Article JGRAPH_5, MML version 4.99.1005

:: JGRAPH_5:th 3
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st |.b1.| <= 1
   holds - 1 <= b1 `1 & b1 `1 <= 1 & - 1 <= b1 `2 & b1 `2 <= 1;

:: JGRAPH_5:th 4
theorem
for b1 being Element of the carrier of TOP-REAL 2
      st |.b1.| <= 1 & b1 `1 <> 0 & b1 `2 <> 0
   holds - 1 < b1 `1 & b1 `1 < 1 & - 1 < b1 `2 & b1 `2 < 1;

:: JGRAPH_5:th 5
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6, b7 being non empty MetrStruct
for b8 being Element of the carrier of b6
for b9 being Element of the carrier of b7
      st b3 <= b1 & b1 <= b2 & b2 <= b4 & b6 = Closed-Interval-MSpace(b1,b2) & b7 = Closed-Interval-MSpace(b3,b4) & b8 = b9 & b8 in the carrier of b6 & b9 in the carrier of b7
   holds Ball(b8,b5) c= Ball(b9,b5);

:: JGRAPH_5:th 7
theorem
for b1, b2 being real set
for b3 being Element of bool the carrier of I[01]
      st 0 <= b1 & b1 <= b2 & b2 <= 1 & b3 = [.b1,b2.]
   holds Closed-Interval-TSpace(b1,b2) = I[01] | b3;

:: JGRAPH_5:th 8
theorem
for b1 being TopStruct
for b2, b3 being non empty TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
      st b5 is being_homeomorphism(b2, b3) & b4 is continuous(b1, b2)
   holds b5 * b4 is continuous(b1, b3);

:: JGRAPH_5:th 9
theorem
for b1, b2, b3 being TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
      st b5 is being_homeomorphism(b2, b3) & b4 is one-to-one
   holds b5 * b4 is one-to-one;

:: JGRAPH_5:th 10
theorem
for b1 being TopStruct
for b2, b3 being non empty TopStruct
for b4 being non empty Element of bool the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 | b4
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b7 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
      st b7 = b6 * b5 & b5 is continuous(b1, b2 | b4) & b6 is continuous(b2, b3)
   holds b7 is continuous(b1, b3);

:: JGRAPH_5:th 11
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of REAL
for b9 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
      st b9 is being_homeomorphism(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b9 . b5 = b7 & b9 . b6 = b8 & b9 . b1 = b3 & b9 . b2 = b4 & b3 <= b4 & b7 <= b8 & b5 in [.b1,b2.] & b6 in [.b1,b2.]
   holds b5 <= b6;

:: JGRAPH_5:th 12
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of REAL
for b9 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
      st b9 is being_homeomorphism(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b9 . b5 = b7 & b9 . b6 = b8 & b9 . b1 = b4 & b9 . b2 = b3 & b3 <= b4 & b8 <= b7 & b5 in [.b1,b2.] & b6 in [.b1,b2.]
   holds b5 <= b6;

:: JGRAPH_5:th 13
theorem
for b1 being Element of NAT holds
   - 0.REAL b1 = 0.REAL b1;

:: JGRAPH_5:th 14
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 Element of REAL
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 &
         b3 <> b4 &
         b5 <> b6 &
         (b1 . b7) `1 = b3 &
         b5 <= (b1 . b7) `2 &
         (b1 . b7) `2 <= b6 &
         (b1 . b8) `1 = b4 &
         b5 <= (b1 . b8) `2 &
         (b1 . b8) `2 <= b6 &
         (b2 . b7) `2 = b5 &
         b3 <= (b2 . b7) `1 &
         (b2 . b7) `1 <= b4 &
         (b2 . b8) `2 = b6 &
         b3 <= (b2 . b8) `1 &
         (b2 . b8) `1 <= b4 &
         (for b9 being Element of the carrier of I[01] holds
            (b3 < (b1 . b9) `1 & (b1 . b9) `1 < b4 & b5 < (b1 . b9) `2 implies b6 <= (b1 . b9) `2) &
             (b3 < (b2 . b9) `1 & (b2 . b9) `1 < b4 & b5 < (b2 . b9) `2 implies b6 <= (b2 . b9) `2))
   holds rng b1 meets rng b2;

:: JGRAPH_5:th 15
theorem
for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b1 is continuous(I[01], TOP-REAL 2) & b1 is one-to-one
   holds ex b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2 st
      b2 . 0 = b1 . 1 & b2 . 1 = b1 . 0 & rng b2 = rng b1 & b2 is continuous(I[01], TOP-REAL 2) & b2 is one-to-one;

:: JGRAPH_5:th 16
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, b7 being Element of bool the carrier of TOP-REAL 2
for b8, b9 being Element of the carrier of I[01]
      st b8 = 0 &
         b9 = 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 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| <= 1} &
         b4 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `2 <= b10 `1 & - (b10 `1) <= b10 `2} &
         b5 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `1 <= b10 `2 & b10 `2 <= - (b10 `1)} &
         b6 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `1 <= b10 `2 & - (b10 `1) <= b10 `2} &
         b7 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `2 <= b10 `1 & b10 `2 <= - (b10 `1)} &
         b1 . b8 in b5 &
         b1 . b9 in b4 &
         b2 . b8 in b6 &
         b2 . b9 in b7 &
         rng b1 c= b3 &
         rng b2 c= b3
   holds rng b1 meets rng b2;

:: JGRAPH_5:th 17
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, b7 being Element of bool the carrier of TOP-REAL 2
for b8, b9 being Element of the carrier of I[01]
      st b8 = 0 &
         b9 = 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 = {b10 where b10 is Element of the carrier of TOP-REAL 2: 1 <= |.b10.|} &
         b4 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `2 <= b10 `1 & - (b10 `1) <= b10 `2} &
         b5 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `1 <= b10 `2 & b10 `2 <= - (b10 `1)} &
         b6 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `1 <= b10 `2 & - (b10 `1) <= b10 `2} &
         b7 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `2 <= b10 `1 & b10 `2 <= - (b10 `1)} &
         b1 . b8 in b5 &
         b1 . b9 in b4 &
         b2 . b8 in b7 &
         b2 . b9 in b6 &
         rng b1 c= b3 &
         rng b2 c= b3
   holds rng b1 meets rng b2;

:: JGRAPH_5:th 18
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, b7 being Element of bool the carrier of TOP-REAL 2
for b8, b9 being Element of the carrier of I[01]
      st b8 = 0 &
         b9 = 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 = {b10 where b10 is Element of the carrier of TOP-REAL 2: 1 <= |.b10.|} &
         b4 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `2 <= b10 `1 & - (b10 `1) <= b10 `2} &
         b5 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `1 <= b10 `2 & b10 `2 <= - (b10 `1)} &
         b6 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `1 <= b10 `2 & - (b10 `1) <= b10 `2} &
         b7 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `2 <= b10 `1 & b10 `2 <= - (b10 `1)} &
         b1 . b8 in b5 &
         b1 . b9 in b4 &
         b2 . b8 in b6 &
         b2 . b9 in b7 &
         rng b1 c= b3 &
         rng b2 c= b3
   holds rng b1 meets rng b2;

:: JGRAPH_5:th 19
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
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: 1 <= |.b4.|} &
         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 . 0 = |[- 1,0]| &
         b1 . 1 = |[1,0]| &
         b2 . 1 = |[0,1]| &
         b2 . 0 = |[0,- 1]| &
         rng b1 c= b3 &
         rng b2 c= b3
   holds rng b1 meets rng b2;

:: JGRAPH_5:th 20
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being Element of bool the carrier of TOP-REAL 2
   st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: 1 <= |.b6.|} &
      |.b1.| = 1 &
      |.b2.| = 1 &
      |.b3.| = 1 &
      |.b4.| = 1 &
      (ex b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
         b6 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
          b6 .: b5 c= b5 &
          b6 . b1 = |[- 1,0]| &
          b6 . b2 = |[0,1]| &
          b6 . b3 = |[1,0]| &
          b6 . b4 = |[0,- 1]|)
for b6, b7 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b6 is continuous(I[01], TOP-REAL 2) & b6 is one-to-one & b7 is continuous(I[01], TOP-REAL 2) & b7 is one-to-one & b6 . 0 = b1 & b6 . 1 = b3 & b7 . 0 = b4 & b7 . 1 = b2 & rng b6 c= b5 & rng b7 c= b5
   holds rng b6 meets rng b7;

:: JGRAPH_5:th 21
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphN . b2
   holds 0 < b3 `2;

:: JGRAPH_5:th 22
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphN . b2
   holds 0 <= b3 `2;

:: JGRAPH_5:th 23
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 & |.b2.| <> 0
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphN . b2
   holds 0 <= b3 `2 & b3 `1 < 0;

:: JGRAPH_5:th 24
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.| <> 0 &
      |.b3.| <> 0 &
      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_5:th 25
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphE . b2
   holds 0 < b3 `1;

:: JGRAPH_5:th 26
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 & |.b2.| <> 0
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphE . b2
   holds 0 <= b3 `1 & b3 `2 < 0;

:: JGRAPH_5:th 27
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.| <> 0 &
      |.b3.| <> 0 &
      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_5:th 28
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphS . b2
   holds b3 `2 < 0;

:: JGRAPH_5:th 29
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_5:th 30
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.| <> 0 &
      |.b3.| <> 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_5:th 31
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
   holds W-bound b1 = - 1 & E-bound b1 = 1 & S-bound b1 = - 1 & N-bound b1 = 1;

:: JGRAPH_5:th 32
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
   holds W-min b1 = |[- 1,0]|;

:: JGRAPH_5:th 33
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
   holds E-max b1 = |[1,0]|;

:: JGRAPH_5:th 34
theorem
for b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of R^1
      st for b2 being Element of the carrier of TOP-REAL 2 holds
           b1 . b2 = proj1 . b2
   holds b1 is continuous(TOP-REAL 2, R^1);

:: JGRAPH_5:th 35
theorem
for b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of R^1
      st for b2 being Element of the carrier of TOP-REAL 2 holds
           b1 . b2 = proj2 . b2
   holds b1 is continuous(TOP-REAL 2, R^1);

:: JGRAPH_5:th 36
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
   holds Upper_Arc b1 c= b1 & Lower_Arc b1 c= b1;

:: JGRAPH_5:th 37
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
   holds Upper_Arc b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 in b1 & 0 <= b2 `2};

:: JGRAPH_5:th 38
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
   holds Lower_Arc b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 in b1 & b2 `2 <= 0};

:: JGRAPH_5:th 39
theorem
for b1, b2, b3, b4 being Element of REAL
      st b1 <= b2 & 0 < b4
   holds ex b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace((b4 * b1) + b3,(b4 * b2) + b3) st
      b5 is being_homeomorphism(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace((b4 * b1) + b3,(b4 * b2) + b3)) &
       (for b6 being Element of REAL
             st b6 in [.b1,b2.]
          holds b5 . b6 = (b4 * b6) + b3);

:: JGRAPH_5:th 40
theorem
for b1, b2, b3, b4 being Element of REAL
      st b1 <= b2 & b4 < 0
   holds ex b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace((b4 * b2) + b3,(b4 * b1) + b3) st
      b5 is being_homeomorphism(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace((b4 * b2) + b3,(b4 * b1) + b3)) &
       (for b6 being Element of REAL
             st b6 in [.b1,b2.]
          holds b5 . b6 = (b4 * b6) + b3);

:: JGRAPH_5:th 41
theorem
ex b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of Closed-Interval-TSpace(- 1,1) st
   b1 is being_homeomorphism(I[01], Closed-Interval-TSpace(- 1,1)) &
    (for b2 being Element of REAL
          st b2 in [.0,1.]
       holds b1 . b2 = ((- 2) * b2) + 1) &
    b1 . 0 = 1 &
    b1 . 1 = - 1;

:: JGRAPH_5:th 42
theorem
ex b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of Closed-Interval-TSpace(- 1,1) st
   b1 is being_homeomorphism(I[01], Closed-Interval-TSpace(- 1,1)) &
    (for b2 being Element of REAL
          st b2 in [.0,1.]
       holds b1 . b2 = (2 * b2) - 1) &
    b1 . 0 = - 1 &
    b1 . 1 = 1;

:: JGRAPH_5:th 43
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
   holds ex b2 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(- 1,1),the carrier of (TOP-REAL 2) | Lower_Arc b1 st
      b2 is being_homeomorphism(Closed-Interval-TSpace(- 1,1), (TOP-REAL 2) | Lower_Arc b1) &
       (for b3 being Element of the carrier of TOP-REAL 2
             st b3 in Lower_Arc b1
          holds b2 . (b3 `1) = b3) &
       b2 . - 1 = W-min b1 &
       b2 . 1 = E-max b1;

:: JGRAPH_5:th 44
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
   holds ex b2 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(- 1,1),the carrier of (TOP-REAL 2) | Upper_Arc b1 st
      b2 is being_homeomorphism(Closed-Interval-TSpace(- 1,1), (TOP-REAL 2) | Upper_Arc b1) &
       (for b3 being Element of the carrier of TOP-REAL 2
             st b3 in Upper_Arc b1
          holds b2 . (b3 `1) = b3) &
       b2 . - 1 = W-min b1 &
       b2 . 1 = E-max b1;

:: JGRAPH_5:th 45
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
   holds ex b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | Lower_Arc b1 st
      b2 is being_homeomorphism(I[01], (TOP-REAL 2) | Lower_Arc b1) &
       (for b3, b4 being Element of the carrier of TOP-REAL 2
       for b5, b6 being Element of REAL
             st b2 . b5 = b3 & b2 . b6 = b4 & b5 in [.0,1.] & b6 in [.0,1.]
          holds    b5 < b6
          iff
             b4 `1 < b3 `1) &
       b2 . 0 = E-max b1 &
       b2 . 1 = W-min b1;

:: JGRAPH_5:th 46
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
   holds ex b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | Upper_Arc b1 st
      b2 is being_homeomorphism(I[01], (TOP-REAL 2) | Upper_Arc b1) &
       (for b3, b4 being Element of the carrier of TOP-REAL 2
       for b5, b6 being Element of REAL
             st b2 . b5 = b3 & b2 . b6 = b4 & b5 in [.0,1.] & b6 in [.0,1.]
          holds    b5 < b6
          iff
             b3 `1 < b4 `1) &
       b2 . 0 = W-min b1 &
       b2 . 1 = E-max b1;

:: JGRAPH_5:th 47
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         b2 in Upper_Arc b3 &
         LE b1,b2,b3
   holds b1 in Upper_Arc b3;

:: JGRAPH_5:th 48
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         LE b1,b2,b3 &
         b1 <> b2 &
         b1 `1 < 0 &
         b2 `1 < 0 &
         b1 `2 < 0 &
         b2 `2 < 0
   holds b2 `1 < b1 `1 & b1 `2 < b2 `2;

:: JGRAPH_5:th 49
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         LE b1,b2,b3 &
         b1 <> b2 &
         b1 `1 < 0 &
         b2 `1 < 0 &
         0 <= b1 `2 &
         0 <= b2 `2
   holds b1 `1 < b2 `1 & b1 `2 < b2 `2;

:: JGRAPH_5:th 50
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         LE b1,b2,b3 &
         b1 <> b2 &
         0 <= b1 `2 &
         0 <= b2 `2
   holds b1 `1 < b2 `1;

:: JGRAPH_5:th 51
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         LE b1,b2,b3 &
         b1 <> b2 &
         b1 `2 <= 0 &
         b2 `2 <= 0 &
         b1 <> W-min b3
   holds b2 `1 < b1 `1;

:: JGRAPH_5:th 52
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         (0 <= b2 `2 or 0 <= b2 `1) &
         LE b1,b2,b3 &
         b1 `2 < 0
   holds 0 <= b1 `1;

:: JGRAPH_5:th 53
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         LE b1,b2,b3 &
         b1 <> b2 &
         0 <= b1 `1 &
         0 <= b2 `1
   holds b2 `2 < b1 `2;

:: JGRAPH_5:th 54
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         b1 in b3 &
         b2 in b3 &
         b1 `1 < 0 &
         b2 `1 < 0 &
         b1 `2 < 0 &
         b2 `2 < 0 &
         (b2 `1 <= b1 `1 or b1 `2 <= b2 `2)
   holds LE b1,b2,b3;

:: JGRAPH_5:th 55
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         b1 in b3 &
         b2 in b3 &
         0 < b1 `1 &
         0 < b2 `1 &
         b1 `2 < 0 &
         b2 `2 < 0 &
         (b2 `1 <= b1 `1 or b2 `2 <= b1 `2)
   holds LE b1,b2,b3;

:: JGRAPH_5:th 56
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         b1 in b3 &
         b2 in b3 &
         b1 `1 < 0 &
         b2 `1 < 0 &
         0 <= b1 `2 &
         0 <= b2 `2 &
         (b1 `1 <= b2 `1 or b1 `2 <= b2 `2)
   holds LE b1,b2,b3;

:: JGRAPH_5:th 57
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         b1 in b3 &
         b2 in b3 &
         0 <= b1 `2 &
         0 <= b2 `2 &
         b1 `1 <= b2 `1
   holds LE b1,b2,b3;

:: JGRAPH_5:th 58
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         b1 in b3 &
         b2 in b3 &
         0 <= b1 `1 &
         0 <= b2 `1 &
         b2 `2 <= b1 `2
   holds LE b1,b2,b3;

:: JGRAPH_5:th 59
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4.| = 1} &
         b1 in b3 &
         b2 in b3 &
         b1 `2 <= 0 &
         b2 `2 <= 0 &
         b2 <> W-min b3 &
         b2 `1 <= b1 `1
   holds LE b1,b2,b3;

:: JGRAPH_5:th 60
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 = b1 -FanMorphS . b2
   holds b3 `2 <= 0;

:: JGRAPH_5:th 61
theorem
for b1 being Element of REAL
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
for b6 being non empty compact Element of bool the carrier of TOP-REAL 2
      st - 1 < b1 &
         b1 < 1 &
         b6 = {b7 where b7 is Element of the carrier of TOP-REAL 2: |.b7.| = 1} &
         LE b2,b3,b6 &
         b4 = b1 -FanMorphS . b2 &
         b5 = b1 -FanMorphS . b3
   holds LE b4,b5,b6;

:: JGRAPH_5:th 62
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6.| = 1} &
         LE b1,b2,b5 &
         LE b2,b3,b5 &
         LE b3,b4,b5 &
         b1 `1 < 0 &
         0 <= b1 `2 &
         b2 `1 < 0 &
         0 <= b2 `2 &
         b3 `1 < 0 &
         0 <= b3 `2 &
         b4 `1 < 0 &
         0 <= b4 `2
   holds ex b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      ex b7, b8, b9, b10 being Element of the carrier of TOP-REAL 2 st
         b6 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
          (for b11 being Element of the carrier of TOP-REAL 2 holds
             |.b6 . b11.| = |.b11.|) &
          b7 = b6 . b1 &
          b8 = b6 . b2 &
          b9 = b6 . b3 &
          b10 = b6 . b4 &
          b7 `1 < 0 &
          b7 `2 < 0 &
          b8 `1 < 0 &
          b8 `2 < 0 &
          b9 `1 < 0 &
          b9 `2 < 0 &
          b10 `1 < 0 &
          b10 `2 < 0 &
          LE b7,b8,b5 &
          LE b8,b9,b5 &
          LE b9,b10,b5;

:: JGRAPH_5:th 63
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6.| = 1} &
         LE b1,b2,b5 &
         LE b2,b3,b5 &
         LE b3,b4,b5 &
         0 <= b1 `2 &
         0 <= b2 `2 &
         0 <= b3 `2 &
         0 < b4 `2
   holds ex b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      ex b7, b8, b9, b10 being Element of the carrier of TOP-REAL 2 st
         b6 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
          (for b11 being Element of the carrier of TOP-REAL 2 holds
             |.b6 . b11.| = |.b11.|) &
          b7 = b6 . b1 &
          b8 = b6 . b2 &
          b9 = b6 . b3 &
          b10 = b6 . b4 &
          b7 `1 < 0 &
          0 <= b7 `2 &
          b8 `1 < 0 &
          0 <= b8 `2 &
          b9 `1 < 0 &
          0 <= b9 `2 &
          b10 `1 < 0 &
          0 <= b10 `2 &
          LE b7,b8,b5 &
          LE b8,b9,b5 &
          LE b9,b10,b5;

:: JGRAPH_5:th 64
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6.| = 1} &
         LE b1,b2,b5 &
         LE b2,b3,b5 &
         LE b3,b4,b5 &
         0 <= b1 `2 &
         0 <= b2 `2 &
         0 <= b3 `2 &
         0 < b4 `2
   holds ex b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      ex b7, b8, b9, b10 being Element of the carrier of TOP-REAL 2 st
         b6 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
          (for b11 being Element of the carrier of TOP-REAL 2 holds
             |.b6 . b11.| = |.b11.|) &
          b7 = b6 . b1 &
          b8 = b6 . b2 &
          b9 = b6 . b3 &
          b10 = b6 . b4 &
          b7 `1 < 0 &
          b7 `2 < 0 &
          b8 `1 < 0 &
          b8 `2 < 0 &
          b9 `1 < 0 &
          b9 `2 < 0 &
          b10 `1 < 0 &
          b10 `2 < 0 &
          LE b7,b8,b5 &
          LE b8,b9,b5 &
          LE b9,b10,b5;

:: JGRAPH_5:th 65
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6.| = 1} &
         LE b1,b2,b5 &
         LE b2,b3,b5 &
         LE b3,b4,b5 &
         (0 <= b1 `2 or 0 <= b1 `1) &
         (0 <= b2 `2 or 0 <= b2 `1) &
         (0 <= b3 `2 or 0 <= b3 `1) &
         (b4 `2 <= 0 implies 0 < b4 `1)
   holds ex b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      ex b7, b8, b9, b10 being Element of the carrier of TOP-REAL 2 st
         b6 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
          (for b11 being Element of the carrier of TOP-REAL 2 holds
             |.b6 . b11.| = |.b11.|) &
          b7 = b6 . b1 &
          b8 = b6 . b2 &
          b9 = b6 . b3 &
          b10 = b6 . b4 &
          0 <= b7 `2 &
          0 <= b8 `2 &
          0 <= b9 `2 &
          0 < b10 `2 &
          LE b7,b8,b5 &
          LE b8,b9,b5 &
          LE b9,b10,b5;

:: JGRAPH_5:th 66
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6.| = 1} &
         LE b1,b2,b5 &
         LE b2,b3,b5 &
         LE b3,b4,b5 &
         (0 <= b1 `2 or 0 <= b1 `1) &
         (0 <= b2 `2 or 0 <= b2 `1) &
         (0 <= b3 `2 or 0 <= b3 `1) &
         (b4 `2 <= 0 implies 0 < b4 `1)
   holds ex b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      ex b7, b8, b9, b10 being Element of the carrier of TOP-REAL 2 st
         b6 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
          (for b11 being Element of the carrier of TOP-REAL 2 holds
             |.b6 . b11.| = |.b11.|) &
          b7 = b6 . b1 &
          b8 = b6 . b2 &
          b9 = b6 . b3 &
          b10 = b6 . b4 &
          b7 `1 < 0 &
          b7 `2 < 0 &
          b8 `1 < 0 &
          b8 `2 < 0 &
          b9 `1 < 0 &
          b9 `2 < 0 &
          b10 `1 < 0 &
          b10 `2 < 0 &
          LE b7,b8,b5 &
          LE b8,b9,b5 &
          LE b9,b10,b5;

:: JGRAPH_5:th 67
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6.| = 1} &
         b4 = W-min b5 &
         LE b1,b2,b5 &
         LE b2,b3,b5 &
         LE b3,b4,b5
   holds ex b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      ex b7, b8, b9, b10 being Element of the carrier of TOP-REAL 2 st
         b6 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
          (for b11 being Element of the carrier of TOP-REAL 2 holds
             |.b6 . b11.| = |.b11.|) &
          b7 = b6 . b1 &
          b8 = b6 . b2 &
          b9 = b6 . b3 &
          b10 = b6 . b4 &
          b7 `1 < 0 &
          b7 `2 < 0 &
          b8 `1 < 0 &
          b8 `2 < 0 &
          b9 `1 < 0 &
          b9 `2 < 0 &
          b10 `1 < 0 &
          b10 `2 < 0 &
          LE b7,b8,b5 &
          LE b8,b9,b5 &
          LE b9,b10,b5;

:: JGRAPH_5:th 68
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6.| = 1} &
         LE b1,b2,b5 &
         LE b2,b3,b5 &
         LE b3,b4,b5
   holds ex b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      ex b7, b8, b9, b10 being Element of the carrier of TOP-REAL 2 st
         b6 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
          (for b11 being Element of the carrier of TOP-REAL 2 holds
             |.b6 . b11.| = |.b11.|) &
          b7 = b6 . b1 &
          b8 = b6 . b2 &
          b9 = b6 . b3 &
          b10 = b6 . b4 &
          b7 `1 < 0 &
          b7 `2 < 0 &
          b8 `1 < 0 &
          b8 `2 < 0 &
          b9 `1 < 0 &
          b9 `2 < 0 &
          b10 `1 < 0 &
          b10 `2 < 0 &
          LE b7,b8,b5 &
          LE b8,b9,b5 &
          LE b9,b10,b5;

:: JGRAPH_5:th 69
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6.| = 1} &
         LE b1,b2,b5 &
         LE b2,b3,b5 &
         LE b3,b4,b5 &
         b1 <> b2 &
         b2 <> b3 &
         b3 <> b4 &
         b1 `1 < 0 &
         b2 `1 < 0 &
         b3 `1 < 0 &
         b4 `1 < 0 &
         b1 `2 < 0 &
         b2 `2 < 0 &
         b3 `2 < 0 &
         b4 `2 < 0
   holds ex b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      b6 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
       (for b7 being Element of the carrier of TOP-REAL 2 holds
          |.b6 . b7.| = |.b7.|) &
       |[- 1,0]| = b6 . b1 &
       |[0,1]| = b6 . b2 &
       |[1,0]| = b6 . b3 &
       |[0,- 1]| = b6 . b4;

:: JGRAPH_5:th 70
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6.| = 1} &
         LE b1,b2,b5 &
         LE b2,b3,b5 &
         LE b3,b4,b5 &
         b1 <> b2 &
         b2 <> b3 &
         b3 <> b4
   holds ex b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
      b6 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2) &
       (for b7 being Element of the carrier of TOP-REAL 2 holds
          |.b6 . b7.| = |.b7.|) &
       |[- 1,0]| = b6 . b1 &
       |[0,1]| = b6 . b2 &
       |[1,0]| = b6 . b3 &
       |[0,- 1]| = b6 . b4;

:: JGRAPH_5:th 71
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
for b6 being Element of bool the carrier of TOP-REAL 2
   st b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: |.b7.| = 1} &
      LE b1,b2,b5 &
      LE b2,b3,b5 &
      LE b3,b4,b5
for b7, b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b7 is continuous(I[01], TOP-REAL 2) &
         b7 is one-to-one &
         b8 is continuous(I[01], TOP-REAL 2) &
         b8 is one-to-one &
         b6 = {b9 where b9 is Element of the carrier of TOP-REAL 2: |.b9.| <= 1} &
         b7 . 0 = b1 &
         b7 . 1 = b3 &
         b8 . 0 = b2 &
         b8 . 1 = b4 &
         rng b7 c= b6 &
         rng b8 c= b6
   holds rng b7 meets rng b8;

:: JGRAPH_5:th 72
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
for b6 being Element of bool the carrier of TOP-REAL 2
   st b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: |.b7.| = 1} &
      LE b1,b2,b5 &
      LE b2,b3,b5 &
      LE b3,b4,b5
for b7, b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b7 is continuous(I[01], TOP-REAL 2) &
         b7 is one-to-one &
         b8 is continuous(I[01], TOP-REAL 2) &
         b8 is one-to-one &
         b6 = {b9 where b9 is Element of the carrier of TOP-REAL 2: |.b9.| <= 1} &
         b7 . 0 = b1 &
         b7 . 1 = b3 &
         b8 . 0 = b4 &
         b8 . 1 = b2 &
         rng b7 c= b6 &
         rng b8 c= b6
   holds rng b7 meets rng b8;

:: JGRAPH_5:th 73
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
for b6 being Element of bool the carrier of TOP-REAL 2
   st b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: |.b7.| = 1} &
      LE b1,b2,b5 &
      LE b2,b3,b5 &
      LE b3,b4,b5
for b7, b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b7 is continuous(I[01], TOP-REAL 2) &
         b7 is one-to-one &
         b8 is continuous(I[01], TOP-REAL 2) &
         b8 is one-to-one &
         b6 = {b9 where b9 is Element of the carrier of TOP-REAL 2: 1 <= |.b9.|} &
         b7 . 0 = b1 &
         b7 . 1 = b3 &
         b8 . 0 = b4 &
         b8 . 1 = b2 &
         rng b7 c= b6 &
         rng b8 c= b6
   holds rng b7 meets rng b8;

:: JGRAPH_5:th 74
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
for b6 being Element of bool the carrier of TOP-REAL 2
   st b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: |.b7.| = 1} &
      LE b1,b2,b5 &
      LE b2,b3,b5 &
      LE b3,b4,b5
for b7, b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
      st b7 is continuous(I[01], TOP-REAL 2) &
         b7 is one-to-one &
         b8 is continuous(I[01], TOP-REAL 2) &
         b8 is one-to-one &
         b6 = {b9 where b9 is Element of the carrier of TOP-REAL 2: 1 <= |.b9.|} &
         b7 . 0 = b1 &
         b7 . 1 = b3 &
         b8 . 0 = b2 &
         b8 . 1 = b4 &
         rng b7 c= b6 &
         rng b8 c= b6
   holds rng b7 meets rng b8;