Article TOPREAL6, MML version 4.99.1005

:: TOPREAL6:th 6
theorem
for b1, b2 being real set
      st 0 <= b1 & 0 <= b2
   holds sqrt (b1 + b2) <= (sqrt b1) + sqrt b2;

:: TOPREAL6:th 7
theorem
for b1, b2 being real set
      st 0 <= b1 & b1 <= b2
   holds abs b1 <= abs b2;

:: TOPREAL6:th 8
theorem
for b1, b2 being real set
      st b1 <= b2 & b2 <= 0
   holds abs b2 <= abs b1;

:: TOPREAL6:th 9
theorem
for b1 being Element of REAL holds
   Product (0 |-> b1) = 1;

:: TOPREAL6:th 10
theorem
for b1 being Element of REAL holds
   Product (1 |-> b1) = b1;

:: TOPREAL6:th 11
theorem
for b1 being Element of REAL holds
   Product (2 |-> b1) = b1 * b1;

:: TOPREAL6:th 12
theorem
for b1 being Element of REAL
for b2 being Element of NAT holds
   Product ((b2 + 1) |-> b1) = (Product (b2 |-> b1)) * b1;

:: TOPREAL6:th 13
theorem
for b1 being Element of REAL
for b2 being Element of NAT holds
      b2 <> 0 & b1 = 0
   iff
      Product (b2 |-> b1) = 0;

:: TOPREAL6:th 14
theorem
for b1 being Element of REAL
for b2, b3 being Element of NAT
      st b1 <> 0 & b2 <= b3
   holds Product ((b3 -' b2) |-> b1) = (Product (b3 |-> b1)) / Product (b2 |-> b1);

:: TOPREAL6:th 15
theorem
for b1 being Element of REAL
for b2, b3 being Element of NAT
      st b1 <> 0 & b2 <= b3
   holds b1 |^ (b3 -' b2) = (b1 |^ b3) / (b1 |^ b2);

:: TOPREAL6:th 16
theorem
for b1, b2 being Element of REAL holds
sqr <*b1,b2*> = <*b1 ^2,b2 ^2*>;

:: TOPREAL6:th 17
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being FinSequence of REAL
      st b2 in dom abs b3 & b1 = b3 . b2
   holds (abs b3) . b2 = abs b1;

:: TOPREAL6:th 18
theorem
for b1, b2 being Element of REAL holds
abs <*b1,b2*> = <*abs b1,abs b2*>;

:: TOPREAL6:th 19
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds (abs (b2 - b1)) + abs (b4 - b3) = (b2 - b1) + (b4 - b3);

:: TOPREAL6:th 20
theorem
for b1, b2 being real set
      st 0 < b2
   holds b1 in ].b1 - b2,b1 + b2.[;

:: TOPREAL6:th 21
theorem
for b1, b2 being real set
      st 0 <= b2
   holds b1 in [.b1 - b2,b1 + b2.];

:: TOPREAL6:th 22
theorem
for b1, b2 being real set
      st b1 < b2
   holds inf ].b1,b2.[ = b1 & sup ].b1,b2.[ = b2;

:: TOPREAL6:th 24
theorem
for b1 being bounded Element of bool REAL holds
   b1 c= [.inf b1,sup b1.];

:: TOPREAL6:funcreg 1
registration
  let a1 be TopStruct;
  let a2 be finite Element of bool the carrier of a1;
  cluster a1 | a2 -> finite strict;
end;

:: TOPREAL6:exreg 1
registration
  cluster non empty finite strict TopSpace-like TopStruct;
end;

:: TOPREAL6:condreg 1
registration
  let a1 be TopStruct;
  cluster empty -> connected (Element of bool the carrier of a1);
end;

:: TOPREAL6:th 25
theorem
for b1, b2 being TopSpace-like TopStruct
      st b1,b2 are_homeomorphic & b1 is connected
   holds b2 is connected;

:: TOPREAL6:th 26
theorem
for b1 being TopSpace-like TopStruct
for b2 being finite Element of bool bool the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 in b2
           holds b3 is compact(b1)
   holds union b2 is compact(b1);

:: TOPREAL6:th 29
theorem
for b1, b2, b3, b4, b5, b6 being set
      st b1 c= b2 & b3 c= b4
   holds product ((b5,b6)-->(b1,b3)) c= product ((b5,b6)-->(b2,b4));

:: TOPREAL6:th 30
theorem
for b1, b2 being Element of bool REAL holds
product ((1,2)-->(b1,b2)) is Element of bool the carrier of TOP-REAL 2;

:: TOPREAL6:th 31
theorem
for b1 being Element of REAL holds
   |.|[0,b1]|.| = abs b1 &
    |.|[b1,0]|.| = abs b1;

:: TOPREAL6:th 32
theorem
for b1 being Element of the carrier of Euclid 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 = 0.REAL 2 & b1 = b2
   holds b2 = <*0,0*> & b2 `1 = 0 & b2 `2 = 0;

:: TOPREAL6:th 33
theorem
for b1, b2 being Element of the carrier of Euclid 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 = 0.REAL 2 & b2 = b3
   holds dist(b1,b2) = |.b3.|;

:: TOPREAL6:th 34
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2 holds
   b1 * b2 = |[b1 * (b2 `1),b1 * (b2 `2)]|;

:: TOPREAL6:th 35
theorem
for b1 being Element of REAL
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b2 = ((1 - b1) * b3) + (b1 * b4) &
         b2 <> b3 &
         0 <= b1
   holds 0 < b1;

:: TOPREAL6:th 36
theorem
for b1 being Element of REAL
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b2 = ((1 - b1) * b3) + (b1 * b4) &
         b2 <> b4 &
         b1 <= 1
   holds b1 < 1;

:: TOPREAL6:th 37
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 in LSeg(b2,b3) & b1 <> b2 & b1 <> b3 & b2 `1 < b3 `1
   holds b2 `1 < b1 `1 & b1 `1 < b3 `1;

:: TOPREAL6:th 38
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 in LSeg(b2,b3) & b1 <> b2 & b1 <> b3 & b2 `2 < b3 `2
   holds b2 `2 < b1 `2 & b1 `2 < b3 `2;

:: TOPREAL6:th 39
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
   ex b3 being Element of the carrier of TOP-REAL 2 st
      b3 `1 < W-bound b1 & b2 <> b3;

:: TOPREAL6:th 40
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
   ex b3 being Element of the carrier of TOP-REAL 2 st
      E-bound b1 < b3 `1 & b2 <> b3;

:: TOPREAL6:th 41
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
   ex b3 being Element of the carrier of TOP-REAL 2 st
      N-bound b1 < b3 `2 & b2 <> b3;

:: TOPREAL6:th 42
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
   ex b3 being Element of the carrier of TOP-REAL 2 st
      b3 `2 < S-bound b1 & b2 <> b3;

:: TOPREAL6:condreg 2
registration
  cluster non horizontal -> non empty (Element of bool the carrier of TOP-REAL 2);
end;

:: TOPREAL6:condreg 3
registration
  cluster non vertical -> non empty (Element of bool the carrier of TOP-REAL 2);
end;

:: TOPREAL6:condreg 4
registration
  cluster being_Region -> open connected (Element of bool the carrier of TOP-REAL 2);
end;

:: TOPREAL6:condreg 5
registration
  cluster open connected -> being_Region (Element of bool the carrier of TOP-REAL 2);
end;

:: TOPREAL6:condreg 6
registration
  cluster empty -> horizontal (Element of bool the carrier of TOP-REAL 2);
end;

:: TOPREAL6:condreg 7
registration
  cluster empty -> vertical (Element of bool the carrier of TOP-REAL 2);
end;

:: TOPREAL6:exreg 2
registration
  cluster non empty convex Element of bool the carrier of TOP-REAL 2;
end;

:: TOPREAL6:funcreg 2
registration
  let a1, a2 be Element of the carrier of TOP-REAL 2;
  cluster LSeg(a1,a2) -> connected convex;
end;

:: TOPREAL6:funcreg 3
registration
  cluster R^2-unit_square -> connected;
end;

:: TOPREAL6:condreg 8
registration
  cluster being_simple_closed_curve -> connected (Element of bool the carrier of TOP-REAL 2);
end;

:: TOPREAL6:th 43
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   LSeg(NE-corner b1,SE-corner b1) c= L~ SpStSeq b1;

:: TOPREAL6:th 44
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   LSeg(SW-corner b1,SE-corner b1) c= L~ SpStSeq b1;

:: TOPREAL6:th 45
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   LSeg(SW-corner b1,NW-corner b1) c= L~ SpStSeq b1;

:: TOPREAL6:th 46
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `1 < W-bound b1} is non empty connected convex Element of bool the carrier of TOP-REAL 2;

:: TOPREAL6:th 47
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of the carrier of Euclid 2
for b4 being real set
      st b3 = b1 & b2 in Ball(b3,b4)
   holds b1 `1 - b4 < b2 `1 & b2 `1 < b1 `1 + b4;

:: TOPREAL6:th 48
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of the carrier of Euclid 2
for b4 being real set
      st b3 = b1 & b2 in Ball(b3,b4)
   holds b1 `2 - b4 < b2 `2 & b2 `2 < b1 `2 + b4;

:: TOPREAL6:th 49
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of the carrier of Euclid 2
for b3 being real set
      st b1 = b2
   holds product ((1,2)-->(].b1 `1 - (b3 / sqrt 2),b1 `1 + (b3 / sqrt 2).[,].b1 `2 - (b3 / sqrt 2),b1 `2 + (b3 / sqrt 2).[)) c= Ball(b2,b3);

:: TOPREAL6:th 50
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of the carrier of Euclid 2
for b3 being real set
      st b1 = b2
   holds Ball(b2,b3) c= product ((1,2)-->(].b1 `1 - b3,b1 `1 + b3.[,].b1 `2 - b3,b1 `2 + b3.[));

:: TOPREAL6:th 51
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of the carrier of Euclid 2
for b3 being Element of bool the carrier of TOP-REAL 2
for b4 being real set
      st b3 = Ball(b2,b4) & b1 = b2
   holds proj1 .: b3 = ].b1 `1 - b4,b1 `1 + b4.[;

:: TOPREAL6:th 52
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of the carrier of Euclid 2
for b3 being Element of bool the carrier of TOP-REAL 2
for b4 being real set
      st b3 = Ball(b2,b4) & b1 = b2
   holds proj2 .: b3 = ].b1 `2 - b4,b1 `2 + b4.[;

:: TOPREAL6:th 53
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of the carrier of Euclid 2
for b3 being non empty Element of bool the carrier of TOP-REAL 2
for b4 being real set
      st b3 = Ball(b2,b4) & b1 = b2
   holds W-bound b3 = b1 `1 - b4;

:: TOPREAL6:th 54
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of the carrier of Euclid 2
for b3 being non empty Element of bool the carrier of TOP-REAL 2
for b4 being real set
      st b3 = Ball(b2,b4) & b1 = b2
   holds E-bound b3 = b1 `1 + b4;

:: TOPREAL6:th 55
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of the carrier of Euclid 2
for b3 being non empty Element of bool the carrier of TOP-REAL 2
for b4 being real set
      st b3 = Ball(b2,b4) & b1 = b2
   holds S-bound b3 = b1 `2 - b4;

:: TOPREAL6:th 56
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of the carrier of Euclid 2
for b3 being non empty Element of bool the carrier of TOP-REAL 2
for b4 being real set
      st b3 = Ball(b2,b4) & b1 = b2
   holds N-bound b3 = b1 `2 + b4;

:: TOPREAL6:th 57
theorem
for b1 being Element of the carrier of Euclid 2
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being real set
      st b2 = Ball(b1,b3)
   holds b2 is not horizontal;

:: TOPREAL6:th 58
theorem
for b1 being Element of the carrier of Euclid 2
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being real set
      st b2 = Ball(b1,b3)
   holds b2 is not vertical;

:: TOPREAL6:th 59
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of Euclid 2
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in Ball(b2,b1)
   holds not |[b3 `1 - (2 * b1),b3 `2]| in Ball(b2,b1);

:: TOPREAL6:th 60
theorem
for b1 being Element of REAL
for b2 being non empty compact Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of Euclid 2
      st b3 = 0.REAL 2 & 0 < b1
   holds b2 c= Ball(b3,((((abs E-bound b2) + abs N-bound b2) + abs W-bound b2) + abs S-bound b2) + b1);

:: TOPREAL6:th 61
theorem
for b1 being real set
for b2 being non empty Reflexive symmetric triangle MetrStruct
for b3 being Element of the carrier of b2
      st b1 < 0
   holds Sphere(b3,b1) = {};

:: TOPREAL6:th 62
theorem
for b1 being non empty Reflexive discerning MetrStruct
for b2 being Element of the carrier of b1 holds
   Sphere(b2,0) = {b2};

:: TOPREAL6:th 63
theorem
for b1 being real set
for b2 being non empty Reflexive symmetric triangle MetrStruct
for b3 being Element of the carrier of b2
      st b1 < 0
   holds cl_Ball(b3,b1) = {};

:: TOPREAL6:th 64
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1 holds
   cl_Ball(b2,0) = {b2};

:: TOPREAL6:th 65
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being real set
for b4 being Element of bool the carrier of TopSpaceMetr b1
      st b4 = cl_Ball(b2,b3)
   holds b4 is closed(TopSpaceMetr b1);

:: TOPREAL6:th 66
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of Euclid b1
for b3 being Element of bool the carrier of TOP-REAL b1
for b4 being real set
      st b3 = cl_Ball(b2,b4)
   holds b3 is closed(TOP-REAL b1);

:: TOPREAL6:th 67
theorem
for b1 being real set
for b2 being non empty Reflexive symmetric triangle MetrStruct
for b3 being Element of the carrier of b2 holds
   cl_Ball(b3,b1) is bounded(b2);

:: TOPREAL6:th 68
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being real set
for b4 being Element of bool the carrier of TopSpaceMetr b1
      st b4 = Sphere(b2,b3)
   holds b4 is closed(TopSpaceMetr b1);

:: TOPREAL6:th 69
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of Euclid b1
for b3 being Element of bool the carrier of TOP-REAL b1
for b4 being real set
      st b3 = Sphere(b2,b4)
   holds b3 is closed(TOP-REAL b1);

:: TOPREAL6:th 70
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being real set holds
   Sphere(b2,b3) is bounded(b1);

:: TOPREAL6:th 71
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
      st b2 is Bounded(b1)
   holds Cl b2 is Bounded(b1);

:: TOPREAL6:th 72
theorem
for b1 being non empty MetrStruct holds
      b1 is bounded
   iff
      for b2 being Element of bool the carrier of b1 holds
         b2 is bounded(b1);

:: TOPREAL6:th 73
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2, b3 being Element of bool the carrier of b1
      st the carrier of b1 = b2 \/ b3 & b1 is not bounded & b2 is bounded(b1)
   holds b3 is not bounded(b1);

:: TOPREAL6:th 74
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
      st 1 <= b1 & the carrier of TOP-REAL b1 = b2 \/ b3 & b2 is Bounded(b1)
   holds b3 is not Bounded(b1);

:: TOPREAL6:th 76
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
      st b2 is Bounded(b1) & b3 is Bounded(b1)
   holds b2 \/ b3 is Bounded(b1);

:: TOPREAL6:funcreg 4
registration
  let a1 be non empty Element of bool REAL;
  cluster Cl a1 -> non empty;
end;

:: TOPREAL6:funcreg 5
registration
  let a1 be bounded_below Element of bool REAL;
  cluster Cl a1 -> bounded_below;
end;

:: TOPREAL6:funcreg 6
registration
  let a1 be bounded_above Element of bool REAL;
  cluster Cl a1 -> bounded_above;
end;

:: TOPREAL6:th 77
theorem
for b1 being non empty bounded_below Element of bool REAL holds
   inf b1 = inf Cl b1;

:: TOPREAL6:th 78
theorem
for b1 being non empty bounded_above Element of bool REAL holds
   sup b1 = sup Cl b1;

:: TOPREAL6:funcreg 7
registration
  cluster R^1 -> strict TopSpace-like being_T2;
end;

:: TOPREAL6:th 79
theorem
for b1 being Element of bool REAL
for b2 being Element of bool the carrier of R^1
      st b1 = b2
   holds    b1 is closed
   iff
      b2 is closed(R^1);

:: TOPREAL6:th 80
theorem
for b1 being Element of bool REAL
for b2 being Element of bool the carrier of R^1
      st b1 = b2
   holds Cl b1 = Cl b2;

:: TOPREAL6:th 81
theorem
for b1 being Element of bool REAL
for b2 being Element of bool the carrier of R^1
      st b1 = b2
   holds    b1 is compact
   iff
      b2 is compact(R^1);

:: TOPREAL6:condreg 9
registration
  cluster finite -> compact (Element of bool REAL);
end;

:: TOPREAL6:funcreg 8
registration
  let a1, a2 be real set;
  cluster [.a1,a2.] -> compact;
end;

:: TOPREAL6:th 82
theorem
for b1, b2 being real set holds
   b1 <> b2
iff
   Cl ].b1,b2.[ = [.b1,b2.];

:: TOPREAL6:exreg 3
registration
  cluster non empty finite complex-membered ext-real-membered real-membered bounded Element of bool REAL;
end;

:: TOPREAL6:th 83
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
      st b2 = b3
   holds    b2 is continuous(b1)
   iff
      b3 is continuous(b1, R^1);

:: TOPREAL6:th 84
theorem
for b1, b2 being Element of bool REAL
for b3 being Function-like quasi_total Relation of the carrier of [:R^1,R^1:],the carrier of TOP-REAL 2
      st for b4, b5 being Element of REAL holds
        b3 . [b4,b5] = <*b4,b5*>
   holds b3 .: [:b1,b2:] = product ((1,2)-->(b1,b2));

:: TOPREAL6:th 85
theorem
for b1 being Function-like quasi_total Relation of the carrier of [:R^1,R^1:],the carrier of TOP-REAL 2
      st for b2, b3 being Element of REAL holds
        b1 . [b2,b3] = <*b2,b3*>
   holds b1 is being_homeomorphism([:R^1,R^1:], TOP-REAL 2);

:: TOPREAL6:th 86
theorem
[:R^1,R^1:],TOP-REAL 2 are_homeomorphic;

:: TOPREAL6:th 87
theorem
for b1, b2 being compact Element of bool REAL holds
product ((1,2)-->(b1,b2)) is compact Element of bool the carrier of TOP-REAL 2;

:: TOPREAL6:th 88
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2) & b1 is closed(TOP-REAL 2)
   holds b1 is compact(TOP-REAL 2);

:: TOPREAL6:th 89
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
   st b1 is Bounded(2)
for b2 being Function-like quasi_total continuous Relation of the carrier of TOP-REAL 2,REAL holds
   Cl (b2 .: b1) c= b2 .: Cl b1;

:: TOPREAL6:th 90
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   proj1 .: Cl b1 c= Cl (proj1 .: b1);

:: TOPREAL6:th 91
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   proj2 .: Cl b1 c= Cl (proj2 .: b1);

:: TOPREAL6:th 92
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds Cl (proj1 .: b1) = proj1 .: Cl b1;

:: TOPREAL6:th 93
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds Cl (proj2 .: b1) = proj2 .: Cl b1;

:: TOPREAL6:th 94
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds W-bound b1 = W-bound Cl b1;

:: TOPREAL6:th 95
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds E-bound b1 = E-bound Cl b1;

:: TOPREAL6:th 96
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds N-bound b1 = N-bound Cl b1;

:: TOPREAL6:th 97
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds S-bound b1 = S-bound Cl b1;

:: TOPREAL6:th 98
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
      st (b2 is Bounded(b1) or b3 is Bounded(b1))
   holds b2 /\ b3 is Bounded(b1);

:: TOPREAL6:th 99
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
      st b2 is not Bounded(b1) & b3 is Bounded(b1)
   holds b2 \ b3 is not Bounded(b1);

:: TOPREAL6:funcnot 1 => TOPREAL6:func 1
definition
  let a1 be Element of NAT;
  let a2, a3 be Element of the carrier of TOP-REAL a1;
  func dist(A2,A3) -> Element of REAL means
    ex b1, b2 being Element of the carrier of Euclid a1 st
       b1 = a2 & b2 = a3 & it = dist(b1,b2);
  commutativity;
::  for a1 being Element of NAT
::  for a2, a3 being Element of the carrier of TOP-REAL a1 holds
::  dist(a2,a3) = dist(a3,a2);
end;

:: TOPREAL6:def 1
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of REAL holds
      b4 = dist(b2,b3)
   iff
      ex b5, b6 being Element of the carrier of Euclid b1 st
         b5 = b2 & b6 = b3 & b4 = dist(b5,b6);

:: TOPREAL6:th 100
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of Euclid 2
      st b5 = |[b1,b2]| & b6 = |[b3,b4]|
   holds dist(b5,b6) = sqrt ((b1 - b3) ^2 + ((b2 - b4) ^2));

:: TOPREAL6:th 101
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
dist(b1,b2) = sqrt ((b1 `1 - (b2 `1)) ^2 + ((b1 `2 - (b2 `2)) ^2));

:: TOPREAL6:th 102
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   dist(b2,b2) = 0;

:: TOPREAL6:th 103
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
dist(b2,b4) <= (dist(b2,b3)) + dist(b3,b4);

:: TOPREAL6:th 104
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
      st b1 <= b5 `1 & b5 `1 <= b2 & b3 <= b5 `2 & b5 `2 <= b4 & b1 <= b6 `1 & b6 `1 <= b2 & b3 <= b6 `2 & b6 `2 <= b4
   holds dist(b5,b6) <= (b2 - b1) + (b4 - b3);