:: JORDAN1K semantic presentation
theorem Th1: :: JORDAN1K:1
theorem Th2: :: JORDAN1K:2
theorem Th3: :: JORDAN1K:3
theorem Th4: :: JORDAN1K:4
theorem Th5: :: JORDAN1K:5
theorem Th6: :: JORDAN1K:6
theorem Th7: :: JORDAN1K:7
theorem Th8: :: JORDAN1K:8
theorem Th9: :: JORDAN1K:9
theorem Th10: :: JORDAN1K:10
theorem Th11: :: JORDAN1K:11
theorem Th12: :: JORDAN1K:12
theorem Th13: :: JORDAN1K:13
theorem Th14: :: JORDAN1K:14
theorem Th15: :: JORDAN1K:15
theorem Th16: :: JORDAN1K:16
theorem Th17: :: JORDAN1K:17
theorem Th18: :: JORDAN1K:18
theorem Th19: :: JORDAN1K:19
theorem Th20: :: JORDAN1K:20
theorem Th21: :: JORDAN1K:21
theorem Th22: :: JORDAN1K:22
theorem Th23: :: JORDAN1K:23
theorem Th24: :: JORDAN1K:24
theorem Th25: :: JORDAN1K:25
theorem Th26: :: JORDAN1K:26
theorem Th27: :: JORDAN1K:27
theorem Th28: :: JORDAN1K:28
theorem Th29: :: JORDAN1K:29
theorem Th30: :: JORDAN1K:30
theorem Th31: :: JORDAN1K:31
theorem Th32: :: JORDAN1K:32
theorem Th33: :: JORDAN1K:33
theorem Th34: :: JORDAN1K:34
theorem Th35: :: JORDAN1K:35
theorem Th36: :: JORDAN1K:36
theorem Th37: :: JORDAN1K:37
definition
let c1 be
Nat;
let c2,
c3 be
Subset of
(TOP-REAL c1);
func dist_min c2,
c3 -> Real means :
Def1:
:: JORDAN1K:def 1
ex
b1,
b2 being
Subset of
(TopSpaceMetr (Euclid a1)) st
(
a2 = b1 &
a3 = b2 &
a4 = min_dist_min b1,
b2 );
existence
ex b1 being Realex b2, b3 being Subset of (TopSpaceMetr (Euclid c1)) st
( c2 = b2 & c3 = b3 & b1 = min_dist_min b2,b3 )
uniqueness
for b1, b2 being Real st ex b3, b4 being Subset of (TopSpaceMetr (Euclid c1)) st
( c2 = b3 & c3 = b4 & b1 = min_dist_min b3,b4 ) & ex b3, b4 being Subset of (TopSpaceMetr (Euclid c1)) st
( c2 = b3 & c3 = b4 & b2 = min_dist_min b3,b4 ) holds
b1 = b2
;
end;
:: deftheorem Def1 defines dist_min JORDAN1K:def 1 :
theorem Th38: :: JORDAN1K:38
theorem Th39: :: JORDAN1K:39
theorem Th40: :: JORDAN1K:40
theorem Th41: :: JORDAN1K:41
theorem Th42: :: JORDAN1K:42
theorem Th43: :: JORDAN1K:43
:: deftheorem Def2 defines dist JORDAN1K:def 2 :
theorem Th44: :: JORDAN1K:44
theorem Th45: :: JORDAN1K:45
theorem Th46: :: JORDAN1K:46
theorem Th47: :: JORDAN1K:47
theorem Th48: :: JORDAN1K:48
theorem Th49: :: JORDAN1K:49
theorem Th50: :: JORDAN1K:50
theorem Th51: :: JORDAN1K:51