:: BROUWER semantic presentation
set c1 = TOP-REAL 2;
Lemma1:
[#] (TOP-REAL 2) = the carrier of (TOP-REAL 2)
by PRE_TOPC:12;
:: deftheorem Def1 defines DiffElems BROUWER:def 1 :
theorem Th1: :: BROUWER:1
theorem Th2: :: BROUWER:2
:: deftheorem Def2 defines Tdisk BROUWER:def 2 :
theorem Th3: :: BROUWER:3
theorem Th4: :: BROUWER:4
theorem Th5: :: BROUWER:5
definition
let c
2 be non
empty Nat;
let c
3 be
Point of
(TOP-REAL c2);
let c
4, c
5 be
Point of
(TOP-REAL c2);
let c
6 be non
negative real number ;
assume that E6:
c
4 is
Point of
(Tdisk c3,c6)
and E7:
c
5 is
Point of
(Tdisk c3,c6)
and E8:
c
4 <> c
5
;
func HC c
3,c
4,c
2,c
5 -> Point of
(TOP-REAL a1) means :
Def3:
:: BROUWER:def 3
( a
6 in (halfline a3,a4) /\ (Sphere a2,a5) & a
6 <> a
3 );
existence
ex b1 being Point of (TOP-REAL c2) st
( b1 in (halfline c4,c5) /\ (Sphere c3,c6) & b1 <> c4 )
uniqueness
for b1, b2 being Point of (TOP-REAL c2) holds
( b1 in (halfline c4,c5) /\ (Sphere c3,c6) & b1 <> c4 & b2 in (halfline c4,c5) /\ (Sphere c3,c6) & b2 <> c4 implies b1 = b2 )
end;
:: deftheorem Def3 defines HC BROUWER:def 3 :
theorem Th6: :: BROUWER:6
theorem Th7: :: BROUWER:7
for b
1 being
real number for b
2 being non
negative real number for b
3 being non
empty Natfor b
4, b
5, b
6 being
Point of
(TOP-REAL b3)for b
7, b
8, b
9 being
Element of
REAL b
3 holds
( b
7 = b
4 & b
8 = b
5 & b
9 = b
6 & b
4 is
Point of
(Tdisk b6,b2) & b
5 is
Point of
(Tdisk b6,b2) & b
4 <> b
5 & b
1 = ((- |((b5 - b4),(b4 - b6))|) + (sqrt ((|((b5 - b4),(b4 - b6))| ^2 ) - ((Sum (sqr (b8 - b7))) * ((Sum (sqr (b7 - b9))) - (b2 ^2 )))))) / (Sum (sqr (b8 - b7))) implies
HC b
4,b
5,b
6,b
2 = ((1 - b1) * b4) + (b1 * b5) )
theorem Th8: :: BROUWER:8
for b
1 being
real number for b
2 being non
negative real number for b
3, b
4, b
5, b
6 being
real number for b
7, b
8, b
9 being
Point of
(TOP-REAL 2) holds
( b
7 is
Point of
(Tdisk b9,b2) & b
8 is
Point of
(Tdisk b9,b2) & b
7 <> b
8 & b
3 = (b8 `1 ) - (b7 `1 ) & b
4 = (b8 `2 ) - (b7 `2 ) & b
5 = (b7 `1 ) - (b9 `1 ) & b
6 = (b7 `2 ) - (b9 `2 ) & b
1 = ((- ((b5 * b3) + (b6 * b4))) + (sqrt ((((b5 * b3) + (b6 * b4)) ^2 ) - (((b3 ^2 ) + (b4 ^2 )) * (((b5 ^2 ) + (b6 ^2 )) - (b2 ^2 )))))) / ((b3 ^2 ) + (b4 ^2 )) implies
HC b
7,b
8,b
9,b
2 = |[((b7 `1 ) + (b1 * b3)),((b7 `2 ) + (b1 * b4))]| )
definition
let c
2 be non
empty Nat;
let c
3 be
Point of
(TOP-REAL c2);
let c
4 be non
negative real number ;
let c
5 be
Point of
(Tdisk c3,c4);
let c
6 be
Function of
(Tdisk c3,c4),
(Tdisk c3,c4);
assume E9:
not c
5 is_a_fixpoint_of c
6
;
func HC c
4,c
5 -> Point of
(Tcircle a2,a3) means :
Def4:
:: BROUWER:def 4
ex b
1, b
2 being
Point of
(TOP-REAL a1) st
( b
1 = a
4 & b
2 = a
5 . a
4 & a
6 = HC b
2,b
1,a
2,a
3 );
existence
ex b1 being Point of (Tcircle c3,c4)ex b2, b3 being Point of (TOP-REAL c2) st
( b2 = c5 & b3 = c6 . c5 & b1 = HC b3,b2,c3,c4 )
uniqueness
for b1, b2 being Point of (Tcircle c3,c4) holds
( ex b3, b4 being Point of (TOP-REAL c2) st
( b3 = c5 & b4 = c6 . c5 & b1 = HC b4,b3,c3,c4 ) & ex b3, b4 being Point of (TOP-REAL c2) st
( b3 = c5 & b4 = c6 . c5 & b2 = HC b4,b3,c3,c4 ) implies b1 = b2 )
;
end;
:: deftheorem Def4 defines HC BROUWER:def 4 :
for b
1 being non
empty Natfor b
2 being
Point of
(TOP-REAL b1)for b
3 being non
negative real number for b
4 being
Point of
(Tdisk b2,b3)for b
5 being
Function of
(Tdisk b2,b3),
(Tdisk b2,b3) holds
( not b
4 is_a_fixpoint_of b
5 implies for b
6 being
Point of
(Tcircle b2,b3) holds
( b
6 = HC b
4,b
5 iff ex b
7, b
8 being
Point of
(TOP-REAL b1) st
( b
7 = b
4 & b
8 = b
5 . b
4 & b
6 = HC b
8,b
7,b
2,b
3 ) ) );
theorem Th9: :: BROUWER:9
theorem Th10: :: BROUWER:10
definition
let c
2 be non
empty Nat;
let c
3 be non
negative real number ;
let c
4 be
Point of
(TOP-REAL c2);
let c
5 be
Function of
(Tdisk c4,c3),
(Tdisk c4,c3);
func BR-map c
4 -> Function of
(Tdisk a3,a2),
(Tcircle a3,a2) means :
Def5:
:: BROUWER:def 5
for b
1 being
Point of
(Tdisk a3,a2) holds a
5 . b
1 = HC b
1,a
4;
existence
ex b1 being Function of (Tdisk c4,c3),(Tcircle c4,c3) st
for b2 being Point of (Tdisk c4,c3) holds b1 . b2 = HC b2,c5
uniqueness
for b1, b2 being Function of (Tdisk c4,c3),(Tcircle c4,c3) holds
( ( for b3 being Point of (Tdisk c4,c3) holds b1 . b3 = HC b3,c5 ) & ( for b3 being Point of (Tdisk c4,c3) holds b2 . b3 = HC b3,c5 ) implies b1 = b2 )
end;
:: deftheorem Def5 defines BR-map BROUWER:def 5 :
theorem Th11: :: BROUWER:11
theorem Th12: :: BROUWER:12
theorem Th13: :: BROUWER:13
theorem Th14: :: BROUWER:14
theorem Th15: :: BROUWER:15