Article TOPREALA, MML version 4.99.1005

:: TOPREALA:th 1
theorem
for b1 being integer set
for b2 being real set holds
   frac (b2 + b1) = frac b2;

:: TOPREALA:th 2
theorem
for b1, b2 being real set
      st b1 <= b2 & b2 < [\b1/] + 1
   holds [\b2/] = [\b1/];

:: TOPREALA:th 3
theorem
for b1, b2 being real set
      st b1 <= b2 & b2 < [\b1/] + 1
   holds frac b1 <= frac b2;

:: TOPREALA:th 4
theorem
for b1, b2 being real set
      st b1 < b2 & b2 < [\b1/] + 1
   holds frac b1 < frac b2;

:: TOPREALA:th 5
theorem
for b1, b2 being real set
      st [\b2/] + 1 <= b1 & b1 <= b2 + 1
   holds [\b1/] = [\b2/] + 1;

:: TOPREALA:th 6
theorem
for b1, b2 being real set
      st [\b2/] + 1 <= b1 & b1 < b2 + 1
   holds frac b1 < frac b2;

:: TOPREALA:th 7
theorem
for b1, b2, b3 being real set
      st b1 <= b2 & b2 < b1 + 1 & b1 <= b3 & b3 < b1 + 1 & frac b2 = frac b3
   holds b2 = b3;

:: TOPREALA:funcreg 1
registration
  let a1 be real set;
  let a2 be real positive set;
  cluster ].a1,a1 + a2.[ -> non empty;
end;

:: TOPREALA:funcreg 2
registration
  let a1 be real set;
  let a2 be real positive set;
  cluster [.a1,a1 + a2.[ -> non empty;
end;

:: TOPREALA:funcreg 3
registration
  let a1 be real set;
  let a2 be real positive set;
  cluster ].a1,a1 + a2.] -> non empty;
end;

:: TOPREALA:funcreg 4
registration
  let a1 be real set;
  let a2 be real positive set;
  cluster [.a1,a1 + a2.] -> non empty;
end;

:: TOPREALA:funcreg 5
registration
  let a1 be real set;
  let a2 be real positive set;
  cluster ].a1 - a2,a1.[ -> non empty;
end;

:: TOPREALA:funcreg 6
registration
  let a1 be real set;
  let a2 be real positive set;
  cluster [.a1 - a2,a1.[ -> non empty;
end;

:: TOPREALA:funcreg 7
registration
  let a1 be real set;
  let a2 be real positive set;
  cluster ].a1 - a2,a1.] -> non empty;
end;

:: TOPREALA:funcreg 8
registration
  let a1 be real set;
  let a2 be real positive set;
  cluster [.a1 - a2,a1.] -> non empty;
end;

:: TOPREALA:funcreg 9
registration
  let a1 be real non positive set;
  let a2 be real positive set;
  cluster ].a1,a2.[ -> non empty;
end;

:: TOPREALA:funcreg 10
registration
  let a1 be real non positive set;
  let a2 be real positive set;
  cluster [.a1,a2.[ -> non empty;
end;

:: TOPREALA:funcreg 11
registration
  let a1 be real non positive set;
  let a2 be real positive set;
  cluster ].a1,a2.] -> non empty;
end;

:: TOPREALA:funcreg 12
registration
  let a1 be real non positive set;
  let a2 be real positive set;
  cluster [.a1,a2.] -> non empty;
end;

:: TOPREALA:funcreg 13
registration
  let a1 be real negative set;
  let a2 be real non negative set;
  cluster ].a1,a2.[ -> non empty;
end;

:: TOPREALA:funcreg 14
registration
  let a1 be real negative set;
  let a2 be real non negative set;
  cluster [.a1,a2.[ -> non empty;
end;

:: TOPREALA:funcreg 15
registration
  let a1 be real negative set;
  let a2 be real non negative set;
  cluster ].a1,a2.] -> non empty;
end;

:: TOPREALA:funcreg 16
registration
  let a1 be real negative set;
  let a2 be real non negative set;
  cluster [.a1,a2.] -> non empty;
end;

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

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

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

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

:: TOPREALA:th 12
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds [.b2,b3.[ c= [.b1,b4.[;

:: TOPREALA:th 13
theorem
for b1, b2, b3, b4 being real set
      st b1 < b2 & b3 <= b4
   holds [.b2,b3.[ c= ].b1,b4.];

:: TOPREALA:th 14
theorem
for b1, b2, b3, b4 being real set
      st b1 < b2 & b3 <= b4
   holds [.b2,b3.[ c= ].b1,b4.[;

:: TOPREALA:th 15
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.] c= [.b1,b4.];

:: TOPREALA:th 16
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 < b4
   holds ].b2,b3.] c= [.b1,b4.[;

:: TOPREALA:th 17
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.] c= ].b1,b4.];

:: TOPREALA:th 18
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 < b4
   holds ].b2,b3.] c= ].b1,b4.[;

:: TOPREALA:th 19
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.[ c= [.b1,b4.];

:: TOPREALA:th 20
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.[ c= [.b1,b4.[;

:: TOPREALA:th 21
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.[ c= ].b1,b4.];

:: TOPREALA:th 22
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set
      st b2 in proj1 b1 & b1 . b2 in b1 .: b3 & b1 is one-to-one
   holds b2 in b3;

:: TOPREALA:th 23
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being natural set
      st b2 + 1 in dom b1 & not b2 in dom b1
   holds b2 = 0;

:: TOPREALA:th 24
theorem
for b1, b2, b3, b4 being set
for b5 being Relation-like Function-like set
      st b1 <> b2 & b5 in product ((b1,b2)-->(b3,b4))
   holds b5 . b1 in b3 & b5 . b2 in b4;

:: TOPREALA:th 25
theorem
for b1, b2 being set holds
<*b1,b2*> = (1,2)-->(b1,b2);

:: TOPREALA:exreg 1
registration
  cluster non empty strict TopSpace-like constituted-FinSeqs TopStruct;
end;

:: TOPREALA:condreg 1
registration
  let a1 be TopSpace-like constituted-FinSeqs TopStruct;
  cluster -> constituted-FinSeqs (SubSpace of a1);
end;

:: TOPREALA:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being open a_neighborhood of b3
for b6 being Element of bool the carrier of b2
      st b3 = b4 & b6 = b5 /\ [#] b2
   holds b6 is open a_neighborhood of b4;

:: TOPREALA:condreg 2
registration
  cluster empty TopSpace-like -> discrete anti-discrete (TopStruct);
end;

:: TOPREALA:condreg 3
registration
  let a1 be TopSpace-like discrete TopStruct;
  let a2 be TopSpace-like TopStruct;
  cluster Function-like quasi_total -> continuous (Relation of the carrier of a1,the carrier of a2);
end;

:: TOPREALA:th 27
theorem
for b1 being TopSpace-like TopStruct
for b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is empty
   holds b3 is continuous(b1, b2);

:: TOPREALA:condreg 4
registration
  let a1 be TopSpace-like TopStruct;
  let a2 be TopStruct;
  cluster empty Function-like quasi_total -> continuous (Relation of the carrier of a1,the carrier of a2);
end;

:: TOPREALA:th 28
theorem
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being non empty SubSpace of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3 holds
   b4 is Function-like quasi_total Relation of the carrier of b1,the carrier of b2;

:: TOPREALA:th 29
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
for b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b1 | b3,the carrier of b2 | b4
      st b6 = b5 | b3
   holds b6 is continuous(b1 | b3, b2 | b4);

:: TOPREALA:th 30
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b2
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 b1,the carrier of b3
      st b4 = b5 & b4 is open(b1, b2)
   holds b5 is open(b1, b3);

:: TOPREALA:th 31
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b1 | b3,the carrier of b2 | b4
      st b6 = b5 | b3 &
         b6 is onto(the carrier of b1 | b3, the carrier of b2 | b4) &
         b5 is open(b1, b2) &
         b5 is one-to-one
   holds b6 is open(b1 | b3, b2 | b4);

:: TOPREALA:th 32
theorem
for b1, b2, b3 being non empty TopSpace-like 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 b4 is open(b1, b2) & b5 is open(b2, b3)
   holds b5 * b4 is open(b1, b3);

:: TOPREALA:th 33
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being open SubSpace of b2
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 b1,the carrier of b3
      st b4 = b5 & b5 is open(b1, b3)
   holds b4 is open(b1, b2);

:: TOPREALA:th 34
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is one-to-one & b3 is onto(the carrier of b1, the carrier of b2)
   holds    b3 is continuous(b1, b2)
   iff
      b3 /" is open(b2, b1);

:: TOPREALA:th 35
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is one-to-one & b3 is onto(the carrier of b1, the carrier of b2)
   holds    b3 is open(b1, b2)
   iff
      b3 /" is continuous(b2, b1);

:: TOPREALA:th 36
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct holds
      b1,b2 are_homeomorphic
   iff
      TopStruct(#the carrier of b1,the topology of b1#),TopStruct(#the carrier of b2,the topology of b2#) are_homeomorphic;

:: TOPREALA:th 37
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is one-to-one & b3 is onto(the carrier of b1, the carrier of b2) & b3 is continuous(b1, b2) & b3 is open(b1, b2)
   holds b3 is being_homeomorphism(b1, b2);

:: TOPREALA:th 38
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL
      st b2 = REAL --> b1
   holds b2 is_continuous_on REAL;

:: TOPREALA:th 39
theorem
for b1, b2, b3 being Function-like Relation of REAL,REAL
      st dom b1 = (dom b2) \/ dom b3 &
         dom b2 is open &
         dom b3 is open &
         b2 is_continuous_on dom b2 &
         b3 is_continuous_on dom b3 &
         (for b4 being set
               st b4 in dom b2
            holds b1 . b4 = b2 . b4) &
         (for b4 being set
               st b4 in dom b3
            holds b1 . b4 = b3 . b4)
   holds b1 is_continuous_on dom b1;

:: TOPREALA:th 40
theorem
for b1 being Element of the carrier of R^1
for b2 being Element of bool REAL
for b3 being Element of bool the carrier of R^1
      st b3 = b2 & b2 is Neighbourhood of b1
   holds b3 is a_neighborhood of b1;

:: TOPREALA:th 41
theorem
for b1 being Element of the carrier of R^1
for b2 being a_neighborhood of b1 holds
   ex b3 being Neighbourhood of b1 st
      b3 c= b2;

:: TOPREALA:th 42
theorem
for b1 being Function-like quasi_total Relation of the carrier of R^1,the carrier of R^1
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of the carrier of R^1
      st b1 = b2 & b2 is_continuous_in b3
   holds b1 is_continuous_at b3;

:: TOPREALA:th 43
theorem
for b1 being Function-like quasi_total Relation of the carrier of R^1,the carrier of R^1
for b2 being Function-like quasi_total Relation of REAL,REAL
      st b1 = b2 & b2 is_continuous_on REAL
   holds b1 is continuous(R^1, R^1);

:: TOPREALA:th 44
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds [.b2,b3.] is closed Element of bool the carrier of Closed-Interval-TSpace(b1,b4);

:: TOPREALA:th 45
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.[ is open Element of bool the carrier of Closed-Interval-TSpace(b1,b4);

:: TOPREALA:th 46
theorem
for b1, b2, b3 being real set
      st b1 <= b2 & b1 <= b3
   holds ].b3,b2.] is open Element of bool the carrier of Closed-Interval-TSpace(b1,b2);

:: TOPREALA:th 47
theorem
for b1, b2, b3 being real set
      st b1 <= b2 & b3 <= b2
   holds [.b1,b3.[ is open Element of bool the carrier of Closed-Interval-TSpace(b1,b2);

:: TOPREALA:th 48
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds the carrier of [:Closed-Interval-TSpace(b1,b2),Closed-Interval-TSpace(b3,b4):] = [:[.b1,b2.],[.b3,b4.]:];

:: TOPREALA:th 49
theorem
for b1, b2 being real set holds
|[b1,b2]| = (1,2)-->(b1,b2);

:: TOPREALA:th 50
theorem
for b1, b2 being real set holds
|[b1,b2]| . 1 = b1 & |[b1,b2]| . 2 = b2;

:: TOPREALA:th 51
theorem
for b1, b2, b3, b4 being real set holds
closed_inside_of_rectangle(b1,b2,b3,b4) = product ((1,2)-->([.b1,b2.],[.b3,b4.]));

:: TOPREALA:th 52
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds |[b1,b3]| in closed_inside_of_rectangle(b1,b2,b3,b4);

:: TOPREALA:funcnot 1 => TOPREALA:func 1
definition
  let a1, a2, a3, a4 be real set;
  func Trectangle(A1,A2,A3,A4) -> SubSpace of TOP-REAL 2 equals
    (TOP-REAL 2) | closed_inside_of_rectangle(a1,a2,a3,a4);
end;

:: TOPREALA:def 1
theorem
for b1, b2, b3, b4 being real set holds
Trectangle(b1,b2,b3,b4) = (TOP-REAL 2) | closed_inside_of_rectangle(b1,b2,b3,b4);

:: TOPREALA:th 54
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds Trectangle(b1,b2,b3,b4) is not empty;

:: TOPREALA:funcreg 17
registration
  let a1, a2 be real non positive set;
  let a3, a4 be real non negative set;
  cluster Trectangle(a1,a3,a2,a4) -> non empty;
end;

:: TOPREALA:funcnot 2 => TOPREALA:func 2
definition
  func R2Homeomorphism -> Function-like quasi_total Relation of the carrier of [:R^1,R^1:],the carrier of TOP-REAL 2 means
    for b1, b2 being real set holds
    it . [b1,b2] = <*b1,b2*>;
end;

:: TOPREALA:def 2
theorem
for b1 being Function-like quasi_total Relation of the carrier of [:R^1,R^1:],the carrier of TOP-REAL 2 holds
      b1 = R2Homeomorphism
   iff
      for b2, b3 being real set holds
      b1 . [b2,b3] = <*b2,b3*>;

:: TOPREALA:th 55
theorem
for b1, b2 being Element of bool REAL holds
R2Homeomorphism .: [:b1,b2:] = product ((1,2)-->(b1,b2));

:: TOPREALA:th 56
theorem
R2Homeomorphism is being_homeomorphism([:R^1,R^1:], TOP-REAL 2);

:: TOPREALA:th 57
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds R2Homeomorphism | the carrier of [:Closed-Interval-TSpace(b1,b2),Closed-Interval-TSpace(b3,b4):] is Function-like quasi_total Relation of the carrier of [:Closed-Interval-TSpace(b1,b2),Closed-Interval-TSpace(b3,b4):],the carrier of Trectangle(b1,b2,b3,b4);

:: TOPREALA:th 58
theorem
for b1, b2, b3, b4 being real set
   st b1 <= b2 & b3 <= b4
for b5 being Function-like quasi_total Relation of the carrier of [:Closed-Interval-TSpace(b1,b2),Closed-Interval-TSpace(b3,b4):],the carrier of Trectangle(b1,b2,b3,b4)
      st b5 = R2Homeomorphism | the carrier of [:Closed-Interval-TSpace(b1,b2),Closed-Interval-TSpace(b3,b4):]
   holds b5 is being_homeomorphism([:Closed-Interval-TSpace(b1,b2),Closed-Interval-TSpace(b3,b4):], Trectangle(b1,b2,b3,b4));

:: TOPREALA:th 59
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds [:Closed-Interval-TSpace(b1,b2),Closed-Interval-TSpace(b3,b4):],Trectangle(b1,b2,b3,b4) are_homeomorphic;

:: TOPREALA:th 60
theorem
for b1, b2, b3, b4 being real set
   st b1 <= b2 & b3 <= b4
for b5 being Element of bool the carrier of Closed-Interval-TSpace(b1,b2)
for b6 being Element of bool the carrier of Closed-Interval-TSpace(b3,b4) holds
   product ((1,2)-->(b5,b6)) is Element of bool the carrier of Trectangle(b1,b2,b3,b4);

:: TOPREALA:th 61
theorem
for b1, b2, b3, b4 being real set
   st b1 <= b2 & b3 <= b4
for b5 being open Element of bool the carrier of Closed-Interval-TSpace(b1,b2)
for b6 being open Element of bool the carrier of Closed-Interval-TSpace(b3,b4) holds
   product ((1,2)-->(b5,b6)) is open Element of bool the carrier of Trectangle(b1,b2,b3,b4);

:: TOPREALA:th 62
theorem
for b1, b2, b3, b4 being real set
   st b1 <= b2 & b3 <= b4
for b5 being closed Element of bool the carrier of Closed-Interval-TSpace(b1,b2)
for b6 being closed Element of bool the carrier of Closed-Interval-TSpace(b3,b4) holds
   product ((1,2)-->(b5,b6)) is closed Element of bool the carrier of Trectangle(b1,b2,b3,b4);