Article TOPREAL5, MML version 4.99.1005

:: TOPREAL5:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b3 is open(b1) & b4 is open(b1) & b3 meets b2 & b4 meets b2 & b2 c= b3 \/ b4 & b3 misses b4
   holds b2 is not connected(b1);

:: TOPREAL5:th 6
theorem
for b1, b2 being real set
      st b1 <= b2
   holds [#] Closed-Interval-TSpace(b1,b2) is connected(Closed-Interval-TSpace(b1,b2));

:: TOPREAL5:th 9
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
      st not b2 in b1 &
         (ex b3, b4 being real set st
            b3 in b1 & b4 in b1 & b3 < b2 & b2 < b4)
   holds b1 is connected(not R^1);

:: TOPREAL5:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5, b6 being Element of REAL
for b7 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of R^1
      st b1 is connected & b7 . b2 = b4 & b7 . b3 = b5 & b4 <= b6 & b6 <= b5
   holds ex b8 being Element of the carrier of b1 st
      b7 . b8 = b6;

:: TOPREAL5:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
for b5, b6, b7 being Element of REAL
for b8 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of R^1
      st b4 is connected(b1) & b8 . b2 = b5 & b8 . b3 = b6 & b5 <= b7 & b7 <= b6 & b2 in b4 & b3 in b4
   holds ex b9 being Element of the carrier of b1 st
      b9 in b4 & b8 . b9 = b7;

:: TOPREAL5:th 12
theorem
for b1, b2, b3, b4 being real set
   st b1 < b2
for b5 being Function-like quasi_total continuous Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of R^1
for b6 being real set
      st b5 . b1 = b3 & b5 . b2 = b4 & b3 < b6 & b6 < b4
   holds ex b7 being Element of REAL st
      b5 . b7 = b6 & b1 < b7 & b7 < b2;

:: TOPREAL5:th 13
theorem
for b1, b2, b3, b4 being real set
   st b1 < b2
for b5 being Function-like quasi_total continuous Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of R^1
for b6 being real set
      st b5 . b1 = b3 & b5 . b2 = b4 & b6 < b3 & b4 < b6
   holds ex b7 being Element of REAL st
      b5 . b7 = b6 & b1 < b7 & b7 < b2;

:: TOPREAL5:th 14
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total continuous Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of R^1
for b4, b5 being real set
      st b1 < b2 & b4 * b5 < 0 & b4 = b3 . b1 & b5 = b3 . b2
   holds ex b6 being Element of REAL st
      b3 . b6 = 0 & b1 < b6 & b6 < b2;

:: TOPREAL5:th 15
theorem
for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of R^1
for b2, b3 being real set
      st b1 is continuous(I[01], R^1) & b1 . 0 <> b1 . 1 & b2 = b1 . 0 & b3 = b1 . 1
   holds ex b4 being Element of REAL st
      0 < b4 & b4 < 1 & b1 . b4 = (b2 + b3) / 2;

:: TOPREAL5:th 16
theorem
for b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of R^1
      st b1 = proj1
   holds b1 is continuous(TOP-REAL 2, R^1);

:: TOPREAL5:th 17
theorem
for b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of R^1
      st b1 = proj2
   holds b1 is continuous(TOP-REAL 2, R^1);

:: TOPREAL5:th 18
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
      st b2 is continuous(I[01], (TOP-REAL 2) | b1)
   holds ex b3 being Function-like quasi_total Relation of the carrier of I[01],the carrier of R^1 st
      b3 is continuous(I[01], R^1) &
       (for b4 being Element of REAL
       for b5 being Element of the carrier of TOP-REAL 2
             st b4 in the carrier of I[01] & b5 = b2 . b4
          holds b5 `1 = b3 . b4);

:: TOPREAL5:th 19
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
      st b2 is continuous(I[01], (TOP-REAL 2) | b1)
   holds ex b3 being Function-like quasi_total Relation of the carrier of I[01],the carrier of R^1 st
      b3 is continuous(I[01], R^1) &
       (for b4 being Element of REAL
       for b5 being Element of the carrier of TOP-REAL 2
             st b4 in the carrier of I[01] & b5 = b2 . b4
          holds b5 `2 = b3 . b4);

:: TOPREAL5:th 20
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
   st b1 is being_simple_closed_curve
for b2 being Element of REAL holds
   ex b3 being Element of the carrier of TOP-REAL 2 st
      b3 in b1 & b3 `2 <> b2;

:: TOPREAL5:th 21
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
   st b1 is being_simple_closed_curve
for b2 being Element of REAL holds
   ex b3 being Element of the carrier of TOP-REAL 2 st
      b3 in b1 & b3 `1 <> b2;

:: TOPREAL5:th 22
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve
   holds S-bound b1 < N-bound b1;

:: TOPREAL5:th 23
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve
   holds W-bound b1 < E-bound b1;

:: TOPREAL5:th 24
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve
   holds S-min b1 <> N-max b1;

:: TOPREAL5:th 25
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve
   holds W-min b1 <> E-max b1;

:: TOPREAL5:condreg 1
registration
  cluster being_simple_closed_curve -> non horizontal non vertical (Element of bool the carrier of TOP-REAL 2);
end;