Article TOPGEN_5, MML version 4.99.1005

:: TOPGEN_5:th 1
theorem
for b1, b2 being Relation-like Function-like set
   st b1 tolerates b2
for b3 being set holds
   (b1 +* b2) " b3 = (b1 " b3) \/ (b2 " b3);

:: TOPGEN_5:th 2
theorem
for b1, b2 being Relation-like Function-like set
   st proj1 b1 misses proj1 b2
for b3 being set holds
   (b1 +* b2) " b3 = (b1 " b3) \/ (b2 " b3);

:: TOPGEN_5:th 3
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set
      st b2 in proj1 b3
   holds (commute (b1 .--> b3)) . b2 = b1 .--> (b3 . b2);

:: TOPGEN_5:th 4
theorem
for b1 being set
for b2 being Relation-like Function-like set holds
      b1 in proj1 commute b2
   iff
      ex b3 being set st
         ex b4 being Relation-like Function-like set st
            b3 in proj1 b2 & b4 = b2 . b3 & b1 in proj1 b4;

:: TOPGEN_5:th 5
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set holds
      b1 in proj1 ((commute b3) . b2)
   iff
      ex b4 being Relation-like Function-like set st
         b1 in proj1 b3 & b4 = b3 . b1 & b2 in proj1 b4;

:: TOPGEN_5:th 6
theorem
for b1, b2 being set
for b3, b4 being Relation-like Function-like set
      st b1 in proj1 b3 & b4 = b3 . b1 & b2 in proj1 b4
   holds ((commute b3) . b2) . b1 = b4 . b2;

:: TOPGEN_5:th 7
theorem
for b1 being set
for b2, b3, b4 being Relation-like Function-like set
      st b4 = b2 \/ b3
   holds (commute b4) . b1 = ((commute b2) . b1) \/ ((commute b3) . b1);

:: TOPGEN_5:condreg 1
registration
  cluster finite -> bounded (Element of bool REAL);
end;

:: TOPGEN_5:th 8
theorem
for b1, b2, b3, b4 being real set
      st b1 < b2 & b3 <= b4
   holds ].b1,b3.[ /\ [.b2,b4.] = [.b2,b3.[;

:: TOPGEN_5:th 9
theorem
for b1, b2, b3, b4 being real set
      st b2 <= b1 & b4 < b3
   holds ].b1,b3.[ /\ [.b2,b4.] = ].b1,b4.];

:: TOPGEN_5:th 10
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b2 < b3 & b3 <= b4
   holds [.b1,b3.[ \/ ].b2,b4.] = [.b1,b4.];

:: TOPGEN_5:th 11
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b2 < b3 & b3 <= b4
   holds [.b1,b3.[ /\ ].b2,b4.] = ].b2,b3.[;

:: TOPGEN_5:th 12
theorem
for b1, b2 being set holds
product <*b1,b2*>,[:b1,b2:] are_equipotent &
 Card product <*b1,b2*> = (Card b1) *` Card b2;

:: TOPGEN_5:sch 1
scheme TOPGEN_5:sch 1
{F1 -> non empty set,
  F2 -> non empty set,
  F3 -> non empty set,
  F4 -> set,
  F5 -> set}:
ex b1 being Function-like quasi_total Relation of F3(),F2() st
   for b2 being Element of F1()
         st b2 in F3()
      holds (P1[b2] implies b1 . b2 = F4(b2)) & (P1[b2] or b1 . b2 = F5(b2))
provided
   F3() c= F1()
and
   for b1 being Element of F1()
         st b1 in F3()
      holds (P1[b1] implies F4(b1) in F2()) & (P1[b1] or F5(b1) in F2());


:: TOPGEN_5:sch 2
scheme TOPGEN_5:sch 2
{F1 -> non empty set,
  F2 -> non empty set,
  F3 -> non empty set,
  F4 -> set,
  F5 -> set,
  F6 -> set}:
ex b1 being Function-like quasi_total Relation of F3(),F2() st
   for b2 being Element of F1()
         st b2 in F3()
      holds (P1[b2] implies b1 . b2 = F4(b2)) & (not (P1[b2]) & P2[b2] implies b1 . b2 = F5(b2)) & (not (P1[b2]) & not (P2[b2]) implies b1 . b2 = F6(b2))
provided
   F3() c= F1()
and
   for b1 being Element of F1()
         st b1 in F3()
      holds (P1[b1] implies F4(b1) in F2()) & (not (P1[b1]) & P2[b1] implies F5(b1) in F2()) & (not (P1[b1]) & not (P2[b1]) implies F6(b1) in F2());


:: TOPGEN_5:th 13
theorem
for b1, b2 being real set holds
|.|[b1,b2]|.| ^2 = b1 ^2 + (b2 ^2);

:: TOPGEN_5:th 14
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3, b4 being closed Element of bool the carrier of b1
for b5 being Function-like quasi_total continuous Relation of the carrier of b1 | b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b1 | b4,the carrier of b2
      st b5 tolerates b6
   holds b5 +* b6 is Function-like quasi_total continuous Relation of the carrier of b1 | (b3 \/ b4),the carrier of b2;

:: TOPGEN_5:th 15
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3, b4 being closed Element of bool the carrier of b1
   st b3 misses b4
for b5 being Function-like quasi_total continuous Relation of the carrier of b1 | b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b1 | b4,the carrier of b2 holds
   b5 +* b6 is Function-like quasi_total continuous Relation of the carrier of b1 | (b3 \/ b4),the carrier of b2;

:: TOPGEN_5:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being open closed Element of bool the carrier of b1
for b4 being Function-like quasi_total continuous Relation of the carrier of b1 | b3,the carrier of b2
for b5 being Function-like quasi_total continuous Relation of the carrier of b1 | (b3 `),the carrier of b2 holds
   b4 +* b5 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;

:: TOPGEN_5:th 17
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being real positive set holds
   b2 in Ball(b2,b3);

:: TOPGEN_5:funcnot 1 => TOPGEN_5:func 1
definition
  func y=0-line -> Element of bool the carrier of TOP-REAL 2 equals
    {|[b1,0]| where b1 is Element of REAL: TRUE};
end;

:: TOPGEN_5:def 1
theorem
y=0-line = {|[b1,0]| where b1 is Element of REAL: TRUE};

:: TOPGEN_5:funcnot 2 => TOPGEN_5:func 2
definition
  func y>=0-plane -> Element of bool the carrier of TOP-REAL 2 equals
    {|[b1,b2]| where b1 is Element of REAL, b2 is Element of REAL: 0 <= b2};
end;

:: TOPGEN_5:def 2
theorem
y>=0-plane = {|[b1,b2]| where b1 is Element of REAL, b2 is Element of REAL: 0 <= b2};

:: TOPGEN_5:th 18
theorem
for b1, b2 being set holds
   <*b1,b2*> in y=0-line
iff
   b1 in REAL & b2 = 0;

:: TOPGEN_5:th 19
theorem
for b1, b2 being real set holds
   |[b1,b2]| in y=0-line
iff
   b2 = 0;

:: TOPGEN_5:th 20
theorem
Card y=0-line = continuum;

:: TOPGEN_5:th 21
theorem
for b1, b2 being set holds
   <*b1,b2*> in y>=0-plane
iff
   b1 in REAL &
    (ex b3 being Element of REAL st
       b2 = b3 & 0 <= b3);

:: TOPGEN_5:th 22
theorem
for b1, b2 being real set holds
   |[b1,b2]| in y>=0-plane
iff
   0 <= b2;

:: TOPGEN_5:funcreg 1
registration
  cluster y=0-line -> non empty;
end;

:: TOPGEN_5:funcreg 2
registration
  cluster y>=0-plane -> non empty;
end;

:: TOPGEN_5:th 23
theorem
y=0-line c= y>=0-plane;

:: TOPGEN_5:th 24
theorem
for b1, b2, b3 being real set
      st 0 < b3
   holds    Ball(|[b1,b2]|,b3) c= y>=0-plane
   iff
      b3 <= b2;

:: TOPGEN_5:th 25
theorem
for b1, b2, b3 being real set
      st 0 < b3 & 0 <= b2
   holds    Ball(|[b1,b2]|,b3) misses y=0-line
   iff
      b3 <= b2;

:: TOPGEN_5:th 26
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4, b5 being real positive set
      st |.b2 - b3.| <= b4 - b5
   holds Ball(b3,b5) c= Ball(b2,b4);

:: TOPGEN_5:th 27
theorem
for b1 being real set
for b2, b3 being real positive set
      st b2 <= b3
   holds Ball(|[b1,b2]|,b2) c= Ball(|[b1,b3]|,b3);

:: TOPGEN_5:th 28
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Neighborhood_System of b1
for b4 being Neighborhood_System of b2
      st b3 = b4
   holds TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#);

:: TOPGEN_5:funcnot 3 => TOPGEN_5:func 3
definition
  func Niemytzki-plane -> non empty strict TopSpace-like TopStruct means
    the carrier of it = y>=0-plane &
     (ex b1 being Neighborhood_System of it st
        (for b2 being Element of REAL holds
            b1 . |[b2,0]| = {(Ball(|[b2,b3]|,b3)) \/ {|[b2,0]|} where b3 is Element of REAL: 0 < b3}) &
         (for b2, b3 being Element of REAL
               st 0 < b3
            holds b1 . |[b2,b3]| = {(Ball(|[b2,b3]|,b4)) /\ y>=0-plane where b4 is Element of REAL: 0 < b4}));
end;

:: TOPGEN_5:def 3
theorem
for b1 being non empty strict TopSpace-like TopStruct holds
      b1 = Niemytzki-plane
   iff
      the carrier of b1 = y>=0-plane &
       (ex b2 being Neighborhood_System of b1 st
          (for b3 being Element of REAL holds
              b2 . |[b3,0]| = {(Ball(|[b3,b4]|,b4)) \/ {|[b3,0]|} where b4 is Element of REAL: 0 < b4}) &
           (for b3, b4 being Element of REAL
                 st 0 < b4
              holds b2 . |[b3,b4]| = {(Ball(|[b3,b4]|,b5)) /\ y>=0-plane where b5 is Element of REAL: 0 < b5}));

:: TOPGEN_5:th 29
theorem
y>=0-plane \ y=0-line is open Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 30
theorem
y=0-line is closed Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 31
theorem
for b1 being real set
for b2 being real positive set holds
   (Ball(|[b1,b2]|,b2)) \/ {|[b1,0]|} is open Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 32
theorem
for b1 being real set
for b2, b3 being real positive set holds
(Ball(|[b1,b2]|,b3)) /\ y>=0-plane is open Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 33
theorem
for b1, b2 being real set
for b3 being real positive set
      st b3 <= b2
   holds Ball(|[b1,b2]|,b3) is open Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 34
theorem
for b1 being Element of the carrier of Niemytzki-plane
for b2 being real positive set holds
   ex b3 being Element of the carrier of TOP-REAL 2 st
      ex b4 being open Element of bool the carrier of Niemytzki-plane st
         b1 in b4 &
          b3 in b4 &
          (for b5 being Element of the carrier of TOP-REAL 2
                st b5 in b4
             holds |.b5 - b3.| < b2);

:: TOPGEN_5:th 35
theorem
for b1, b2 being real set
for b3 being real positive set holds
   ex b4, b5 being rational set st
      |[b4,b5]| in Ball(|[b1,b2]|,b3) &
       |[b4,b5]| <> |[b1,b2]|;

:: TOPGEN_5:th 36
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
   st b1 = (y>=0-plane \ y=0-line) /\ product <*RAT,RAT*>
for b2 being set holds
   Cl (b1 \ {b2}) = [#] Niemytzki-plane;

:: TOPGEN_5:th 37
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
   st b1 = y>=0-plane \ y=0-line
for b2 being set holds
   Cl (b1 \ {b2}) = [#] Niemytzki-plane;

:: TOPGEN_5:th 38
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
      st b1 = y>=0-plane \ y=0-line
   holds Cl b1 = [#] Niemytzki-plane;

:: TOPGEN_5:th 39
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
      st b1 = y=0-line
   holds Cl b1 = b1 & Int b1 = {};

:: TOPGEN_5:th 40
theorem
(y>=0-plane \ y=0-line) /\ product <*RAT,RAT*> is dense Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 41
theorem
(y>=0-plane \ y=0-line) /\ product <*RAT,RAT*> is dense-in-itself Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 42
theorem
y>=0-plane \ y=0-line is dense Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 43
theorem
y>=0-plane \ y=0-line is dense-in-itself Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 44
theorem
y=0-line is nowhere_dense Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 45
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
      st b1 = y=0-line
   holds Der b1 is empty;

:: TOPGEN_5:th 46
theorem
for b1 being Element of bool y=0-line holds
   b1 is closed Element of bool the carrier of Niemytzki-plane;

:: TOPGEN_5:th 47
theorem
RAT is dense Element of bool the carrier of Sorgenfrey-line;

:: TOPGEN_5:th 48
theorem
Sorgenfrey-line is separable;

:: TOPGEN_5:th 49
theorem
Niemytzki-plane is separable;

:: TOPGEN_5:th 50
theorem
Niemytzki-plane is being_T1;

:: TOPGEN_5:th 51
theorem
Niemytzki-plane is not being_T4;

:: TOPGEN_5:attrnot 1 => TOPGEN_5:attr 1
definition
  let a1 be TopSpace-like TopStruct;
  attr a1 is Tychonoff means
    a1 is being_T1 &
     (for b1 being closed Element of bool the carrier of a1
     for b2 being Element of the carrier of a1
           st b2 in b1 `
        holds ex b3 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of I[01] st
           b3 . b2 = 0 & b3 .: b1 c= {1});
end;

:: TOPGEN_5:dfs 4
definiens
  let a1 be TopSpace-like TopStruct;
To prove
     a1 is Tychonoff
it is sufficient to prove
  thus a1 is being_T1 &
     (for b1 being closed Element of bool the carrier of a1
     for b2 being Element of the carrier of a1
           st b2 in b1 `
        holds ex b3 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of I[01] st
           b3 . b2 = 0 & b3 .: b1 c= {1});

:: TOPGEN_5:def 4
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is Tychonoff
   iff
      b1 is being_T1 &
       (for b2 being closed Element of bool the carrier of b1
       for b3 being Element of the carrier of b1
             st b3 in b2 `
          holds ex b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of I[01] st
             b4 . b3 = 0 & b4 .: b2 c= {1});

:: TOPGEN_5:condreg 2
registration
  cluster TopSpace-like Tychonoff -> being_T3 being_T1 (TopStruct);
end;

:: TOPGEN_5:condreg 3
registration
  cluster non empty TopSpace-like being_T4 being_T1 -> Tychonoff (TopStruct);
end;

:: TOPGEN_5:th 52
theorem
for b1 being TopSpace-like being_T1 TopStruct
   st b1 is Tychonoff
for b2 being prebasis of b1
for b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
      st b3 in b4 & b4 in b2
   holds ex b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of I[01] st
      b5 . b3 = 0 & b5 .: (b4 `) c= {1};

:: TOPGEN_5:condreg 4
registration
  let a1 be set;
  let a2 be non empty real-membered set;
  cluster -> real-valued (Relation of a1,a2);
end;

:: TOPGEN_5:th 53
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty SubSpace of R^1
for b3, b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b5 being Element of bool the carrier of b1
      st for b6 being Element of the carrier of b1 holds
              b6 in b5
           iff
              b3 . b6 <= b4 . b6
   holds b5 is closed(b1);

:: TOPGEN_5:th 54
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty SubSpace of R^1
for b3, b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
ex b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 st
   for b6 being Element of the carrier of b1 holds
      b5 . b6 = max(b3 . b6,b4 . b6);

:: TOPGEN_5:th 55
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of R^1
for b3 being non empty finite set
for b4 being Function-yielding ManySortedSet of b3
      st for b5 being set
              st b5 in b3
           holds b4 . b5 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
   holds ex b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 st
      for b6 being Element of the carrier of b1
      for b7 being non empty finite Element of bool REAL
            st b7 = proj2 ((commute b4) . b6)
         holds b5 . b6 = upper_bound b7;

:: TOPGEN_5:th 56
theorem
for b1 being non empty TopSpace-like being_T1 TopStruct
for b2 being prebasis of b1
      st for b3 being Element of the carrier of b1
        for b4 being Element of bool the carrier of b1
              st b3 in b4 & b4 in b2
           holds ex b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of I[01] st
              b5 . b3 = 0 & b5 .: (b4 `) c= {1}
   holds b1 is Tychonoff;

:: TOPGEN_5:th 57
theorem
Sorgenfrey-line is being_T1;

:: TOPGEN_5:th 58
theorem
for b1 being real set holds
   halfline b1 is closed Element of bool the carrier of Sorgenfrey-line;

:: TOPGEN_5:th 59
theorem
for b1 being real set holds
   left_closed_halfline b1 is closed Element of bool the carrier of Sorgenfrey-line;

:: TOPGEN_5:th 60
theorem
for b1 being real set holds
   right_closed_halfline b1 is closed Element of bool the carrier of Sorgenfrey-line;

:: TOPGEN_5:th 61
theorem
for b1, b2 being real set holds
[.b1,b2.[ is closed Element of bool the carrier of Sorgenfrey-line;

:: TOPGEN_5:th 62
theorem
for b1 being real set
for b2 being rational set
      st b1 < b2
   holds ex b3 being Function-like quasi_total continuous Relation of the carrier of Sorgenfrey-line,the carrier of I[01] st
      for b4 being Element of the carrier of Sorgenfrey-line holds
         (b4 in [.b1,b2.[ implies b3 . b4 = 0) &
          (b4 in [.b1,b2.[ or b3 . b4 = 1);

:: TOPGEN_5:th 63
theorem
Sorgenfrey-line is Tychonoff;

:: TOPGEN_5:funcnot 4 => TOPGEN_5:func 4
definition
  let a1 be real set;
  let a2 be real positive set;
  func +(A1,A2) -> Function-like quasi_total Relation of the carrier of Niemytzki-plane,the carrier of I[01] means
    it . |[a1,0]| = 0 &
     (for b1 being real set
     for b2 being real non negative set holds
        ((b1 = a1 implies b2 <> 0) &
          not |[b1,b2]| in Ball(|[a1,a2]|,a2) implies it . |[b1,b2]| = 1) &
         (|[b1,b2]| in Ball(|[a1,a2]|,a2) implies it . |[b1,b2]| = |.|[a1,0]| - |[b1,b2]|.| ^2 / ((2 * a2) * b2)));
end;

:: TOPGEN_5:def 5
theorem
for b1 being real set
for b2 being real positive set
for b3 being Function-like quasi_total Relation of the carrier of Niemytzki-plane,the carrier of I[01] holds
      b3 = +(b1,b2)
   iff
      b3 . |[b1,0]| = 0 &
       (for b4 being real set
       for b5 being real non negative set holds
          ((b4 = b1 implies b5 <> 0) &
            not |[b4,b5]| in Ball(|[b1,b2]|,b2) implies b3 . |[b4,b5]| = 1) &
           (|[b4,b5]| in Ball(|[b1,b2]|,b2) implies b3 . |[b4,b5]| = |.|[b1,0]| - |[b4,b5]|.| ^2 / ((2 * b2) * b5)));

:: TOPGEN_5:th 64
theorem
for b1 being Element of the carrier of TOP-REAL 2
   st 0 <= b1 `2
for b2 being real set
for b3 being real positive set
      st (+(b2,b3)) . b1 = 0
   holds b1 = |[b2,0]|;

:: TOPGEN_5:th 65
theorem
for b1, b2 being real set
for b3 being real positive set
      st b1 <> b2
   holds (+(b1,b3)) . |[b2,0]| = 1;

:: TOPGEN_5:th 66
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being real set
for b3, b4 being real positive set
      st b3 <= 1 &
         |.b1 - |[b2,b4 * b3]|.| = b4 * b3 &
         b1 `2 <> 0
   holds (+(b2,b4)) . b1 = b3;

:: TOPGEN_5:th 67
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being real set
for b4 being real positive set
      st 0 <= b3 &
         b3 <= 1 &
         |.b1 - |[b2,b4 * b3]|.| < b4 * b3
   holds (+(b2,b4)) . b1 < b3;

:: TOPGEN_5:th 68
theorem
for b1 being Element of the carrier of TOP-REAL 2
   st 0 <= b1 `2
for b2, b3 being real set
for b4 being real positive set
      st 0 <= b3 &
         b3 < 1 &
         b4 * b3 < |.b1 - |[b2,b4 * b3]|.|
   holds b3 < (+(b2,b4)) . b1;

:: TOPGEN_5:th 69
theorem
for b1 being Element of the carrier of TOP-REAL 2
   st 0 <= b1 `2
for b2, b3, b4 being real set
for b5 being real positive set
      st 0 <= b3 & b4 <= 1 & (+(b2,b5)) . b1 in ].b3,b4.[
   holds ex b6 being real positive set st
      b6 <= b1 `2 &
       Ball(b1,b6) c= (+(b2,b5)) " ].b3,b4.[;

:: TOPGEN_5:th 70
theorem
for b1 being real set
for b2, b3 being real positive set holds
Ball(|[b1,b3 * b2]|,b3 * b2) c= (+(b1,b3)) " ].0,b2.[;

:: TOPGEN_5:th 71
theorem
for b1 being real set
for b2, b3 being real positive set holds
(Ball(|[b1,b3 * b2]|,b3 * b2)) \/ {|[b1,0]|} c= (+(b1,b3)) " [.0,b2.[;

:: TOPGEN_5:th 72
theorem
for b1 being Element of the carrier of TOP-REAL 2
   st 0 <= b1 `2
for b2, b3 being real set
for b4 being real positive set
      st 0 < (+(b2,b4)) . b1 & (+(b2,b4)) . b1 < b3 & b3 <= 1
   holds b1 in Ball(|[b2,b4 * b3]|,b4 * b3);

:: TOPGEN_5:th 73
theorem
for b1 being Element of the carrier of TOP-REAL 2
   st 0 < b1 `2
for b2, b3 being real set
for b4 being real positive set
      st 0 <= b3 & b3 < (+(b2,b4)) . b1
   holds ex b5 being real positive set st
      b5 <= b1 `2 &
       Ball(b1,b5) c= (+(b2,b4)) " ].b3,1.];

:: TOPGEN_5:th 74
theorem
for b1 being Element of the carrier of TOP-REAL 2
   st b1 `2 = 0
for b2 being real set
for b3 being real positive set
      st (+(b2,b3)) . b1 = 1
   holds ex b4 being real positive set st
      (Ball(|[b1 `1,b4]|,b4)) \/ {b1} c= (+(b2,b3)) " {1};

:: TOPGEN_5:th 75
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Basis of b1 holds
   {b4 /\ [#] b2 where b4 is Element of bool the carrier of b1: b4 in b3 & b4 meets [#] b2} is Basis of b2;

:: TOPGEN_5:th 76
theorem
{].b1,b2.[ where b1 is Element of REAL, b2 is Element of REAL: b1 < b2} is Basis of R^1;

:: TOPGEN_5:th 77
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being set
      st b2 in b4 &
         b3 in b4 &
         b4 \/ {b2 \/ b3} is Basis of b1
   holds b4 is Basis of b1;

:: TOPGEN_5:th 78
theorem
({[.0,b1.[ where b1 is Element of REAL: 0 < b1 & b1 <= 1} \/ {].b1,1.] where b1 is Element of REAL: 0 <= b1 & b1 < 1}) \/ {].b1,b2.[ where b1 is Element of REAL, b2 is Element of REAL: 0 <= b1 & b1 < b2 & b2 <= 1} is Basis of I[01];

:: TOPGEN_5:th 79
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 I[01] holds
      b2 is continuous(b1, I[01])
   iff
      for b3, b4 being real set
            st 0 <= b3 & b3 < 1 & 0 < b4 & b4 <= 1
         holds b2 " [.0,b4.[ is open(b1) & b2 " ].b3,1.] is open(b1);

:: TOPGEN_5:funcreg 3
registration
  let a1 be real set;
  let a2 be real positive set;
  cluster +(a1,a2) -> Function-like quasi_total continuous;
end;

:: TOPGEN_5:th 80
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
for b2 being Element of REAL
for b3 being real positive set
      st b1 = (Ball(|[b2,b3]|,b3)) \/ {|[b2,0]|}
   holds ex b4 being Function-like quasi_total continuous Relation of the carrier of Niemytzki-plane,the carrier of I[01] st
      b4 . |[b2,0]| = 0 &
       (for b5, b6 being real set holds
       (|[b5,b6]| in b1 ` implies b4 . |[b5,b6]| = 1) &
        (|[b5,b6]| in b1 \ {|[b2,0]|} implies b4 . |[b5,b6]| = |.|[b2,0]| - |[b5,b6]|.| ^2 / ((2 * b3) * b6)));

:: TOPGEN_5:funcnot 5 => TOPGEN_5:func 5
definition
  let a1, a2 be real set;
  let a3 be real positive set;
  func +(A1,A2,A3) -> Function-like quasi_total Relation of the carrier of Niemytzki-plane,the carrier of I[01] means
    for b1 being real set
    for b2 being real non negative set holds
       (|[b1,b2]| in Ball(|[a1,a2]|,a3) or it . |[b1,b2]| = 1) &
        (|[b1,b2]| in Ball(|[a1,a2]|,a3) implies it . |[b1,b2]| = |.|[a1,a2]| - |[b1,b2]|.| / a3);
end;

:: TOPGEN_5:def 6
theorem
for b1, b2 being real set
for b3 being real positive set
for b4 being Function-like quasi_total Relation of the carrier of Niemytzki-plane,the carrier of I[01] holds
      b4 = +(b1,b2,b3)
   iff
      for b5 being real set
      for b6 being real non negative set holds
         (|[b5,b6]| in Ball(|[b1,b2]|,b3) or b4 . |[b5,b6]| = 1) &
          (|[b5,b6]| in Ball(|[b1,b2]|,b3) implies b4 . |[b5,b6]| = |.|[b1,b2]| - |[b5,b6]|.| / b3);

:: TOPGEN_5:th 81
theorem
for b1 being Element of the carrier of TOP-REAL 2
   st 0 <= b1 `2
for b2 being real set
for b3 being real non negative set
for b4 being real positive set holds
      (+(b2,b3,b4)) . b1 = 0
   iff
      b1 = |[b2,b3]|;

:: TOPGEN_5:th 82
theorem
for b1 being real set
for b2 being real non negative set
for b3, b4 being real positive set
      st b4 <= 1
   holds (+(b1,b2,b3)) " [.0,b4.[ = (Ball(|[b1,b2]|,b3 * b4)) /\ y>=0-plane;

:: TOPGEN_5:th 83
theorem
for b1 being Element of the carrier of TOP-REAL 2
   st 0 < b1 `2
for b2 being real set
for b3 being real non negative set
for b4, b5 being real positive set
      st b3 < (+(b2,b4,b5)) . b1
   holds b5 * b3 < |.|[b2,b4]| - b1.| &
    (Ball(b1,|.|[b2,b4]| - b1.| - (b5 * b3))) /\ y>=0-plane c= (+(b2,b4,b5)) " ].b3,1.];

:: TOPGEN_5:th 84
theorem
for b1 being Element of the carrier of TOP-REAL 2
   st b1 `2 = 0
for b2 being real set
for b3 being real non negative set
for b4, b5 being real positive set
      st b3 < (+(b2,b4,b5)) . b1
   holds b5 * b3 < |.|[b2,b4]| - b1.| &
    (ex b6 being real positive set st
       b6 = (|.|[b2,b4]| - b1.| - (b5 * b3)) / 2 &
        (Ball(|[b1 `1,b6]|,b6)) \/ {b1} c= (+(b2,b4,b5)) " ].b3,1.]);

:: TOPGEN_5:funcreg 4
registration
  let a1 be real set;
  let a2, a3 be real positive set;
  cluster +(a1,a2,a3) -> Function-like quasi_total continuous;
end;

:: TOPGEN_5:th 85
theorem
for b1 being Element of bool the carrier of Niemytzki-plane
for b2, b3 being Element of REAL
for b4 being real positive set
      st 0 < b3 &
         b1 = (Ball(|[b2,b3]|,b4)) /\ y>=0-plane
   holds ex b5 being Function-like quasi_total continuous Relation of the carrier of Niemytzki-plane,the carrier of I[01] st
      b5 . |[b2,b3]| = 0 &
       (for b6, b7 being real set holds
       (|[b6,b7]| in b1 ` implies b5 . |[b6,b7]| = 1) &
        (|[b6,b7]| in b1 implies b5 . |[b6,b7]| = |.|[b2,b3]| - |[b6,b7]|.| / b4));

:: TOPGEN_5:th 86
theorem
Niemytzki-plane is being_T1;

:: TOPGEN_5:th 87
theorem
Niemytzki-plane is Tychonoff;