier of TOP-REAL 2: - 1 < b6 `1 & b6 `1 < 1 & - 1 < b6 `2 & b6 `2 < 1} &
         (b1 . b4) `1 = - 1 &
         (b1 . b5) `1 = 1 &
         - 1 <= (b1 . b4) `2 &
         (b1 . b4) `2 <= 1 &
         - 1 <= (b1 . b5) `2 &
         (b1 . b5) `2 <= 1 &
         (b2 . b4) `2 = - 1 &
         (b2 . b5) `2 = 1 &
         - 1 <= (b2 . b4) `1 &
         (b2 . b4) `1 <= 1 &
         - 1 <= (b2 . b5) `1 &
         (b2 . b5) `1 <= 1 &
         rng b1 misses b3 &
         rng b2 misses b3
   holds rng b1 meets rng b2;

:: JGRAPH_2: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
      st for b6 being Element of the carrier of TOP-REAL 2 holds
           b5 . b6 = |[(b1 * (b6 `1)) + b2,(b3 * (b6 `2)) + b4]|
   holds b5 is continuous(TOP-REAL 2, TOP-REAL 2);

:: JGRAPH_2:funcnot 2 => JGRAPH_2:func 2
definition
  let a1, a2, a3, a4 be real set;
  func AffineMap(A1,A2,A3,A4) -> Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 means
    for b1 being Element of the carrier of TOP-REAL 2 holds
       it . b1 = |[(a1 * (b1 `1)) + a2,(a3 * (b1 `2)) + a4]|;
end;

:: JGRAPH_2:def 2
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 holds
      b5 = AffineMap(b1,b2,b3,b4)
   iff
      for b6 being Element of the carrier of TOP-REAL 2 holds
         b5 . b6 = |[(b1 * (b6 `1)) + b2,(b3 * (b6 `2)) + b4]|;

:: JGRAPH_2:funcreg 1
registration
  let a1, a2, a3, a4 be real set;
  cluster AffineMap(a1,a2,a3,a4) -> Function-like quasi_total continuous;
end;

:: JGRAPH_2:th 54
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & 0 < b3
   holds AffineMap(b1,b2,b3,b4) is one-to-one;

:: JGRAPH_2:th 55
theorem
for b1, b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b3, b4, b5, b6 being real set
for b7, b8 being Element of the carrier of I[01]
      st b7 = 0 &
         b8 = 1 &
         b1 is continuous(I[01], TOP-REAL 2) &
         b1 is one-to-one &
         b2 is continuous(I[01], TOP-REAL 2) &
         b2 is one-to-one &
         (b1 . b7) `1 = b3 &
         (b1 . b8) `1 = b4 &
         b5 <= (b1 . b7) `2 &
         (b1 . b7) `2 <= b6 &
         b5 <= (b1 . b8) `2 &
         (b1 . b8) `2 <= b6 &
         (b2 . b7) `2 = b5 &
         (b2 . b8) `2 = b6 &
         b3 <= (b2 . b7) `1 &
         (b2 . b7) `1 <= b4 &
         b3 <= (b2 . b8) `1 &
         (b2 . b8) `1 <= b4 &
         b3 < b4 &
         b5 < b6 &
         (for b9 being Element of the carrier of I[01]
               st b3 < (b1 . b9) `1 & (b1 . b9) `1 < b4 & b5 < (b1 . b9) `2
            holds b6 <= (b1 . b9) `2) &
         (for b9 being Element of the carrier of I[01]
               st b3 < (b2 . b9) `1 & (b2 . b9) `1 < b4 & b5 < (b2 . b9) `2
            holds b6 <= (b2 . b9) `2)
   holds rng b1 meets rng b2;

:: JGRAPH_2:th 56
theorem
{b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `2 <= b1 `1} is closed Element of bool the carrier of TOP-REAL 2 &
 {b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `1 <= b1 `2} is closed Element of bool the carrier of TOP-REAL 2;

:: JGRAPH_2:th 57
theorem
{b1 where b1 is Element of the carrier of TOP-REAL 2: - (b1 `1) <= b1 `2} is closed Element of bool the carrier of TOP-REAL 2 &
 {b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `2 <= - (b1 `1)} is closed Element of bool the carrier of TOP-REAL 2;

:: JGRAPH_2:th 58
theorem
{b1 where b1 is Element of the carrier of TOP-REAL 2: - (b1 `2) <= b1 `1} is closed Element of bool the carrier of TOP-REAL 2 &
 {b1 where b1 is Element of the carrier of TOP-REAL 2: b1 `1 <= - (b1 `2)} is closed Element of bool the carrier of TOP-REAL 2;