:: JORDAN6 semantic presentation
theorem Th1: :: JORDAN6:1
canceled;
theorem Th2: :: JORDAN6:2
for b
1, b
2 being
real number holds
( b
1 <= b
2 implies ( b
1 <= (b1 + b2) / 2 &
(b1 + b2) / 2
<= b
2 ) )
theorem Th3: :: JORDAN6:3
theorem Th4: :: JORDAN6:4
theorem Th5: :: JORDAN6:5
theorem Th6: :: JORDAN6:6
theorem Th7: :: JORDAN6:7
theorem Th8: :: JORDAN6:8
theorem Th9: :: JORDAN6:9
theorem Th10: :: JORDAN6:10
theorem Th11: :: JORDAN6:11
theorem Th12: :: JORDAN6:12
theorem Th13: :: JORDAN6:13
theorem Th14: :: JORDAN6:14
theorem Th15: :: JORDAN6:15
definition
let c
1 be
Subset of
(TOP-REAL 2);
let c
2, c
3 be
Point of
(TOP-REAL 2);
func x_Middle c
1,c
2,c
3 -> Point of
(TOP-REAL 2) means :
Def1:
:: JORDAN6:def 1
for b
1 being
Subset of
(TOP-REAL 2) holds
( b
1 = { b2 where B is Point of (TOP-REAL 2) : b2 `1 = ((a2 `1 ) + (a3 `1 )) / 2 } implies a
4 = First_Point a
1,a
2,a
3,b
1 );
existence
ex b1 being Point of (TOP-REAL 2) st
for b2 being Subset of (TOP-REAL 2) holds
( b2 = { b3 where B is Point of (TOP-REAL 2) : b3 `1 = ((c2 `1 ) + (c3 `1 )) / 2 } implies b1 = First_Point c1,c2,c3,b2 )
uniqueness
for b1, b2 being Point of (TOP-REAL 2) holds
( ( for b3 being Subset of (TOP-REAL 2) holds
( b3 = { b4 where B is Point of (TOP-REAL 2) : b4 `1 = ((c2 `1 ) + (c3 `1 )) / 2 } implies b1 = First_Point c1,c2,c3,b3 ) ) & ( for b3 being Subset of (TOP-REAL 2) holds
( b3 = { b4 where B is Point of (TOP-REAL 2) : b4 `1 = ((c2 `1 ) + (c3 `1 )) / 2 } implies b2 = First_Point c1,c2,c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines x_Middle JORDAN6:def 1 :
definition
let c
1 be
Subset of
(TOP-REAL 2);
let c
2, c
3 be
Point of
(TOP-REAL 2);
func y_Middle c
1,c
2,c
3 -> Point of
(TOP-REAL 2) means :
Def2:
:: JORDAN6:def 2
for b
1 being
Subset of
(TOP-REAL 2) holds
( b
1 = { b2 where B is Point of (TOP-REAL 2) : b2 `2 = ((a2 `2 ) + (a3 `2 )) / 2 } implies a
4 = First_Point a
1,a
2,a
3,b
1 );
existence
ex b1 being Point of (TOP-REAL 2) st
for b2 being Subset of (TOP-REAL 2) holds
( b2 = { b3 where B is Point of (TOP-REAL 2) : b3 `2 = ((c2 `2 ) + (c3 `2 )) / 2 } implies b1 = First_Point c1,c2,c3,b2 )
uniqueness
for b1, b2 being Point of (TOP-REAL 2) holds
( ( for b3 being Subset of (TOP-REAL 2) holds
( b3 = { b4 where B is Point of (TOP-REAL 2) : b4 `2 = ((c2 `2 ) + (c3 `2 )) / 2 } implies b1 = First_Point c1,c2,c3,b3 ) ) & ( for b3 being Subset of (TOP-REAL 2) holds
( b3 = { b4 where B is Point of (TOP-REAL 2) : b4 `2 = ((c2 `2 ) + (c3 `2 )) / 2 } implies b2 = First_Point c1,c2,c3,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines y_Middle JORDAN6:def 2 :
theorem Th16: :: JORDAN6:16
theorem Th17: :: JORDAN6:17
theorem Th18: :: JORDAN6:18
theorem Th19: :: JORDAN6:19
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
( b
1 is_an_arc_of b
2,b
3 &
LE b
4,b
5,b
1,b
2,b
3 implies
LE b
5,b
4,b
1,b
3,b
2 )
definition
let c
1 be
Subset of
(TOP-REAL 2);
let c
2, c
3, c
4 be
Point of
(TOP-REAL 2);
func L_Segment c
1,c
2,c
3,c
4 -> Subset of
(TOP-REAL 2) equals :: JORDAN6:def 3
{ b1 where B is Point of (TOP-REAL 2) : LE b1,a4,a1,a2,a3 } ;
coherence
{ b1 where B is Point of (TOP-REAL 2) : LE b1,c4,c1,c2,c3 } is Subset of (TOP-REAL 2)
end;
:: deftheorem Def3 defines L_Segment JORDAN6:def 3 :
definition
let c
1 be
Subset of
(TOP-REAL 2);
let c
2, c
3, c
4 be
Point of
(TOP-REAL 2);
func R_Segment c
1,c
2,c
3,c
4 -> Subset of
(TOP-REAL 2) equals :: JORDAN6:def 4
{ b1 where B is Point of (TOP-REAL 2) : LE a4,b1,a1,a2,a3 } ;
coherence
{ b1 where B is Point of (TOP-REAL 2) : LE c4,b1,c1,c2,c3 } is Subset of (TOP-REAL 2)
end;
:: deftheorem Def4 defines R_Segment JORDAN6:def 4 :
theorem Th20: :: JORDAN6:20
theorem Th21: :: JORDAN6:21
theorem Th22: :: JORDAN6:22
theorem Th23: :: JORDAN6:23
canceled;
theorem Th24: :: JORDAN6:24
canceled;
theorem Th25: :: JORDAN6:25
theorem Th26: :: JORDAN6:26
theorem Th27: :: JORDAN6:27
theorem Th28: :: JORDAN6:28
definition
let c
1 be
Subset of
(TOP-REAL 2);
let c
2, c
3, c
4, c
5 be
Point of
(TOP-REAL 2);
func Segment c
1,c
2,c
3,c
4,c
5 -> Subset of
(TOP-REAL 2) equals :: JORDAN6:def 5
(R_Segment a1,a2,a3,a4) /\ (L_Segment a1,a2,a3,a5);
correctness
coherence
(R_Segment c1,c2,c3,c4) /\ (L_Segment c1,c2,c3,c5) is Subset of (TOP-REAL 2);
;
end;
:: deftheorem Def5 defines Segment JORDAN6:def 5 :
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
Segment b
1,b
2,b
3,b
4,b
5 = (R_Segment b1,b2,b3,b4) /\ (L_Segment b1,b2,b3,b5);
theorem Th29: :: JORDAN6:29
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
Segment b
1,b
2,b
3,b
4,b
5 = { b6 where B is Point of (TOP-REAL 2) : ( LE b4,b6,b1,b2,b3 & LE b6,b5,b1,b2,b3 ) }
theorem Th30: :: JORDAN6:30
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
( b
1 is_an_arc_of b
2,b
3 &
LE b
5,b
4,b
1,b
3,b
2 implies
LE b
4,b
5,b
1,b
2,b
3 )
theorem Th31: :: JORDAN6:31
theorem Th32: :: JORDAN6:32
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
( b
1 is_an_arc_of b
2,b
3 & b
4 in b
1 & b
5 in b
1 implies
Segment b
1,b
2,b
3,b
4,b
5 = Segment b
1,b
3,b
2,b
5,b
4 )
:: deftheorem Def6 defines Vertical_Line JORDAN6:def 6 :
:: deftheorem Def7 defines Horizontal_Line JORDAN6:def 7 :
theorem Th33: :: JORDAN6:33
theorem Th34: :: JORDAN6:34
theorem Th35: :: JORDAN6:35
theorem Th36: :: JORDAN6:36
canceled;
theorem Th37: :: JORDAN6:37
canceled;
theorem Th38: :: JORDAN6:38
canceled;
theorem Th39: :: JORDAN6:39
canceled;
theorem Th40: :: JORDAN6:40
for b
1 being
Subset of
(TOP-REAL 2) holds
not ( b
1 is_simple_closed_curve & ( for b
2, b
3 being non
empty Subset of
(TOP-REAL 2) holds
not ( b
2 is_an_arc_of W-min b
1,
E-max b
1 & b
3 is_an_arc_of E-max b
1,
W-min b
1 & b
2 /\ b
3 = {(W-min b1),(E-max b1)} & b
2 \/ b
3 = b
1 &
(First_Point b2,(W-min b1),(E-max b1),(Vertical_Line (((W-bound b1) + (E-bound b1)) / 2))) `2 > (Last_Point b3,(E-max b1),(W-min b1),(Vertical_Line (((W-bound b1) + (E-bound b1)) / 2))) `2 ) ) )
theorem Th41: :: JORDAN6:41
Lemma26:
for b1, b2 being real number holds ].b1,b2.[ misses {b1,b2}
by RCOMP_1:46;
Lemma27:
for b1, b2, b3 being real number holds
( b3 in ].b1,b2.[ iff ( b1 < b3 & b3 < b2 ) )
by RCOMP_1:47;
theorem Th42: :: JORDAN6:42
canceled;
theorem Th43: :: JORDAN6:43
canceled;
theorem Th44: :: JORDAN6:44
canceled;
theorem Th45: :: JORDAN6:45
canceled;
theorem Th46: :: JORDAN6:46
theorem Th47: :: JORDAN6:47
theorem Th48: :: JORDAN6:48
theorem Th49: :: JORDAN6:49
theorem Th50: :: JORDAN6:50
theorem Th51: :: JORDAN6:51
canceled;
theorem Th52: :: JORDAN6:52
theorem Th53: :: JORDAN6:53
theorem Th54: :: JORDAN6:54
theorem Th55: :: JORDAN6:55
theorem Th56: :: JORDAN6:56
theorem Th57: :: JORDAN6:57
theorem Th58: :: JORDAN6:58
theorem Th59: :: JORDAN6:59
theorem Th60: :: JORDAN6:60
theorem Th61: :: JORDAN6:61
Lemma39:
for b1 being Function of I[01] ,R^1
for b2, b3, b4 being real number holds
not ( b1 is continuous & b1 . 0[01] < b4 & b4 < b1 . 1[01] & b2 = b1 . 0 & b3 = b1 . 1 & ( for b5 being Real holds
not ( 0 < b5 & b5 < 1 & b1 . b5 = b4 ) ) )
theorem Th62: :: JORDAN6:62
canceled;
theorem Th63: :: JORDAN6:63
canceled;
theorem Th64: :: JORDAN6:64
E41:
now
let c
1 be
Simple_closed_curve;
let c
2, c
3 be non
empty Subset of
(TOP-REAL 2);
assume E42:
( ex b
1 being non
empty Subset of
(TOP-REAL 2) st
( c
2 is_an_arc_of W-min c
1,
E-max c
1 & b
1 is_an_arc_of E-max c
1,
W-min c
1 & c
2 /\ b
1 = {(W-min c1),(E-max c1)} & c
2 \/ b
1 = c
1 &
(First_Point c2,(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (Last_Point b1,(E-max c1),(W-min c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 ) & ex b
1 being non
empty Subset of
(TOP-REAL 2) st
( c
3 is_an_arc_of W-min c
1,
E-max c
1 & b
1 is_an_arc_of E-max c
1,
W-min c
1 & c
3 /\ b
1 = {(W-min c1),(E-max c1)} & c
3 \/ b
1 = c
1 &
(First_Point c3,(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (Last_Point b1,(E-max c1),(W-min c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 ) )
;
then consider c
4 being non
empty Subset of
(TOP-REAL 2) such that E43:
( c
2 is_an_arc_of W-min c
1,
E-max c
1 & c
4 is_an_arc_of E-max c
1,
W-min c
1 & c
2 /\ c
4 = {(W-min c1),(E-max c1)} & c
2 \/ c
4 = c
1 &
(First_Point c2,(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (Last_Point c4,(E-max c1),(W-min c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 )
;
E44:
c
4 is_an_arc_of W-min c
1,
E-max c
1
by E43, JORDAN5B:14;
consider c
5 being non
empty Subset of
(TOP-REAL 2) such that E45:
( c
3 is_an_arc_of W-min c
1,
E-max c
1 & c
5 is_an_arc_of E-max c
1,
W-min c
1 & c
3 /\ c
5 = {(W-min c1),(E-max c1)} & c
3 \/ c
5 = c
1 &
(First_Point c3,(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (Last_Point c5,(E-max c1),(W-min c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 )
by E42;
E46:
c
5 is_an_arc_of W-min c
1,
E-max c
1
by E45, JORDAN5B:14;
now
assume E47:
( c
2 = c
5 & c
4 = c
3 )
;
E48:
(W-min c1) `1 = W-bound c
1
by EUCLID:56;
E49:
(E-max c1) `1 = E-bound c
1
by EUCLID:56;
then
(W-min c1) `1 < (E-max c1) `1
by E48, TOPREAL5:23;
then E50:
(
(W-min c1) `1 <= ((W-bound c1) + (E-bound c1)) / 2 &
((W-bound c1) + (E-bound c1)) / 2
<= (E-max c1) `1 )
by E48, E49, TOPREAL3:3;
then
( c
4 meets Vertical_Line (((W-bound c1) + (E-bound c1)) / 2) & c
4 /\ (Vertical_Line (((W-bound c1) + (E-bound c1)) / 2)) is
closed )
by E44, Th64;
then E51:
(First_Point c2,(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (First_Point c4,(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2
by E43, JORDAN5C:18;
( c
5 meets Vertical_Line (((W-bound c1) + (E-bound c1)) / 2) & c
5 /\ (Vertical_Line (((W-bound c1) + (E-bound c1)) / 2)) is
closed )
by E46, E50, Th64;
hence
not verum
by E45, E47, E51, JORDAN5C:18;
end;
hence
c
2 = c
3
by E43, E44, E45, E46, Th61;
end;
definition
let c
1 be
Subset of
(TOP-REAL 2);
assume E42:
c
1 is_simple_closed_curve
;
func Upper_Arc c
1 -> non
empty Subset of
(TOP-REAL 2) means :
Def8:
:: JORDAN6:def 8
( a
2 is_an_arc_of W-min a
1,
E-max a
1 & ex b
1 being non
empty Subset of
(TOP-REAL 2) st
( b
1 is_an_arc_of E-max a
1,
W-min a
1 & a
2 /\ b
1 = {(W-min a1),(E-max a1)} & a
2 \/ b
1 = a
1 &
(First_Point a2,(W-min a1),(E-max a1),(Vertical_Line (((W-bound a1) + (E-bound a1)) / 2))) `2 > (Last_Point b1,(E-max a1),(W-min a1),(Vertical_Line (((W-bound a1) + (E-bound a1)) / 2))) `2 ) );
existence
ex b1 being non empty Subset of (TOP-REAL 2) st
( b1 is_an_arc_of W-min c1, E-max c1 & ex b2 being non empty Subset of (TOP-REAL 2) st
( b2 is_an_arc_of E-max c1, W-min c1 & b1 /\ b2 = {(W-min c1),(E-max c1)} & b1 \/ b2 = c1 & (First_Point b1,(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (Last_Point b2,(E-max c1),(W-min c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 ) )
uniqueness
for b1, b2 being non empty Subset of (TOP-REAL 2) holds
( b1 is_an_arc_of W-min c1, E-max c1 & ex b3 being non empty Subset of (TOP-REAL 2) st
( b3 is_an_arc_of E-max c1, W-min c1 & b1 /\ b3 = {(W-min c1),(E-max c1)} & b1 \/ b3 = c1 & (First_Point b1,(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (Last_Point b3,(E-max c1),(W-min c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 ) & b2 is_an_arc_of W-min c1, E-max c1 & ex b3 being non empty Subset of (TOP-REAL 2) st
( b3 is_an_arc_of E-max c1, W-min c1 & b2 /\ b3 = {(W-min c1),(E-max c1)} & b2 \/ b3 = c1 & (First_Point b2,(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (Last_Point b3,(E-max c1),(W-min c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 ) implies b1 = b2 )
by E42, Lemma41;
end;
:: deftheorem Def8 defines Upper_Arc JORDAN6:def 8 :
definition
let c
1 be
Subset of
(TOP-REAL 2);
assume E43:
c
1 is_simple_closed_curve
;
then E44:
Upper_Arc c
1 is_an_arc_of W-min c
1,
E-max c
1
by Def8;
func Lower_Arc c
1 -> non
empty Subset of
(TOP-REAL 2) means :
Def9:
:: JORDAN6:def 9
( a
2 is_an_arc_of E-max a
1,
W-min a
1 &
(Upper_Arc a1) /\ a
2 = {(W-min a1),(E-max a1)} &
(Upper_Arc a1) \/ a
2 = a
1 &
(First_Point (Upper_Arc a1),(W-min a1),(E-max a1),(Vertical_Line (((W-bound a1) + (E-bound a1)) / 2))) `2 > (Last_Point a2,(E-max a1),(W-min a1),(Vertical_Line (((W-bound a1) + (E-bound a1)) / 2))) `2 );
existence
ex b1 being non empty Subset of (TOP-REAL 2) st
( b1 is_an_arc_of E-max c1, W-min c1 & (Upper_Arc c1) /\ b1 = {(W-min c1),(E-max c1)} & (Upper_Arc c1) \/ b1 = c1 & (First_Point (Upper_Arc c1),(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (Last_Point b1,(E-max c1),(W-min c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 )
by E43, Def8;
uniqueness
for b1, b2 being non empty Subset of (TOP-REAL 2) holds
( b1 is_an_arc_of E-max c1, W-min c1 & (Upper_Arc c1) /\ b1 = {(W-min c1),(E-max c1)} & (Upper_Arc c1) \/ b1 = c1 & (First_Point (Upper_Arc c1),(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (Last_Point b1,(E-max c1),(W-min c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 & b2 is_an_arc_of E-max c1, W-min c1 & (Upper_Arc c1) /\ b2 = {(W-min c1),(E-max c1)} & (Upper_Arc c1) \/ b2 = c1 & (First_Point (Upper_Arc c1),(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 > (Last_Point b2,(E-max c1),(W-min c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2))) `2 implies b1 = b2 )
end;
:: deftheorem Def9 defines Lower_Arc JORDAN6:def 9 :
theorem Th65: :: JORDAN6:65
for b
1 being
Subset of
(TOP-REAL 2) holds
( b
1 is_simple_closed_curve implies (
Upper_Arc b
1 is_an_arc_of W-min b
1,
E-max b
1 &
Upper_Arc b
1 is_an_arc_of E-max b
1,
W-min b
1 &
Lower_Arc b
1 is_an_arc_of E-max b
1,
W-min b
1 &
Lower_Arc b
1 is_an_arc_of W-min b
1,
E-max b
1 &
(Upper_Arc b1) /\ (Lower_Arc b1) = {(W-min b1),(E-max b1)} &
(Upper_Arc b1) \/ (Lower_Arc b1) = b
1 &
(First_Point (Upper_Arc b1),(W-min b1),(E-max b1),(Vertical_Line (((W-bound b1) + (E-bound b1)) / 2))) `2 > (Last_Point (Lower_Arc b1),(E-max b1),(W-min b1),(Vertical_Line (((W-bound b1) + (E-bound b1)) / 2))) `2 ) )
theorem Th66: :: JORDAN6:66
theorem Th67: :: JORDAN6:67
theorem Th68: :: JORDAN6:68
theorem Th69: :: JORDAN6:69
theorem Th70: :: JORDAN6:70
definition
let c
1 be
Subset of
(TOP-REAL 2);
let c
2, c
3 be
Point of
(TOP-REAL 2);
pred LE c
2,c
3,c
1 means :
Def10:
:: JORDAN6:def 10
not ( not ( a
2 in Upper_Arc a
1 & a
3 in Lower_Arc a
1 & not a
3 = W-min a
1 ) & not ( a
2 in Upper_Arc a
1 & a
3 in Upper_Arc a
1 &
LE a
2,a
3,
Upper_Arc a
1,
W-min a
1,
E-max a
1 ) & not ( a
2 in Lower_Arc a
1 & a
3 in Lower_Arc a
1 & not a
3 = W-min a
1 &
LE a
2,a
3,
Lower_Arc a
1,
E-max a
1,
W-min a
1 ) );
end;
:: deftheorem Def10 defines LE JORDAN6:def 10 :
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3 being
Point of
(TOP-REAL 2) holds
(
LE b
2,b
3,b
1 iff not ( not ( b
2 in Upper_Arc b
1 & b
3 in Lower_Arc b
1 & not b
3 = W-min b
1 ) & not ( b
2 in Upper_Arc b
1 & b
3 in Upper_Arc b
1 &
LE b
2,b
3,
Upper_Arc b
1,
W-min b
1,
E-max b
1 ) & not ( b
2 in Lower_Arc b
1 & b
3 in Lower_Arc b
1 & not b
3 = W-min b
1 &
LE b
2,b
3,
Lower_Arc b
1,
E-max b
1,
W-min b
1 ) ) );
theorem Th71: :: JORDAN6:71
theorem Th72: :: JORDAN6:72
theorem Th73: :: JORDAN6:73
theorem Th74: :: JORDAN6:74
theorem Th75: :: JORDAN6:75
theorem Th76: :: JORDAN6:76
definition
let c
1 be
Simple_closed_curve;
func Lower_Middle_Point c
1 -> Point of
(TOP-REAL 2) equals :: JORDAN6:def 11
First_Point (Lower_Arc a1),
(W-min a1),
(E-max a1),
(Vertical_Line (((W-bound a1) + (E-bound a1)) / 2));
coherence
First_Point (Lower_Arc c1),(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2)) is Point of (TOP-REAL 2)
;
func Upper_Middle_Point c
1 -> Point of
(TOP-REAL 2) equals :: JORDAN6:def 12
First_Point (Upper_Arc a1),
(W-min a1),
(E-max a1),
(Vertical_Line (((W-bound a1) + (E-bound a1)) / 2));
coherence
First_Point (Upper_Arc c1),(W-min c1),(E-max c1),(Vertical_Line (((W-bound c1) + (E-bound c1)) / 2)) is Point of (TOP-REAL 2)
;
end;
:: deftheorem Def11 defines Lower_Middle_Point JORDAN6:def 11 :
:: deftheorem Def12 defines Upper_Middle_Point JORDAN6:def 12 :
theorem Th77: :: JORDAN6:77
theorem Th78: :: JORDAN6:78
theorem Th79: :: JORDAN6:79
theorem Th80: :: JORDAN6:80
theorem Th81: :: JORDAN6:81
theorem Th82: :: JORDAN6:82
theorem Th83: :: JORDAN6:83
theorem Th84: :: JORDAN6:84
theorem Th85: :: JORDAN6:85