Article TOPREAL9, MML version 4.99.1005

:: TOPREAL9:th 1
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
(b2 - b3) - b4 = (b2 - b4) - b3;

:: TOPREAL9:th 2
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 + b3 = b2 + b4
   holds b3 = b4;

:: TOPREAL9:th 3
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
      st b1 is not empty
   holds b2 <> b2 + 1.REAL b1;

:: TOPREAL9:th 4
theorem
for b1 being Element of NAT
for b2 being real set
for b3, b4 being Element of the carrier of TOP-REAL b1
for b5 being set
      st b5 = ((1 - b2) * b3) + (b2 * b4)
   holds (b5 = b3 & b2 <> 0 implies b3 = b4) &
    (b2 <> 0 & b3 <> b4 or b5 = b3) &
    (b5 = b4 & b2 <> 1 implies b3 = b4) &
    (b2 <> 1 & b3 <> b4 or b5 = b4);

:: TOPREAL9:th 5
theorem
for b1 being FinSequence of REAL holds
   |.b1.| ^2 = Sum sqr b1;

:: TOPREAL9:th 6
theorem
for b1 being real set
for b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3, b4, b5 being Element of the carrier of b2
      st b3 <> b4 & b3 in cl_Ball(b5,b1) & b4 in cl_Ball(b5,b1)
   holds 0 < b1;

:: TOPREAL9:funcnot 1 => TOPREAL9:func 1
definition
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real set;
  func Ball(A2,A3) -> Element of bool the carrier of TOP-REAL a1 equals
    {b1 where b1 is Element of the carrier of TOP-REAL a1: |.b1 - a2.| < a3};
end;

:: TOPREAL9:def 1
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being real set holds
   Ball(b2,b3) = {b4 where b4 is Element of the carrier of TOP-REAL b1: |.b4 - b2.| < b3};

:: TOPREAL9:funcnot 2 => TOPREAL9:func 2
definition
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real set;
  func cl_Ball(A2,A3) -> Element of bool the carrier of TOP-REAL a1 equals
    {b1 where b1 is Element of the carrier of TOP-REAL a1: |.b1 - a2.| <= a3};
end;

:: TOPREAL9:def 2
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being real set holds
   cl_Ball(b2,b3) = {b4 where b4 is Element of the carrier of TOP-REAL b1: |.b4 - b2.| <= b3};

:: TOPREAL9:funcnot 3 => TOPREAL9:func 3
definition
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real set;
  func Sphere(A2,A3) -> Element of bool the carrier of TOP-REAL a1 equals
    {b1 where b1 is Element of the carrier of TOP-REAL a1: |.b1 - a2.| = a3};
end;

:: TOPREAL9:def 3
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being real set holds
   Sphere(b2,b3) = {b4 where b4 is Element of the carrier of TOP-REAL b1: |.b4 - b2.| = b3};

:: TOPREAL9:th 7
theorem
for b1 being Element of NAT
for b2 being real set
for b3, b4 being Element of the carrier of TOP-REAL b1 holds
   b3 in Ball(b4,b2)
iff
   |.b3 - b4.| < b2;

:: TOPREAL9:th 8
theorem
for b1 being Element of NAT
for b2 being real set
for b3, b4 being Element of the carrier of TOP-REAL b1 holds
   b3 in cl_Ball(b4,b2)
iff
   |.b3 - b4.| <= b2;

:: TOPREAL9:th 9
theorem
for b1 being Element of NAT
for b2 being real set
for b3, b4 being Element of the carrier of TOP-REAL b1 holds
   b3 in Sphere(b4,b2)
iff
   |.b3 - b4.| = b2;

:: TOPREAL9:th 10
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
      st b3 in Ball(0.REAL b1,b2)
   holds |.b3.| < b2;

:: TOPREAL9:th 11
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
      st b3 in cl_Ball(0.REAL b1,b2)
   holds |.b3.| <= b2;

:: TOPREAL9:th 12
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
      st b3 in Sphere(0.REAL b1,b2)
   holds |.b3.| = b2;

:: TOPREAL9:th 13
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
      st b3 = b4
   holds Ball(b4,b2) = Ball(b3,b2);

:: TOPREAL9:th 14
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
      st b3 = b4
   holds cl_Ball(b4,b2) = cl_Ball(b3,b2);

:: TOPREAL9:th 15
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
      st b3 = b4
   holds Sphere(b4,b2) = Sphere(b3,b2);

:: TOPREAL9:th 16
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1 holds
   Ball(b3,b2) c= cl_Ball(b3,b2);

:: TOPREAL9:th 17
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1 holds
   Sphere(b3,b2) c= cl_Ball(b3,b2);

:: TOPREAL9:th 18
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1 holds
   (Ball(b3,b2)) \/ Sphere(b3,b2) = cl_Ball(b3,b2);

:: TOPREAL9:th 19
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1 holds
   Ball(b3,b2) misses Sphere(b3,b2);

:: TOPREAL9:funcreg 1
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real non positive set;
  cluster Ball(a2,a3) -> empty;
end;

:: TOPREAL9:funcreg 2
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real positive set;
  cluster Ball(a2,a3) -> non empty;
end;

:: TOPREAL9:th 20
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
      st Ball(b3,b2) is not empty
   holds 0 < b2;

:: TOPREAL9:th 21
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
      st Ball(b3,b2) is empty
   holds b2 <= 0;

:: TOPREAL9:funcreg 3
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real negative set;
  cluster cl_Ball(a2,a3) -> empty;
end;

:: TOPREAL9:funcreg 4
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real non negative set;
  cluster cl_Ball(a2,a3) -> non empty;
end;

:: TOPREAL9:th 22
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
      st cl_Ball(b3,b2) is not empty
   holds 0 <= b2;

:: TOPREAL9:th 23
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
      st cl_Ball(b3,b2) is empty
   holds b2 < 0;

:: TOPREAL9:th 24
theorem
for b1 being Element of NAT
for b2, b3, b4 being real set
for b5, b6, b7 being Element of the carrier of TOP-REAL b1
      st b2 + b3 = 1 & (abs b2) + abs b3 = 1 & b3 <> 0 & b5 in cl_Ball(b6,b4) & b7 in Ball(b6,b4)
   holds (b2 * b5) + (b3 * b7) in Ball(b6,b4);

:: TOPREAL9:funcreg 5
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real set;
  cluster Ball(a2,a3) -> open Bounded;
end;

:: TOPREAL9:funcreg 6
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real set;
  cluster cl_Ball(a2,a3) -> closed Bounded;
end;

:: TOPREAL9:funcreg 7
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real set;
  cluster Sphere(a2,a3) -> closed Bounded;
end;

:: TOPREAL9:funcreg 8
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real set;
  cluster Ball(a2,a3) -> convex;
end;

:: TOPREAL9:funcreg 9
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real set;
  cluster cl_Ball(a2,a3) -> convex;
end;

:: TOPREAL9:attrnot 1 => TOPREAL9:attr 1
definition
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of the carrier of TOP-REAL a1,the carrier of TOP-REAL a1;
  attr a2 is homogeneous means
    for b1 being real set
    for b2 being Element of the carrier of TOP-REAL a1 holds
       a2 . (b1 * b2) = b1 * (a2 . b2);
end;

:: TOPREAL9:dfs 4
definiens
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of the carrier of TOP-REAL a1,the carrier of TOP-REAL a1;
To prove
     a2 is homogeneous
it is sufficient to prove
  thus for b1 being real set
    for b2 being Element of the carrier of TOP-REAL a1 holds
       a2 . (b1 * b2) = b1 * (a2 . b2);

:: TOPREAL9:def 4
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL b1,the carrier of TOP-REAL b1 holds
      b2 is homogeneous(b1)
   iff
      for b3 being real set
      for b4 being Element of the carrier of TOP-REAL b1 holds
         b2 . (b3 * b4) = b3 * (b2 . b4);

:: TOPREAL9:attrnot 2 => TOPREAL9:attr 2
definition
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of the carrier of TOP-REAL a1,the carrier of TOP-REAL a1;
  attr a2 is additive means
    for b1, b2 being Element of the carrier of TOP-REAL a1 holds
    a2 . (b1 + b2) = (a2 . b1) + (a2 . b2);
end;

:: TOPREAL9:dfs 5
definiens
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of the carrier of TOP-REAL a1,the carrier of TOP-REAL a1;
To prove
     a2 is additive
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of TOP-REAL a1 holds
    a2 . (b1 + b2) = (a2 . b1) + (a2 . b2);

:: TOPREAL9:def 5
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL b1,the carrier of TOP-REAL b1 holds
      b2 is additive(b1)
   iff
      for b3, b4 being Element of the carrier of TOP-REAL b1 holds
      b2 . (b3 + b4) = (b2 . b3) + (b2 . b4);

:: TOPREAL9:funcreg 10
registration
  let a1 be Element of NAT;
  cluster (TOP-REAL a1) --> 0.REAL a1 -> Function-like quasi_total homogeneous additive;
end;

:: TOPREAL9:exreg 1
registration
  let a1 be Element of NAT;
  cluster Relation-like Function-like quasi_total continuous homogeneous additive Relation of the carrier of TOP-REAL a1,the carrier of TOP-REAL a1;
end;

:: TOPREAL9:funcreg 11
registration
  let a1, a2 be real set;
  cluster AffineMap(a1,0,a2,0) -> Function-like quasi_total homogeneous additive;
end;

:: TOPREAL9:th 25
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total homogeneous additive Relation of the carrier of TOP-REAL b1,the carrier of TOP-REAL b1
for b3 being convex Element of bool the carrier of TOP-REAL b1 holds
   b2 .: b3 is convex(b1);

:: TOPREAL9:funcnot 4 => TOPREAL9:func 4
definition
  let a1 be Element of NAT;
  let a2, a3 be Element of the carrier of TOP-REAL a1;
  func halfline(A2,A3) -> Element of bool the carrier of TOP-REAL a1 equals
    {((1 - b1) * a2) + (b1 * a3) where b1 is Element of REAL: 0 <= b1};
end;

:: TOPREAL9:def 6
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
halfline(b2,b3) = {((1 - b4) * b2) + (b4 * b3) where b4 is Element of REAL: 0 <= b4};

:: TOPREAL9:th 26
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being set holds
      b4 in halfline(b2,b3)
   iff
      ex b5 being real set st
         b4 = ((1 - b5) * b2) + (b5 * b3) &
          0 <= b5;

:: TOPREAL9:funcreg 12
registration
  let a1 be Element of NAT;
  let a2, a3 be Element of the carrier of TOP-REAL a1;
  cluster halfline(a2,a3) -> non empty;
end;

:: TOPREAL9:th 27
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
b2 in halfline(b2,b3);

:: TOPREAL9:th 28
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
b2 in halfline(b3,b2);

:: TOPREAL9:th 29
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   halfline(b2,b2) = {b2};

:: TOPREAL9:th 30
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 in halfline(b3,b4)
   holds halfline(b3,b2) c= halfline(b3,b4);

:: TOPREAL9:th 31
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 in halfline(b3,b4) & b2 <> b3
   holds halfline(b3,b4) = halfline(b3,b2);

:: TOPREAL9:th 32
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
LSeg(b2,b3) c= halfline(b2,b3);

:: TOPREAL9:funcreg 13
registration
  let a1 be Element of NAT;
  let a2, a3 be Element of the carrier of TOP-REAL a1;
  cluster halfline(a2,a3) -> convex;
end;

:: TOPREAL9:th 33
theorem
for b1 being Element of NAT
for b2 being real set
for b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st b3 in Sphere(b4,b2) & b5 in Ball(b4,b2)
   holds (LSeg(b3,b5)) /\ Sphere(b4,b2) = {b3};

:: TOPREAL9:th 34
theorem
for b1 being Element of NAT
for b2 being real set
for b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st b3 in Sphere(b4,b2) & b5 in Sphere(b4,b2)
   holds (LSeg(b3,b5)) \ {b3,b5} c= Ball(b4,b2);

:: TOPREAL9:th 35
theorem
for b1 being Element of NAT
for b2 being real set
for b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st b3 in Sphere(b4,b2) & b5 in Sphere(b4,b2)
   holds (LSeg(b3,b5)) /\ Sphere(b4,b2) = {b3,b5};

:: TOPREAL9:th 36
theorem
for b1 being Element of NAT
for b2 being real set
for b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st b3 in Sphere(b4,b2) & b5 in Sphere(b4,b2)
   holds (halfline(b3,b5)) /\ Sphere(b4,b2) = {b3,b5};

:: TOPREAL9:th 37
theorem
for b1 being Element of NAT
for b2, b3 being real set
for b4, b5, b6 being Element of the carrier of TOP-REAL b1
for b7, b8, b9 being Element of REAL b1
      st b7 = b4 &
         b8 = b5 &
         b9 = b6 &
         b4 <> b5 &
         b4 in Ball(b6,b2) &
         b3 = ((- (2 * |(b5 - b4,b4 - b6)|)) + sqrt delta(Sum sqr (b8 - b7),2 * |(b5 - b4,b4 - b6)|,(Sum sqr (b7 - b9)) - (b2 ^2))) / (2 * Sum sqr (b8 - b7))
   holds ex b10 being Element of the carrier of TOP-REAL b1 st
      {b10} = (halfline(b4,b5)) /\ Sphere(b6,b2) &
       b10 = ((1 - b3) * b4) + (b3 * b5);

:: TOPREAL9:th 38
theorem
for b1 being Element of NAT
for b2, b3 being real set
for b4, b5, b6 being Element of the carrier of TOP-REAL b1
for b7, b8, b9 being Element of REAL b1
      st b7 = ((1 / 2) * b4) + ((1 / 2) * b5) &
         b8 = b5 &
         b9 = b6 &
         b4 <> b5 &
         b4 in Sphere(b6,b2) &
         b5 in cl_Ball(b6,b2)
   holds ex b10 being Element of the carrier of TOP-REAL b1 st
      b10 <> b4 &
       {b4,b10} = (halfline(b4,b5)) /\ Sphere(b6,b2) &
       (b5 in Sphere(b6,b2) implies b10 = b5) &
       (not b5 in Sphere(b6,b2) &
        b3 = ((- (2 * |(b5 - (((1 / 2) * b4) + ((1 / 2) * b5)),(((1 / 2) * b4) + ((1 / 2) * b5)) - b6)|)) + sqrt delta(Sum sqr (b8 - b7),2 * |(b5 - (((1 / 2) * b4) + ((1 / 2) * b5)),(((1 / 2) * b4) + ((1 / 2) * b5)) - b6)|,(Sum sqr (b7 - b9)) - (b2 ^2))) / (2 * Sum sqr (b8 - b7)) implies b10 = ((1 - b3) * (((1 / 2) * b4) + ((1 / 2) * b5))) + (b3 * b5));

:: TOPREAL9:funcreg 14
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real negative set;
  cluster Sphere(a2,a3) -> empty;
end;

:: TOPREAL9:funcreg 15
registration
  let a1 be non empty Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real non negative set;
  cluster Sphere(a2,a3) -> non empty;
end;

:: TOPREAL9:th 39
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
      st Sphere(b3,b2) is not empty
   holds 0 <= b2;

:: TOPREAL9:th 40
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
      st b1 is not empty & Sphere(b3,b2) is empty
   holds b2 < 0;

:: TOPREAL9:th 41
theorem
for b1, b2 being real set
for b3, b4 being Element of the carrier of TOP-REAL 2 holds
((b1 * b3) + (b2 * b4)) `1 = (b1 * (b3 `1)) + (b2 * (b4 `1));

:: TOPREAL9:th 42
theorem
for b1, b2 being real set
for b3, b4 being Element of the carrier of TOP-REAL 2 holds
((b1 * b3) + (b2 * b4)) `2 = (b1 * (b3 `2)) + (b2 * (b4 `2));

:: TOPREAL9:th 43
theorem
for b1, b2, b3 being real set
for b4 being Element of the carrier of TOP-REAL 2 holds
      b4 in circle(b1,b2,b3)
   iff
      |.b4 - |[b1,b2]|.| = b3;

:: TOPREAL9:th 44
theorem
for b1, b2, b3 being real set
for b4 being Element of the carrier of TOP-REAL 2 holds
      b4 in closed_inside_of_circle(b1,b2,b3)
   iff
      |.b4 - |[b1,b2]|.| <= b3;

:: TOPREAL9:th 45
theorem
for b1, b2, b3 being real set
for b4 being Element of the carrier of TOP-REAL 2 holds
      b4 in inside_of_circle(b1,b2,b3)
   iff
      |.b4 - |[b1,b2]|.| < b3;

:: TOPREAL9:funcreg 16
registration
  let a1, a2 be real set;
  let a3 be real positive set;
  cluster inside_of_circle(a1,a2,a3) -> non empty;
end;

:: TOPREAL9:funcreg 17
registration
  let a1, a2 be real set;
  let a3 be real non negative set;
  cluster closed_inside_of_circle(a1,a2,a3) -> non empty;
end;

:: TOPREAL9:th 46
theorem
for b1, b2, b3 being real set holds
circle(b1,b2,b3) c= closed_inside_of_circle(b1,b2,b3);

:: TOPREAL9:th 47
theorem
for b1, b2, b3 being real set
for b4 being Element of the carrier of Euclid 2
      st b4 = |[b1,b2]|
   holds cl_Ball(b4,b3) = closed_inside_of_circle(b1,b2,b3);

:: TOPREAL9:th 48
theorem
for b1, b2, b3 being real set
for b4 being Element of the carrier of Euclid 2
      st b4 = |[b1,b2]|
   holds Ball(b4,b3) = inside_of_circle(b1,b2,b3);

:: TOPREAL9:th 49
theorem
for b1, b2, b3 being real set
for b4 being Element of the carrier of Euclid 2
      st b4 = |[b1,b2]|
   holds Sphere(b4,b3) = circle(b1,b2,b3);

:: TOPREAL9:th 50
theorem
for b1, b2, b3 being real set holds
Ball(|[b1,b2]|,b3) = inside_of_circle(b1,b2,b3);

:: TOPREAL9:th 51
theorem
for b1, b2, b3 being real set holds
cl_Ball(|[b1,b2]|,b3) = closed_inside_of_circle(b1,b2,b3);

:: TOPREAL9:th 52
theorem
for b1, b2, b3 being real set holds
Sphere(|[b1,b2]|,b3) = circle(b1,b2,b3);

:: TOPREAL9:th 53
theorem
for b1, b2, b3 being real set holds
inside_of_circle(b1,b2,b3) c= closed_inside_of_circle(b1,b2,b3);

:: TOPREAL9:th 54
theorem
for b1, b2, b3 being real set holds
inside_of_circle(b1,b2,b3) misses circle(b1,b2,b3);

:: TOPREAL9:th 55
theorem
for b1, b2, b3 being real set holds
(inside_of_circle(b1,b2,b3)) \/ circle(b1,b2,b3) = closed_inside_of_circle(b1,b2,b3);

:: TOPREAL9:th 56
theorem
for b1 being real set
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in Sphere(0.REAL 2,b1)
   holds b2 `1 ^2 + (b2 `2 ^2) = b1 ^2;

:: TOPREAL9:th 57
theorem
for b1, b2, b3 being real set
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 <> b5 & b4 in closed_inside_of_circle(b1,b2,b3) & b5 in closed_inside_of_circle(b1,b2,b3)
   holds 0 < b3;

:: TOPREAL9:th 58
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
for b7, b8, b9 being Element of REAL 2
      st b7 = b5 &
         b8 = b6 &
         b9 = |[b1,b2]| &
         b3 = ((- (2 * |(b6 - b5,b5 - |[b1,b2]|)|)) + sqrt delta(Sum sqr (b8 - b7),2 * |(b6 - b5,b5 - |[b1,b2]|)|,(Sum sqr (b7 - b9)) - (b4 ^2))) / (2 * Sum sqr (b8 - b7)) &
         b5 <> b6 &
         b5 in inside_of_circle(b1,b2,b4)
   holds ex b10 being Element of the carrier of TOP-REAL 2 st
      {b10} = (halfline(b5,b6)) /\ circle(b1,b2,b4) &
       b10 = ((1 - b3) * b5) + (b3 * b6);

:: TOPREAL9:th 59
theorem
for b1, b2, b3 being real set
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 in circle(b1,b2,b3) & b5 in inside_of_circle(b1,b2,b3)
   holds (LSeg(b4,b5)) /\ circle(b1,b2,b3) = {b4};

:: TOPREAL9:th 60
theorem
for b1, b2, b3 being real set
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 in circle(b1,b2,b3) & b5 in circle(b1,b2,b3)
   holds (LSeg(b4,b5)) \ {b4,b5} c= inside_of_circle(b1,b2,b3);

:: TOPREAL9:th 61
theorem
for b1, b2, b3 being real set
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 in circle(b1,b2,b3) & b5 in circle(b1,b2,b3)
   holds (LSeg(b4,b5)) /\ circle(b1,b2,b3) = {b4,b5};

:: TOPREAL9:th 62
theorem
for b1, b2, b3 being real set
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 in circle(b1,b2,b3) & b5 in circle(b1,b2,b3)
   holds (halfline(b4,b5)) /\ circle(b1,b2,b3) = {b4,b5};

:: TOPREAL9:th 63
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
for b7, b8, b9 being Element of REAL 2
      st b7 = ((1 / 2) * b5) + ((1 / 2) * b6) &
         b8 = b6 &
         b9 = |[b1,b2]| &
         b5 <> b6 &
         b5 in circle(b1,b2,b3) &
         b6 in closed_inside_of_circle(b1,b2,b3)
   holds ex b10 being Element of the carrier of TOP-REAL 2 st
      b10 <> b5 &
       {b5,b10} = (halfline(b5,b6)) /\ circle(b1,b2,b3) &
       (b6 in Sphere(|[b1,b2]|,b3) implies b10 = b6) &
       (not b6 in Sphere(|[b1,b2]|,b3) &
        b4 = ((- (2 * |(b6 - (((1 / 2) * b5) + ((1 / 2) * b6)),(((1 / 2) * b5) + ((1 / 2) * b6)) - |[b1,b2]|)|)) + sqrt delta(Sum sqr (b8 - b7),2 * |(b6 - (((1 / 2) * b5) + ((1 / 2) * b6)),(((1 / 2) * b5) + ((1 / 2) * b6)) - |[b1,b2]|)|,(Sum sqr (b7 - b9)) - (b3 ^2))) / (2 * Sum sqr (b8 - b7)) implies b10 = ((1 - b4) * (((1 / 2) * b5) + ((1 / 2) * b6))) + (b4 * b6));

:: TOPREAL9:funcreg 18
registration
  let a1, a2, a3 be real set;
  cluster inside_of_circle(a1,a2,a3) -> convex;
end;

:: TOPREAL9:funcreg 19
registration
  let a1, a2, a3 be real set;
  cluster closed_inside_of_circle(a1,a2,a3) -> convex;
end;