Article JGRAPH_4, MML version 4.99.1005

:: JGRAPH_4:th 6
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
for b3 being Element of bool the carrier of b1
for b4 being real set
      st b2 is continuous(b1, R^1) &
         b3 = {b5 where b5 is Element of the carrier of b1: b4 < b2 /. b5}
   holds b3 is open(b1);

:: JGRAPH_4:th 7
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
for b3 being Element of bool the carrier of b1
for b4 being Element of REAL
      st b2 is continuous(b1, R^1) &
         b3 = {b5 where b5 is Element of the carrier of b1: b2 /. b5 < b4}
   holds b3 is open(b1);

:: JGRAPH_4:th 8
theorem
for b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
      st b1 is continuous(TOP-REAL 2, TOP-REAL 2) &
         b1 is one-to-one &
         rng b1 = [#] TOP-REAL 2 &
         (for b2 being Element of the carrier of TOP-REAL 2 holds
            ex b3 being non empty compact Element of bool the carrier of TOP-REAL 2 st
               b3 = b1 .: b3 &
                (ex b4 being Element of bool the carrier of TOP-REAL 2 st
                   b2 in b4 & b4 is open(TOP-REAL 2) & b4 c= b3 & b1 . b2 in b4))
   holds b1 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2);

:: JGRAPH_4:th 9
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
for b4, b5 being real set
      st b2 is continuous(b1, R^1) &
         b3 is continuous(b1, R^1) &
         b5 <> 0 &
         (for b6 being Element of the carrier of b1 holds
            b3 . b6 <> 0)
   holds ex b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b7 being Element of the carrier of b1
       for b8, b9 being real set
             st b2 . b7 = b8 & b3 . b7 = b9
          holds b6 . b7 = ((b8 / b9) - b4) / b5) &
       b6 is continuous(b1, R^1);

:: JGRAPH_4:th 10
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
for b4, b5 being Element of REAL
      st b2 is continuous(b1, R^1) &
         b3 is continuous(b1, R^1) &
         b5 <> 0 &
         (for b6 being Element of the carrier of b1 holds
            b3 . b6 <> 0)
   holds ex b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b7 being Element of the carrier of b1
       for b8, b9 being Element of REAL
             st b2 . b7 = b8 & b3 . b7 = b9
          holds b6 . b7 = b9 * (((b8 / b9) - b4) / b5)) &
       b6 is continuous(b1, R^1);

:: JGRAPH_4:th 11
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 ^2) &
       b3 is continuous(b1, R^1);

:: JGRAPH_4:th 12
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 = abs b5) &
       b3 is continuous(b1, R^1);

:: JGRAPH_4:th 13
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) &
       b3 is continuous(b1, R^1);

:: JGRAPH_4:th 14
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
for b4, b5 being real set
      st b2 is continuous(b1, R^1) &
         b3 is continuous(b1, R^1) &
         b5 <> 0 &
         (for b6 being Element of the carrier of b1 holds
            b3 . b6 <> 0)
   holds ex b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b7 being Element of the carrier of b1
       for b8, b9 being real set
             st b2 . b7 = b8 & b3 . b7 = b9
          holds b6 . b7 = b9 * - sqrt abs (1 - ((((b8 / b9) - b4) / b5) ^2))) &
       b6 is continuous(b1, R^1);

:: JGRAPH_4:th 15
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
for b4, b5 being real set
      st b2 is continuous(b1, R^1) &
         b3 is continuous(b1, R^1) &
         b5 <> 0 &
         (for b6 being Element of the carrier of b1 holds
            b3 . b6 <> 0)
   holds ex b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      (for b7 being Element of the carrier of b1
       for b8, b9 being real set
             st b2 . b7 = b8 & b3 . b7 = b9
          holds b6 . b7 = b9 * sqrt abs (1 - ((((b8 / b9) - b4) / b5) ^2))) &
       b6 is continuous(b1, R^1);

:: JGRAPH_4:funcnot 1 => JGRAPH_4:func 1
definition
  let a1 be Element of NAT;
  func A1 NormF -> Function-like quasi_total Relation of the carrier of TOP-REAL a1,the carrier of R^1 means
    for b1 being Element of the carrier of TOP-REAL a1 holds
       it . b1 = |.b1.|;
end;

:: JGRAPH_4:def 1
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL b1,the carrier of R^1 holds
      b2 = b1 NormF
   iff
      for b3 being Element of the carrier of TOP-REAL b1 holds
         b2 . b3 = |.b3.|;

:: JGRAPH_4:th 16
theorem
for b1 being Element of NAT holds
   dom (b1 NormF) = the carrier of TOP-REAL b1 & dom (b1 NormF) = REAL b1;

:: JGRAPH_4:th 19
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL b1,the carrier of R^1
      st b2 = b1 NormF
   holds b2 is continuous(TOP-REAL b1, R^1);

:: JGRAPH_4:th 20
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL b1) | b2,the carrier of R^1
      st for b4 being Element of the carrier of (TOP-REAL b1) | b2 holds
           b3 . b4 = b1 NormF . b4
   holds b3 is continuous((TOP-REAL b1) | b2, R^1);

:: JGRAPH_4:th 21
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of Euclid b1
for b3 being Element of REAL
for b4 being Element of bool the carrier of TOP-REAL b1
      st b4 = cl_Ball(b2,b3)
   holds b4 is Bounded(b1) & b4 is closed(TOP-REAL b1);

:: JGRAPH_4:th 22
theorem
for b1 being Element of the carrier of Euclid 2
for b2 being Element of REAL
for b3 being Element of bool the carrier of TOP-REAL 2
      st b3 = cl_Ball(b1,b2)
   holds b3 is compact(TOP-REAL 2);

:: JGRAPH_4:funcnot 2 => JGRAPH_4:func 2
definition
  let a1 be real set;
  let a2 be Element of the carrier of TOP-REAL 2;
  func FanW(A1,A2) -> Element of the carrier of TOP-REAL 2 equals
    |.a2.| * |[- sqrt (1 - ((((a2 `2 / |.a2.|) - a1) / (1 - a1)) ^2)),((a2 `2 / |.a2.|) - a1) / (1 - a1)]|
    if a1 <= a2 `2 / |.a2.| & a2 `1 < 0,
|.a2.| * |[- sqrt (1 - ((((a2 `2 / |.a2.|) - a1) / (1 + a1)) ^2)),((a2 `2 / |.a2.|) - a1) / (1 + a1)]|
    if a2 `2 / |.a2.| < a1 & a2 `1 < 0
    otherwise a2;
end;

:: JGRAPH_4:def 2
theorem
for b1 being real set
for b2 being Element of the carrier of TOP-REAL 2 holds
   (b1 <= b2 `2 / |.b2.| & b2 `1 < 0 implies FanW(b1,b2) = |.b2.| * |[- sqrt (1 - ((((b2 `2 / |.b2.|) - b1) / (1 - b1)) ^2)),((b2 `2 / |.b2.|) - b1) / (1 - b1)]|) &
    (b2 `2 / |.b2.| < b1 & b2 `1 < 0 implies FanW(b1,b2) = |.b2.| * |[- sqrt (1 - ((((b2 `2 / |.b2.|) - b1) / (1 + b1)) ^2)),((b2 `2 / |.b2.|) - b1) / (1 + b1)]|) &
    ((b1 <= b2 `2 / |.b2.| implies 0 <= b2 `1) &
     (b1 <= b2 `2 / |.b2.| or 0 <= b2 `1) implies FanW(b1,b2) = b2);

:: JGRAPH_4:funcnot 3 => JGRAPH_4:func 3
definition
  let a1 be real set;
  func A1 -FanMorphW -> 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 = FanW(a1,b1);
end;

:: JGRAPH_4:def 3
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 holds
      b2 = b1 -FanMorphW
   iff
      for b3 being Element of the carrier of TOP-REAL 2 holds
         b2 . b3 = FanW(b1,b3);

:: JGRAPH_4:th 23
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being real set holds
   (b2 <= b1 `2 / |.b1.| & b1 `1 < 0 implies b2 -FanMorphW . b1 = |[|.b1.| * - sqrt (1 - ((((b1 `2 / |.b1.|) - b2) / (1 - b2)) ^2)),|.b1.| * (((b1 `2 / |.b1.|) - b2) / (1 - b2))]|) &
    (0 <= b1 `1 implies b2 -FanMorphW . b1 = b1);

:: JGRAPH_4:th 24
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of REAL
      st b1 `2 / |.b1.| <= b2 & b1 `1 < 0
   holds b2 -FanMorphW . b1 = |[|.b1.| * - sqrt (1 - ((((b1 `2 / |.b1.|) - b2) / (1 + b2)) ^2)),|.b1.| * (((b1 `2 / |.b1.|) - b2) / (1 + b2))]|;

:: JGRAPH_4:th 25
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of REAL
      st - 1 < b2 & b2 < 1
   holds (b2 <= b1 `2 / |.b1.| & b1 `1 <= 0 & b1 <> 0.REAL 2 implies b2 -FanMorphW . b1 = |[|.b1.| * - sqrt (1 - ((((b1 `2 / |.b1.|) - b2) / (1 - b2)) ^2)),|.b1.| * (((b1 `2 / |.b1.|) - b2) / (1 - b2))]|) &
    (b1 `2 / |.b1.| <= b2 & b1 `1 <= 0 & b1 <> 0.REAL 2 implies b2 -FanMorphW . b1 = |[|.b1.| * - sqrt (1 - ((((b1 `2 / |.b1.|) - b2) / (1 + b2)) ^2)),|.b1.| * (((b1 `2 / |.b1.|) - b2) / (1 + b2))]|);

:: JGRAPH_4:th 26
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * (((b4 `2 / |.b4.|) - b1) / (1 - b1))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b4 `1 <= 0 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 27
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * (((b4 `2 / |.b4.|) - b1) / (1 + b1))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b4 `1 <= 0 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 28
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * - sqrt (1 - ((((b4 `2 / |.b4.|) - b1) / (1 - b1)) ^2))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b4 `1 <= 0 & b1 <= b4 `2 / |.b4.| & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 29
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * - sqrt (1 - ((((b4 `2 / |.b4.|) - b1) / (1 + b1)) ^2))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b4 `1 <= 0 & b4 `2 / |.b4.| <= b1 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 30
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphW | b2 &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 <= 0 & b5 <> 0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b1 <= b5 `2 / |.b5.| & b5 `1 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 31
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphW | b2 &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 <= 0 & b5 <> 0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `2 / |.b5.| <= b1 & b5 `1 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 32
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b1 * |.b3.| <= b3 `2 & b3 `1 <= 0}
   holds b2 is closed(TOP-REAL 2);

:: JGRAPH_4:th 33
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 <= b1 * |.b3.| & b3 `1 <= 0}
   holds b2 is closed(TOP-REAL 2);

:: JGRAPH_4:th 34
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphW | b2 &
         b3 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 35
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphW | b2 &
         b3 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `1 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 36
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
      st b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 <= 0 & b3 <> 0.REAL 2}
   holds b2 is closed((TOP-REAL 2) | b1);

:: JGRAPH_4:th 37
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of (TOP-REAL 2) | b2
for b4 being Function-like quasi_total Relation of the carrier of ((TOP-REAL 2) | b2) | b3,the carrier of (TOP-REAL 2) | b2
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphW | b3 &
         b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous(((TOP-REAL 2) | b2) | b3, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 38
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
      st b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: 0 <= b3 `1 & b3 <> 0.REAL 2}
   holds b2 is closed((TOP-REAL 2) | b1);

:: JGRAPH_4:th 39
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of (TOP-REAL 2) | b2
for b4 being Function-like quasi_total Relation of the carrier of ((TOP-REAL 2) | b2) | b3,the carrier of (TOP-REAL 2) | b2
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphW | b3 &
         b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `1 & b5 <> 0.REAL 2}
   holds b4 is continuous(((TOP-REAL 2) | b2) | b3, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 40
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2 holds
   |.b1 -FanMorphW . b2.| = |.b2.|;

:: JGRAPH_4:th 41
theorem
for b1 being Element of REAL
for b2, b3 being set
      st - 1 < b1 &
         b1 < 1 &
         b2 in b3 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: b4 `1 <= 0 & b4 <> 0.REAL 2}
   holds b1 -FanMorphW . b2 in b3;

:: JGRAPH_4:th 42
theorem
for b1 being Element of REAL
for b2, b3 being set
      st - 1 < b1 &
         b1 < 1 &
         b2 in b3 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: 0 <= b4 `1 & b4 <> 0.REAL 2}
   holds b1 -FanMorphW . b2 in b3;

:: JGRAPH_4:sch 1
scheme JGRAPH_4:sch 1
{F1 -> non empty Element of bool the carrier of TOP-REAL 2}:
{b1 where b1 is Element of the carrier of TOP-REAL 2: P1[b1] & b1 <> 0.REAL 2} c= the carrier of (TOP-REAL 2) | F1()
provided
   F1() = (the carrier of TOP-REAL 2) \ {0.REAL 2};


:: JGRAPH_4:th 43
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st - 1 < b1 & b1 < 1 & b2 ` = {0.REAL 2}
   holds ex b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b2 st
      b3 = b1 -FanMorphW | b2 &
       b3 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 44
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds ex b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      b2 = b1 -FanMorphW & b2 is continuous(TOP-REAL 2, TOP-REAL 2);

:: JGRAPH_4:th 45
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds b1 -FanMorphW is one-to-one;

:: JGRAPH_4:th 46
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds b1 -FanMorphW is Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 &
    rng (b1 -FanMorphW) = the carrier of TOP-REAL 2;

:: JGRAPH_4:th 47
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
      st - 1 < b1 & b1 < 1
   holds ex b3 being non empty compact Element of bool the carrier of TOP-REAL 2 st
      b3 = b1 -FanMorphW .: b3 &
       (ex b4 being Element of bool the carrier of TOP-REAL 2 st
          b2 in b4 & b4 is open(TOP-REAL 2) & b4 c= b3 & b1 -FanMorphW . b2 in b4);

:: JGRAPH_4:th 48
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds ex b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      b2 = b1 -FanMorphW & b2 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2);

:: JGRAPH_4:th 49
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & b2 `1 < 0 & b1 <= b2 `2 / |.b2.|
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphW . b2
   holds b3 `1 < 0 & 0 <= b3 `2;

:: JGRAPH_4:th 50
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & b2 `1 < 0 & b2 `2 / |.b2.| < b1
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphW . b2
   holds b3 `1 < 0 & b3 `2 < 0;

:: JGRAPH_4:th 51
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      b2 `1 < 0 &
      b1 <= b2 `2 / |.b2.| &
      b3 `1 < 0 &
      b1 <= b3 `2 / |.b3.| &
      b2 `2 / |.b2.| < b3 `2 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphW . b2 & b5 = b1 -FanMorphW . b3
   holds b4 `2 / |.b4.| < b5 `2 / |.b5.|;

:: JGRAPH_4:th 52
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      b2 `1 < 0 &
      b2 `2 / |.b2.| < b1 &
      b3 `1 < 0 &
      b3 `2 / |.b3.| < b1 &
      b2 `2 / |.b2.| < b3 `2 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphW . b2 & b5 = b1 -FanMorphW . b3
   holds b4 `2 / |.b4.| < b5 `2 / |.b5.|;

:: JGRAPH_4:th 53
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      b2 `1 < 0 &
      b3 `1 < 0 &
      b2 `2 / |.b2.| < b3 `2 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphW . b2 & b5 = b1 -FanMorphW . b3
   holds b4 `2 / |.b4.| < b5 `2 / |.b5.|;

:: JGRAPH_4:th 54
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & b2 `1 < 0 & b2 `2 / |.b2.| = b1
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphW . b2
   holds b3 `1 < 0 & b3 `2 = 0;

:: JGRAPH_4:th 55
theorem
for b1 being real set holds
   0.REAL 2 = b1 -FanMorphW . 0.REAL 2;

:: JGRAPH_4:funcnot 4 => JGRAPH_4:func 4
definition
  let a1 be real set;
  let a2 be Element of the carrier of TOP-REAL 2;
  func FanN(A1,A2) -> Element of the carrier of TOP-REAL 2 equals
    |.a2.| * |[((a2 `1 / |.a2.|) - a1) / (1 - a1),sqrt (1 - ((((a2 `1 / |.a2.|) - a1) / (1 - a1)) ^2))]|
    if a1 <= a2 `1 / |.a2.| & 0 < a2 `2,
|.a2.| * |[((a2 `1 / |.a2.|) - a1) / (1 + a1),sqrt (1 - ((((a2 `1 / |.a2.|) - a1) / (1 + a1)) ^2))]|
    if a2 `1 / |.a2.| < a1 & 0 < a2 `2
    otherwise a2;
end;

:: JGRAPH_4:def 4
theorem
for b1 being real set
for b2 being Element of the carrier of TOP-REAL 2 holds
   (b1 <= b2 `1 / |.b2.| & 0 < b2 `2 implies FanN(b1,b2) = |.b2.| * |[((b2 `1 / |.b2.|) - b1) / (1 - b1),sqrt (1 - ((((b2 `1 / |.b2.|) - b1) / (1 - b1)) ^2))]|) &
    (b2 `1 / |.b2.| < b1 & 0 < b2 `2 implies FanN(b1,b2) = |.b2.| * |[((b2 `1 / |.b2.|) - b1) / (1 + b1),sqrt (1 - ((((b2 `1 / |.b2.|) - b1) / (1 + b1)) ^2))]|) &
    ((b1 <= b2 `1 / |.b2.| implies b2 `2 <= 0) &
     (b1 <= b2 `1 / |.b2.| or b2 `2 <= 0) implies FanN(b1,b2) = b2);

:: JGRAPH_4:funcnot 5 => JGRAPH_4:func 5
definition
  let a1 be real set;
  func A1 -FanMorphN -> 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 = FanN(a1,b1);
end;

:: JGRAPH_4:def 5
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 holds
      b2 = b1 -FanMorphN
   iff
      for b3 being Element of the carrier of TOP-REAL 2 holds
         b2 . b3 = FanN(b1,b3);

:: JGRAPH_4:th 56
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being real set holds
   (b2 <= b1 `1 / |.b1.| & 0 < b1 `2 implies b2 -FanMorphN . b1 = |[|.b1.| * (((b1 `1 / |.b1.|) - b2) / (1 - b2)),|.b1.| * sqrt (1 - ((((b1 `1 / |.b1.|) - b2) / (1 - b2)) ^2))]|) &
    (b1 `2 <= 0 implies b2 -FanMorphN . b1 = b1);

:: JGRAPH_4:th 57
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of REAL
      st b1 `1 / |.b1.| <= b2 & 0 < b1 `2
   holds b2 -FanMorphN . b1 = |[|.b1.| * (((b1 `1 / |.b1.|) - b2) / (1 + b2)),|.b1.| * sqrt (1 - ((((b1 `1 / |.b1.|) - b2) / (1 + b2)) ^2))]|;

:: JGRAPH_4:th 58
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of REAL
      st - 1 < b2 & b2 < 1
   holds (b2 <= b1 `1 / |.b1.| & 0 <= b1 `2 & b1 <> 0.REAL 2 implies b2 -FanMorphN . b1 = |[|.b1.| * (((b1 `1 / |.b1.|) - b2) / (1 - b2)),|.b1.| * sqrt (1 - ((((b1 `1 / |.b1.|) - b2) / (1 - b2)) ^2))]|) &
    (b1 `1 / |.b1.| <= b2 & 0 <= b1 `2 & b1 <> 0.REAL 2 implies b2 -FanMorphN . b1 = |[|.b1.| * (((b1 `1 / |.b1.|) - b2) / (1 + b2)),|.b1.| * sqrt (1 - ((((b1 `1 / |.b1.|) - b2) / (1 + b2)) ^2))]|);

:: JGRAPH_4:th 59
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * (((b4 `1 / |.b4.|) - b1) / (1 - b1))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds 0 <= b4 `2 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 60
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * (((b4 `1 / |.b4.|) - b1) / (1 + b1))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds 0 <= b4 `2 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 61
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * sqrt (1 - ((((b4 `1 / |.b4.|) - b1) / (1 - b1)) ^2))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds 0 <= b4 `2 & b1 <= b4 `1 / |.b4.| & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 62
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * sqrt (1 - ((((b4 `1 / |.b4.|) - b1) / (1 + b1)) ^2))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds 0 <= b4 `2 & b4 `1 / |.b4.| <= b1 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 63
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphN | b2 &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `2 & b5 <> 0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b1 <= b5 `1 / |.b5.| & 0 <= b5 `2 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 64
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphN | b2 &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `2 & b5 <> 0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 / |.b5.| <= b1 & 0 <= b5 `2 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 65
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b1 * |.b3.| <= b3 `1 & 0 <= b3 `2}
   holds b2 is closed(TOP-REAL 2);

:: JGRAPH_4:th 66
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 <= b1 * |.b3.| & 0 <= b3 `2}
   holds b2 is closed(TOP-REAL 2);

:: JGRAPH_4:th 67
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphN | b2 &
         b3 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `2 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 68
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphN | b2 &
         b3 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `2 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 69
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
      st b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: 0 <= b3 `2 & b3 <> 0.REAL 2}
   holds b2 is closed((TOP-REAL 2) | b1);

:: JGRAPH_4:th 70
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
      st b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 <= 0 & b3 <> 0.REAL 2}
   holds b2 is closed((TOP-REAL 2) | b1);

:: JGRAPH_4:th 71
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of (TOP-REAL 2) | b2
for b4 being Function-like quasi_total Relation of the carrier of ((TOP-REAL 2) | b2) | b3,the carrier of (TOP-REAL 2) | b2
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphN | b3 &
         b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `2 & b5 <> 0.REAL 2}
   holds b4 is continuous(((TOP-REAL 2) | b2) | b3, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 72
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of (TOP-REAL 2) | b2
for b4 being Function-like quasi_total Relation of the carrier of ((TOP-REAL 2) | b2) | b3,the carrier of (TOP-REAL 2) | b2
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphN | b3 &
         b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `2 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous(((TOP-REAL 2) | b2) | b3, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 73
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2 holds
   |.b1 -FanMorphN . b2.| = |.b2.|;

:: JGRAPH_4:th 74
theorem
for b1 being Element of REAL
for b2, b3 being set
      st - 1 < b1 &
         b1 < 1 &
         b2 in b3 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: 0 <= b4 `2 & b4 <> 0.REAL 2}
   holds b1 -FanMorphN . b2 in b3;

:: JGRAPH_4:th 75
theorem
for b1 being Element of REAL
for b2, b3 being set
      st - 1 < b1 &
         b1 < 1 &
         b2 in b3 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: b4 `2 <= 0 & b4 <> 0.REAL 2}
   holds b1 -FanMorphN . b2 in b3;

:: JGRAPH_4:th 76
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st - 1 < b1 & b1 < 1 & b2 ` = {0.REAL 2}
   holds ex b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b2 st
      b3 = b1 -FanMorphN | b2 &
       b3 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 77
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds ex b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      b2 = b1 -FanMorphN & b2 is continuous(TOP-REAL 2, TOP-REAL 2);

:: JGRAPH_4:th 78
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds b1 -FanMorphN is one-to-one;

:: JGRAPH_4:th 79
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds b1 -FanMorphN is Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 &
    rng (b1 -FanMorphN) = the carrier of TOP-REAL 2;

:: JGRAPH_4:th 80
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
      st - 1 < b1 & b1 < 1
   holds ex b3 being non empty compact Element of bool the carrier of TOP-REAL 2 st
      b3 = b1 -FanMorphN .: b3 &
       (ex b4 being Element of bool the carrier of TOP-REAL 2 st
          b2 in b4 & b4 is open(TOP-REAL 2) & b4 c= b3 & b1 -FanMorphN . b2 in b4);

:: JGRAPH_4:th 81
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds ex b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      b2 = b1 -FanMorphN & b2 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2);

:: JGRAPH_4:th 82
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & 0 < b2 `2 & b1 <= b2 `1 / |.b2.|
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphN . b2
   holds 0 < b3 `2 & 0 <= b3 `1;

:: JGRAPH_4:th 83
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & 0 < b2 `2 & b2 `1 / |.b2.| < b1
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphN . b2
   holds 0 < b3 `2 & b3 `1 < 0;

:: JGRAPH_4:th 84
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      0 < b2 `2 &
      b1 <= b2 `1 / |.b2.| &
      0 < b3 `2 &
      b1 <= b3 `1 / |.b3.| &
      b2 `1 / |.b2.| < b3 `1 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphN . b2 & b5 = b1 -FanMorphN . b3
   holds b4 `1 / |.b4.| < b5 `1 / |.b5.|;

:: JGRAPH_4:th 85
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      0 < b2 `2 &
      b2 `1 / |.b2.| < b1 &
      0 < b3 `2 &
      b3 `1 / |.b3.| < b1 &
      b2 `1 / |.b2.| < b3 `1 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphN . b2 & b5 = b1 -FanMorphN . b3
   holds b4 `1 / |.b4.| < b5 `1 / |.b5.|;

:: JGRAPH_4:th 86
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      0 < b2 `2 &
      0 < b3 `2 &
      b2 `1 / |.b2.| < b3 `1 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphN . b2 & b5 = b1 -FanMorphN . b3
   holds b4 `1 / |.b4.| < b5 `1 / |.b5.|;

:: JGRAPH_4:th 87
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & 0 < b2 `2 & b2 `1 / |.b2.| = b1
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphN . b2
   holds 0 < b3 `2 & b3 `1 = 0;

:: JGRAPH_4:th 88
theorem
for b1 being real set holds
   0.REAL 2 = b1 -FanMorphN . 0.REAL 2;

:: JGRAPH_4:funcnot 6 => JGRAPH_4:func 6
definition
  let a1 be real set;
  let a2 be Element of the carrier of TOP-REAL 2;
  func FanE(A1,A2) -> Element of the carrier of TOP-REAL 2 equals
    |.a2.| * |[sqrt (1 - ((((a2 `2 / |.a2.|) - a1) / (1 - a1)) ^2)),((a2 `2 / |.a2.|) - a1) / (1 - a1)]|
    if a1 <= a2 `2 / |.a2.| & 0 < a2 `1,
|.a2.| * |[sqrt (1 - ((((a2 `2 / |.a2.|) - a1) / (1 + a1)) ^2)),((a2 `2 / |.a2.|) - a1) / (1 + a1)]|
    if a2 `2 / |.a2.| < a1 & 0 < a2 `1
    otherwise a2;
end;

:: JGRAPH_4:def 6
theorem
for b1 being real set
for b2 being Element of the carrier of TOP-REAL 2 holds
   (b1 <= b2 `2 / |.b2.| & 0 < b2 `1 implies FanE(b1,b2) = |.b2.| * |[sqrt (1 - ((((b2 `2 / |.b2.|) - b1) / (1 - b1)) ^2)),((b2 `2 / |.b2.|) - b1) / (1 - b1)]|) &
    (b2 `2 / |.b2.| < b1 & 0 < b2 `1 implies FanE(b1,b2) = |.b2.| * |[sqrt (1 - ((((b2 `2 / |.b2.|) - b1) / (1 + b1)) ^2)),((b2 `2 / |.b2.|) - b1) / (1 + b1)]|) &
    ((b1 <= b2 `2 / |.b2.| implies b2 `1 <= 0) &
     (b1 <= b2 `2 / |.b2.| or b2 `1 <= 0) implies FanE(b1,b2) = b2);

:: JGRAPH_4:funcnot 7 => JGRAPH_4:func 7
definition
  let a1 be real set;
  func A1 -FanMorphE -> 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 = FanE(a1,b1);
end;

:: JGRAPH_4:def 7
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 holds
      b2 = b1 -FanMorphE
   iff
      for b3 being Element of the carrier of TOP-REAL 2 holds
         b2 . b3 = FanE(b1,b3);

:: JGRAPH_4:th 89
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being real set holds
   (b2 <= b1 `2 / |.b1.| & 0 < b1 `1 implies b2 -FanMorphE . b1 = |[|.b1.| * sqrt (1 - ((((b1 `2 / |.b1.|) - b2) / (1 - b2)) ^2)),|.b1.| * (((b1 `2 / |.b1.|) - b2) / (1 - b2))]|) &
    (b1 `1 <= 0 implies b2 -FanMorphE . b1 = b1);

:: JGRAPH_4:th 90
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of REAL
      st b1 `2 / |.b1.| <= b2 & 0 < b1 `1
   holds b2 -FanMorphE . b1 = |[|.b1.| * sqrt (1 - ((((b1 `2 / |.b1.|) - b2) / (1 + b2)) ^2)),|.b1.| * (((b1 `2 / |.b1.|) - b2) / (1 + b2))]|;

:: JGRAPH_4:th 91
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of REAL
      st - 1 < b2 & b2 < 1
   holds (b2 <= b1 `2 / |.b1.| & 0 <= b1 `1 & b1 <> 0.REAL 2 implies b2 -FanMorphE . b1 = |[|.b1.| * sqrt (1 - ((((b1 `2 / |.b1.|) - b2) / (1 - b2)) ^2)),|.b1.| * (((b1 `2 / |.b1.|) - b2) / (1 - b2))]|) &
    (b1 `2 / |.b1.| <= b2 & 0 <= b1 `1 & b1 <> 0.REAL 2 implies b2 -FanMorphE . b1 = |[|.b1.| * sqrt (1 - ((((b1 `2 / |.b1.|) - b2) / (1 + b2)) ^2)),|.b1.| * (((b1 `2 / |.b1.|) - b2) / (1 + b2))]|);

:: JGRAPH_4:th 92
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * (((b4 `2 / |.b4.|) - b1) / (1 - b1))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds 0 <= b4 `1 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 93
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * (((b4 `2 / |.b4.|) - b1) / (1 + b1))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds 0 <= b4 `1 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 94
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * sqrt (1 - ((((b4 `2 / |.b4.|) - b1) / (1 - b1)) ^2))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds 0 <= b4 `1 & b1 <= b4 `2 / |.b4.| & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 95
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * sqrt (1 - ((((b4 `2 / |.b4.|) - b1) / (1 + b1)) ^2))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds 0 <= b4 `1 & b4 `2 / |.b4.| <= b1 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 96
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphE | b2 &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `1 & b5 <> 0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b1 <= b5 `2 / |.b5.| & 0 <= b5 `1 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 97
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphE | b2 &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `1 & b5 <> 0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `2 / |.b5.| <= b1 & 0 <= b5 `1 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 98
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b1 * |.b3.| <= b3 `2 & 0 <= b3 `1}
   holds b2 is closed(TOP-REAL 2);

:: JGRAPH_4:th 99
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `2 <= b1 * |.b3.| & 0 <= b3 `1}
   holds b2 is closed(TOP-REAL 2);

:: JGRAPH_4:th 100
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphE | b2 &
         b3 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `1 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 101
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphE | b2 &
         b3 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 102
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of (TOP-REAL 2) | b2
for b4 being Function-like quasi_total Relation of the carrier of ((TOP-REAL 2) | b2) | b3,the carrier of (TOP-REAL 2) | b2
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphE | b3 &
         b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `1 & b5 <> 0.REAL 2}
   holds b4 is continuous(((TOP-REAL 2) | b2) | b3, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 103
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of (TOP-REAL 2) | b2
for b4 being Function-like quasi_total Relation of the carrier of ((TOP-REAL 2) | b2) | b3,the carrier of (TOP-REAL 2) | b2
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphE | b3 &
         b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous(((TOP-REAL 2) | b2) | b3, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 104
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2 holds
   |.b1 -FanMorphE . b2.| = |.b2.|;

:: JGRAPH_4:th 105
theorem
for b1 being Element of REAL
for b2, b3 being set
      st - 1 < b1 &
         b1 < 1 &
         b2 in b3 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: 0 <= b4 `1 & b4 <> 0.REAL 2}
   holds b1 -FanMorphE . b2 in b3;

:: JGRAPH_4:th 106
theorem
for b1 being Element of REAL
for b2, b3 being set
      st - 1 < b1 &
         b1 < 1 &
         b2 in b3 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: b4 `1 <= 0 & b4 <> 0.REAL 2}
   holds b1 -FanMorphE . b2 in b3;

:: JGRAPH_4:th 107
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st - 1 < b1 & b1 < 1 & b2 ` = {0.REAL 2}
   holds ex b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b2 st
      b3 = b1 -FanMorphE | b2 &
       b3 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 108
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds ex b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      b2 = b1 -FanMorphE & b2 is continuous(TOP-REAL 2, TOP-REAL 2);

:: JGRAPH_4:th 109
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds b1 -FanMorphE is one-to-one;

:: JGRAPH_4:th 110
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds b1 -FanMorphE is Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 &
    rng (b1 -FanMorphE) = the carrier of TOP-REAL 2;

:: JGRAPH_4:th 111
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
      st - 1 < b1 & b1 < 1
   holds ex b3 being non empty compact Element of bool the carrier of TOP-REAL 2 st
      b3 = b1 -FanMorphE .: b3 &
       (ex b4 being Element of bool the carrier of TOP-REAL 2 st
          b2 in b4 & b4 is open(TOP-REAL 2) & b4 c= b3 & b1 -FanMorphE . b2 in b4);

:: JGRAPH_4:th 112
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds ex b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      b2 = b1 -FanMorphE & b2 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2);

:: JGRAPH_4:th 113
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & 0 < b2 `1 & b1 <= b2 `2 / |.b2.|
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphE . b2
   holds 0 < b3 `1 & 0 <= b3 `2;

:: JGRAPH_4:th 114
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & 0 < b2 `1 & b2 `2 / |.b2.| < b1
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphE . b2
   holds 0 < b3 `1 & b3 `2 < 0;

:: JGRAPH_4:th 115
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      0 < b2 `1 &
      b1 <= b2 `2 / |.b2.| &
      0 < b3 `1 &
      b1 <= b3 `2 / |.b3.| &
      b2 `2 / |.b2.| < b3 `2 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphE . b2 & b5 = b1 -FanMorphE . b3
   holds b4 `2 / |.b4.| < b5 `2 / |.b5.|;

:: JGRAPH_4:th 116
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      0 < b2 `1 &
      b2 `2 / |.b2.| < b1 &
      0 < b3 `1 &
      b3 `2 / |.b3.| < b1 &
      b2 `2 / |.b2.| < b3 `2 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphE . b2 & b5 = b1 -FanMorphE . b3
   holds b4 `2 / |.b4.| < b5 `2 / |.b5.|;

:: JGRAPH_4:th 117
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      0 < b2 `1 &
      0 < b3 `1 &
      b2 `2 / |.b2.| < b3 `2 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphE . b2 & b5 = b1 -FanMorphE . b3
   holds b4 `2 / |.b4.| < b5 `2 / |.b5.|;

:: JGRAPH_4:th 118
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & 0 < b2 `1 & b2 `2 / |.b2.| = b1
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphE . b2
   holds 0 < b3 `1 & b3 `2 = 0;

:: JGRAPH_4:th 119
theorem
for b1 being real set holds
   0.REAL 2 = b1 -FanMorphE . 0.REAL 2;

:: JGRAPH_4:funcnot 8 => JGRAPH_4:func 8
definition
  let a1 be real set;
  let a2 be Element of the carrier of TOP-REAL 2;
  func FanS(A1,A2) -> Element of the carrier of TOP-REAL 2 equals
    |.a2.| * |[((a2 `1 / |.a2.|) - a1) / (1 - a1),- sqrt (1 - ((((a2 `1 / |.a2.|) - a1) / (1 - a1)) ^2))]|
    if a1 <= a2 `1 / |.a2.| & a2 `2 < 0,
|.a2.| * |[((a2 `1 / |.a2.|) - a1) / (1 + a1),- sqrt (1 - ((((a2 `1 / |.a2.|) - a1) / (1 + a1)) ^2))]|
    if a2 `1 / |.a2.| < a1 & a2 `2 < 0
    otherwise a2;
end;

:: JGRAPH_4:def 8
theorem
for b1 being real set
for b2 being Element of the carrier of TOP-REAL 2 holds
   (b1 <= b2 `1 / |.b2.| & b2 `2 < 0 implies FanS(b1,b2) = |.b2.| * |[((b2 `1 / |.b2.|) - b1) / (1 - b1),- sqrt (1 - ((((b2 `1 / |.b2.|) - b1) / (1 - b1)) ^2))]|) &
    (b2 `1 / |.b2.| < b1 & b2 `2 < 0 implies FanS(b1,b2) = |.b2.| * |[((b2 `1 / |.b2.|) - b1) / (1 + b1),- sqrt (1 - ((((b2 `1 / |.b2.|) - b1) / (1 + b1)) ^2))]|) &
    ((b1 <= b2 `1 / |.b2.| implies 0 <= b2 `2) &
     (b1 <= b2 `1 / |.b2.| or 0 <= b2 `2) implies FanS(b1,b2) = b2);

:: JGRAPH_4:funcnot 9 => JGRAPH_4:func 9
definition
  let a1 be real set;
  func A1 -FanMorphS -> 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 = FanS(a1,b1);
end;

:: JGRAPH_4:def 9
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 holds
      b2 = b1 -FanMorphS
   iff
      for b3 being Element of the carrier of TOP-REAL 2 holds
         b2 . b3 = FanS(b1,b3);

:: JGRAPH_4:th 120
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being real set holds
   (b2 <= b1 `1 / |.b1.| & b1 `2 < 0 implies b2 -FanMorphS . b1 = |[|.b1.| * (((b1 `1 / |.b1.|) - b2) / (1 - b2)),|.b1.| * - sqrt (1 - ((((b1 `1 / |.b1.|) - b2) / (1 - b2)) ^2))]|) &
    (0 <= b1 `2 implies b2 -FanMorphS . b1 = b1);

:: JGRAPH_4:th 121
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of REAL
      st b1 `1 / |.b1.| <= b2 & b1 `2 < 0
   holds b2 -FanMorphS . b1 = |[|.b1.| * (((b1 `1 / |.b1.|) - b2) / (1 + b2)),|.b1.| * - sqrt (1 - ((((b1 `1 / |.b1.|) - b2) / (1 + b2)) ^2))]|;

:: JGRAPH_4:th 122
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of REAL
      st - 1 < b2 & b2 < 1
   holds (b2 <= b1 `1 / |.b1.| & b1 `2 <= 0 & b1 <> 0.REAL 2 implies b2 -FanMorphS . b1 = |[|.b1.| * (((b1 `1 / |.b1.|) - b2) / (1 - b2)),|.b1.| * - sqrt (1 - ((((b1 `1 / |.b1.|) - b2) / (1 - b2)) ^2))]|) &
    (b1 `1 / |.b1.| <= b2 & b1 `2 <= 0 & b1 <> 0.REAL 2 implies b2 -FanMorphS . b1 = |[|.b1.| * (((b1 `1 / |.b1.|) - b2) / (1 + b2)),|.b1.| * - sqrt (1 - ((((b1 `1 / |.b1.|) - b2) / (1 + b2)) ^2))]|);

:: JGRAPH_4:th 123
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * (((b4 `1 / |.b4.|) - b1) / (1 - b1))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b4 `2 <= 0 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 124
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * (((b4 `1 / |.b4.|) - b1) / (1 + b1))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b4 `2 <= 0 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 125
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * - sqrt (1 - ((((b4 `1 / |.b4.|) - b1) / (1 - b1)) ^2))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b4 `2 <= 0 & b1 <= b4 `1 / |.b4.| & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 126
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of R^1
      st - 1 < b1 &
         b1 < 1 &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b3 . b4 = |.b4.| * - sqrt (1 - ((((b4 `1 / |.b4.|) - b1) / (1 + b1)) ^2))) &
         (for b4 being Element of the carrier of TOP-REAL 2
               st b4 in the carrier of (TOP-REAL 2) | b2
            holds b4 `2 <= 0 & b4 `1 / |.b4.| <= b1 & b4 <> 0.REAL 2)
   holds b3 is continuous((TOP-REAL 2) | b2, R^1);

:: JGRAPH_4:th 127
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphS | b2 &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `2 <= 0 & b5 <> 0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b1 <= b5 `1 / |.b5.| & b5 `2 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 128
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphS | b2 &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `2 <= 0 & b5 <> 0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 / |.b5.| <= b1 & b5 `2 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 129
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b1 * |.b3.| <= b3 `1 & b3 `2 <= 0}
   holds b2 is closed(TOP-REAL 2);

:: JGRAPH_4:th 130
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: b3 `1 <= b1 * |.b3.| & b3 `2 <= 0}
   holds b2 is closed(TOP-REAL 2);

:: JGRAPH_4:th 131
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphS | b2 &
         b3 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `2 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 132
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b3
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphS | b2 &
         b3 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `2 & b5 <> 0.REAL 2}
   holds b4 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b3);

:: JGRAPH_4:th 133
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of (TOP-REAL 2) | b2
for b4 being Function-like quasi_total Relation of the carrier of ((TOP-REAL 2) | b2) | b3,the carrier of (TOP-REAL 2) | b2
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphS | b3 &
         b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `2 <= 0 & b5 <> 0.REAL 2}
   holds b4 is continuous(((TOP-REAL 2) | b2) | b3, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 134
theorem
for b1 being Element of REAL
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of (TOP-REAL 2) | b2
for b4 being Function-like quasi_total Relation of the carrier of ((TOP-REAL 2) | b2) | b3,the carrier of (TOP-REAL 2) | b2
      st - 1 < b1 &
         b1 < 1 &
         b4 = b1 -FanMorphS | b3 &
         b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
         b3 = {b5 where b5 is Element of the carrier of TOP-REAL 2: 0 <= b5 `2 & b5 <> 0.REAL 2}
   holds b4 is continuous(((TOP-REAL 2) | b2) | b3, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 135
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2 holds
   |.b1 -FanMorphS . b2.| = |.b2.|;

:: JGRAPH_4:th 136
theorem
for b1 being Element of REAL
for b2, b3 being set
      st - 1 < b1 &
         b1 < 1 &
         b2 in b3 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: b4 `2 <= 0 & b4 <> 0.REAL 2}
   holds b1 -FanMorphS . b2 in b3;

:: JGRAPH_4:th 137
theorem
for b1 being Element of REAL
for b2, b3 being set
      st - 1 < b1 &
         b1 < 1 &
         b2 in b3 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: 0 <= b4 `2 & b4 <> 0.REAL 2}
   holds b1 -FanMorphS . b2 in b3;

:: JGRAPH_4:th 138
theorem
for b1 being Element of REAL
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st - 1 < b1 & b1 < 1 & b2 ` = {0.REAL 2}
   holds ex b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b2 st
      b3 = b1 -FanMorphS | b2 &
       b3 is continuous((TOP-REAL 2) | b2, (TOP-REAL 2) | b2);

:: JGRAPH_4:th 139
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds b1 -FanMorphS is continuous(TOP-REAL 2, TOP-REAL 2);

:: JGRAPH_4:th 140
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds b1 -FanMorphS is one-to-one;

:: JGRAPH_4:th 141
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds b1 -FanMorphS is Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 &
    rng (b1 -FanMorphS) = the carrier of TOP-REAL 2;

:: JGRAPH_4:th 142
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
      st - 1 < b1 & b1 < 1
   holds ex b3 being non empty compact Element of bool the carrier of TOP-REAL 2 st
      b3 = b1 -FanMorphS .: b3 &
       (ex b4 being Element of bool the carrier of TOP-REAL 2 st
          b2 in b4 & b4 is open(TOP-REAL 2) & b4 c= b3 & b1 -FanMorphS . b2 in b4);

:: JGRAPH_4:th 143
theorem
for b1 being Element of REAL
      st - 1 < b1 & b1 < 1
   holds ex b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      b2 = b1 -FanMorphS & b2 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2);

:: JGRAPH_4:th 144
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & b2 `2 < 0 & b1 <= b2 `1 / |.b2.|
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphS . b2
   holds b3 `2 < 0 & 0 <= b3 `1;

:: JGRAPH_4:th 145
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & b2 `2 < 0 & b2 `1 / |.b2.| < b1
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphS . b2
   holds b3 `2 < 0 & b3 `1 < 0;

:: JGRAPH_4:th 146
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      b2 `2 < 0 &
      b1 <= b2 `1 / |.b2.| &
      b3 `2 < 0 &
      b1 <= b3 `1 / |.b3.| &
      b2 `1 / |.b2.| < b3 `1 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphS . b2 & b5 = b1 -FanMorphS . b3
   holds b4 `1 / |.b4.| < b5 `1 / |.b5.|;

:: JGRAPH_4:th 147
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      b2 `2 < 0 &
      b2 `1 / |.b2.| < b1 &
      b3 `2 < 0 &
      b3 `1 / |.b3.| < b1 &
      b2 `1 / |.b2.| < b3 `1 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphS . b2 & b5 = b1 -FanMorphS . b3
   holds b4 `1 / |.b4.| < b5 `1 / |.b5.|;

:: JGRAPH_4:th 148
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 &
      b1 < 1 &
      b2 `2 < 0 &
      b3 `2 < 0 &
      b2 `1 / |.b2.| < b3 `1 / |.b3.|
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 = b1 -FanMorphS . b2 & b5 = b1 -FanMorphS . b3
   holds b4 `1 / |.b4.| < b5 `1 / |.b5.|;

:: JGRAPH_4:th 149
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2
   st - 1 < b1 & b1 < 1 & b2 `2 < 0 & b2 `1 / |.b2.| = b1
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphS . b2
   holds b3 `2 < 0 & b3 `1 = 0;

:: JGRAPH_4:th 150
theorem
for b1 being real set holds
   0.REAL 2 = b1 -FanMorphS . 0.REAL 2;