Article BROUWER, MML version 4.99.1005

:: BROUWER:funcnot 1 => BROUWER:func 1
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  func DiffElems(A1,A2) -> Element of bool the carrier of [:a1,a2:] equals
    {[b1,b2] where b1 is Element of the carrier of a1, b2 is Element of the carrier of a2: b1 <> b2};
end;

:: BROUWER:def 1
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
DiffElems(b1,b2) = {[b3,b4] where b3 is Element of the carrier of b1, b4 is Element of the carrier of b2: b3 <> b4};

:: BROUWER:th 1
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being set holds
      b3 in DiffElems(b1,b2)
   iff
      ex b4 being Element of the carrier of b1 st
         ex b5 being Element of the carrier of b2 st
            b3 = [b4,b5] & b4 <> b5;

:: BROUWER:funcreg 1
registration
  let a1 be non empty non trivial TopSpace-like TopStruct;
  let a2 be non empty TopSpace-like TopStruct;
  cluster DiffElems(a1,a2) -> non empty;
end;

:: BROUWER:funcreg 2
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty non trivial TopSpace-like TopStruct;
  cluster DiffElems(a1,a2) -> non empty;
end;

:: BROUWER:th 2
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   cl_Ball(b2,0) = {b2};

:: BROUWER:funcnot 2 => BROUWER: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 Tdisk(A2,A3) -> SubSpace of TOP-REAL a1 equals
    (TOP-REAL a1) | cl_Ball(a2,a3);
end;

:: BROUWER: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
   Tdisk(b2,b3) = (TOP-REAL b1) | cl_Ball(b2,b3);

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

:: BROUWER:th 3
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1 holds
   the carrier of Tdisk(b3,b2) = cl_Ball(b3,b2);

:: BROUWER:funcreg 4
registration
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be real set;
  cluster Tdisk(a2,a3) -> convex;
end;

:: BROUWER:th 4
theorem
for b1 being Element of NAT
for b2 being non negative real set
for b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st b3 <> b4 & b3 is Element of the carrier of Tdisk(b5,b2) & b3 is not Element of the carrier of Tcircle(b5,b2)
   holds ex b6 being Element of the carrier of Tcircle(b5,b2) st
      {b6} = (halfline(b3,b4)) /\ Sphere(b5,b2);

:: BROUWER:th 5
theorem
for b1 being Element of NAT
for b2 being non negative real set
for b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st b3 <> b4 & b3 in the carrier of Tcircle(b5,b2) & b4 is Element of the carrier of Tdisk(b5,b2)
   holds ex b6 being Element of the carrier of Tcircle(b5,b2) st
      b6 <> b3 &
       {b3,b6} = (halfline(b3,b4)) /\ Sphere(b5,b2);

:: BROUWER:funcnot 3 => BROUWER:func 3
definition
  let a1 be non empty Element of NAT;
  let a2, a3, a4 be Element of the carrier of TOP-REAL a1;
  let a5 be non negative real set;
  assume a3 is Element of the carrier of Tdisk(a2,a5) & a4 is Element of the carrier of Tdisk(a2,a5) & a3 <> a4;
  func HC(A3,A4,A2,A5) -> Element of the carrier of TOP-REAL a1 means
    it in (halfline(a3,a4)) /\ Sphere(a2,a5) & it <> a3;
end;

:: BROUWER:def 3
theorem
for b1 being non empty Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
for b5 being non negative real set
   st b3 is Element of the carrier of Tdisk(b2,b5) & b4 is Element of the carrier of Tdisk(b2,b5) & b3 <> b4
for b6 being Element of the carrier of TOP-REAL b1 holds
      b6 = HC(b3,b4,b2,b5)
   iff
      b6 in (halfline(b3,b4)) /\ Sphere(b2,b5) & b6 <> b3;

:: BROUWER:th 6
theorem
for b1 being non negative real set
for b2 being non empty Element of NAT
for b3, b4, b5 being Element of the carrier of TOP-REAL b2
      st b3 is Element of the carrier of Tdisk(b4,b1) & b5 is Element of the carrier of Tdisk(b4,b1) & b3 <> b5
   holds HC(b3,b5,b4,b1) is Element of the carrier of Tcircle(b4,b1);

:: BROUWER:th 7
theorem
for b1 being real set
for b2 being non negative real set
for b3 being non empty Element of NAT
for b4, b5, b6 being Element of the carrier of TOP-REAL b3
for b7, b8, b9 being Element of REAL b3
      st b7 = b4 &
         b8 = b5 &
         b9 = b6 &
         b4 is Element of the carrier of Tdisk(b6,b2) &
         b5 is Element of the carrier of Tdisk(b6,b2) &
         b4 <> b5 &
         b1 = ((- |(b5 - b4,b4 - b6)|) + sqrt (|(b5 - b4,b4 - b6)| ^2 - ((Sum sqr (b8 - b7)) * ((Sum sqr (b7 - b9)) - (b2 ^2))))) / Sum sqr (b8 - b7)
   holds HC(b4,b5,b6,b2) = ((1 - b1) * b4) + (b1 * b5);

:: BROUWER:th 8
theorem
for b1 being real set
for b2 being non negative real set
for b3, b4, b5, b6 being real set
for b7, b8, b9 being Element of the carrier of TOP-REAL 2
      st b7 is Element of the carrier of Tdisk(b9,b2) &
         b8 is Element of the carrier of Tdisk(b9,b2) &
         b7 <> b8 &
         b3 = b8 `1 - (b7 `1) &
         b4 = b8 `2 - (b7 `2) &
         b5 = b7 `1 - (b9 `1) &
         b6 = b7 `2 - (b9 `2) &
         b1 = ((- ((b5 * b3) + (b6 * b4))) + sqrt (((b5 * b3) + (b6 * b4)) ^2 - ((b3 ^2 + (b4 ^2)) * ((b5 ^2 + (b6 ^2)) - (b2 ^2))))) / (b3 ^2 + (b4 ^2))
   holds HC(b7,b8,b9,b2) = |[b7 `1 + (b1 * b3),b7 `2 + (b1 * b4)]|;

:: BROUWER:funcnot 4 => BROUWER:func 4
definition
  let a1 be non empty Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be non negative real set;
  let a4 be Element of the carrier of Tdisk(a2,a3);
  let a5 be Function-like quasi_total Relation of the carrier of Tdisk(a2,a3),the carrier of Tdisk(a2,a3);
  assume not a4 is_a_fixpoint_of a5;
  func HC(A4,A5) -> Element of the carrier of Tcircle(a2,a3) means
    ex b1, b2 being Element of the carrier of TOP-REAL a1 st
       b1 = a4 & b2 = a5 . a4 & it = HC(b2,b1,a2,a3);
end;

:: BROUWER:def 4
theorem
for b1 being non empty Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being non negative real set
for b4 being Element of the carrier of Tdisk(b2,b3)
for b5 being Function-like quasi_total Relation of the carrier of Tdisk(b2,b3),the carrier of Tdisk(b2,b3)
   st not b4 is_a_fixpoint_of b5
for b6 being Element of the carrier of Tcircle(b2,b3) holds
      b6 = HC(b4,b5)
   iff
      ex b7, b8 being Element of the carrier of TOP-REAL b1 st
         b7 = b4 & b8 = b5 . b4 & b6 = HC(b8,b7,b2,b3);

:: BROUWER:th 9
theorem
for b1 being non negative real set
for b2 being non empty Element of NAT
for b3 being Element of the carrier of TOP-REAL b2
for b4 being Element of the carrier of Tdisk(b3,b1)
for b5 being Function-like quasi_total Relation of the carrier of Tdisk(b3,b1),the carrier of Tdisk(b3,b1)
      st not b4 is_a_fixpoint_of b5 & b4 is Element of the carrier of Tcircle(b3,b1)
   holds HC(b4,b5) = b4;

:: BROUWER:th 10
theorem
for b1 being positive real set
for b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty SubSpace of Tdisk(b2,b1)
      st b3 = Tcircle(b2,b1)
   holds not b3 is_a_retract_of Tdisk(b2,b1);

:: BROUWER:funcnot 5 => BROUWER:func 5
definition
  let a1 be non empty Element of NAT;
  let a2 be non negative real set;
  let a3 be Element of the carrier of TOP-REAL a1;
  let a4 be Function-like quasi_total Relation of the carrier of Tdisk(a3,a2),the carrier of Tdisk(a3,a2);
  func BR-map A4 -> Function-like quasi_total Relation of the carrier of Tdisk(a3,a2),the carrier of Tcircle(a3,a2) means
    for b1 being Element of the carrier of Tdisk(a3,a2) holds
       it . b1 = HC(b1,a4);
end;

:: BROUWER:def 5
theorem
for b1 being non empty Element of NAT
for b2 being non negative real set
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Function-like quasi_total Relation of the carrier of Tdisk(b3,b2),the carrier of Tdisk(b3,b2)
for b5 being Function-like quasi_total Relation of the carrier of Tdisk(b3,b2),the carrier of Tcircle(b3,b2) holds
      b5 = BR-map b4
   iff
      for b6 being Element of the carrier of Tdisk(b3,b2) holds
         b5 . b6 = HC(b6,b4);

:: BROUWER:th 11
theorem
for b1 being non negative real set
for b2 being non empty Element of NAT
for b3 being Element of the carrier of TOP-REAL b2
for b4 being Element of the carrier of Tdisk(b3,b1)
for b5 being Function-like quasi_total Relation of the carrier of Tdisk(b3,b1),the carrier of Tdisk(b3,b1)
      st not b4 is_a_fixpoint_of b5 & b4 is Element of the carrier of Tcircle(b3,b1)
   holds (BR-map b5) . b4 = b4;

:: BROUWER:th 12
theorem
for b1 being non negative real set
for b2 being non empty Element of NAT
for b3 being Element of the carrier of TOP-REAL b2
for b4 being Function-like quasi_total continuous Relation of the carrier of Tdisk(b3,b1),the carrier of Tdisk(b3,b1)
      st b4 has_no_fixpoint
   holds (BR-map b4) | Sphere(b3,b1) = id Tcircle(b3,b1);

:: BROUWER:th 13
theorem
for b1 being positive real set
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Function-like quasi_total continuous Relation of the carrier of Tdisk(b2,b1),the carrier of Tdisk(b2,b1)
      st b3 has_no_fixpoint
   holds BR-map b3 is continuous(Tdisk(b2,b1), Tcircle(b2,b1));

:: BROUWER:th 14
theorem
for b1 being non negative real set
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Function-like quasi_total continuous Relation of the carrier of Tdisk(b2,b1),the carrier of Tdisk(b2,b1) holds
   b3 has_a_fixpoint;

:: BROUWER:th 15
theorem
for b1 being non negative real set
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Function-like quasi_total continuous Relation of the carrier of Tdisk(b2,b1),the carrier of Tdisk(b2,b1) holds
   ex b4 being Element of the carrier of Tdisk(b2,b1) st
      b3 . b4 = b4;