:: JORDAN1K semantic presentation

theorem Th1: :: JORDAN1K:1
for b1 being set
for b2 being non empty set
for b3 being Function of b1,b2 holds
( b3 is onto implies for b4 being Element of b2 holds
ex b5 being set st
( b5 in b1 & b4 = b3 . b5 ) )
proof end;

theorem Th2: :: JORDAN1K:2
for b1 being set
for b2 being non empty set
for b3 being Function of b1,b2 holds
( b3 is onto implies for b4 being Element of b2 holds
ex b5 being Element of b1 st b4 = b3 . b5 )
proof end;

theorem Th3: :: JORDAN1K:3
for b1 being set
for b2 being non empty set
for b3 being Function of b1,b2
for b4 being Subset of b1 holds
( b3 is onto implies (b3 .: b4) ` c= b3 .: (b4 ` ) )
proof end;

theorem Th4: :: JORDAN1K:4
for b1 being set
for b2 being non empty set
for b3 being Function of b1,b2
for b4 being Subset of b1 holds
( b3 is one-to-one implies b3 .: (b4 ` ) c= (b3 .: b4) ` )
proof end;

theorem Th5: :: JORDAN1K:5
for b1 being set
for b2 being non empty set
for b3 being Function of b1,b2
for b4 being Subset of b1 holds
( b3 is bijective implies (b3 .: b4) ` = b3 .: (b4 ` ) )
proof end;

theorem Th6: :: JORDAN1K:6
for b1 being TopSpace
for b2 being Subset of b1 holds
( b2 is_a_component_of {} b1 iff b2 is empty )
proof end;

theorem Th7: :: JORDAN1K:7
for b1 being non empty TopSpace
for b2, b3, b4 being Subset of b1 holds
( b2 c= b3 & b2 is_a_component_of b4 & b3 is_a_component_of b4 implies b2 = b3 )
proof end;

theorem Th8: :: JORDAN1K:8
for b1 being Nat holds
( b1 >= 1 implies for b2 being Subset of (Euclid b1) holds
not ( b2 is bounded & b2 ` is bounded ) )
proof end;

theorem Th9: :: JORDAN1K:9
for b1 being non empty MetrSpace
for b2 being non empty Subset of (TopSpaceMetr b1)
for b3 being Point of (TopSpaceMetr b1) holds (dist_min b2) . b3 >= 0
proof end;

theorem Th10: :: JORDAN1K:10
for b1 being Real
for b2 being non empty MetrSpace
for b3 being non empty Subset of (TopSpaceMetr b2)
for b4 being Point of b2 holds
( ( for b5 being Point of b2 holds
( b5 in b3 implies dist b4,b5 >= b1 ) ) implies (dist_min b3) . b4 >= b1 )
proof end;

theorem Th11: :: JORDAN1K:11
for b1 being non empty MetrSpace
for b2, b3 being non empty Subset of (TopSpaceMetr b1) holds min_dist_min b2,b3 >= 0
proof end;

theorem Th12: :: JORDAN1K:12
for b1 being non empty MetrSpace
for b2, b3 being compact Subset of (TopSpaceMetr b1) holds
( b2 meets b3 implies min_dist_min b2,b3 = 0 )
proof end;

theorem Th13: :: JORDAN1K:13
for b1 being Real
for b2 being non empty MetrSpace
for b3, b4 being non empty Subset of (TopSpaceMetr b2) holds
( ( for b5, b6 being Point of b2 holds
( b5 in b3 & b6 in b4 implies dist b5,b6 >= b1 ) ) implies min_dist_min b3,b4 >= b1 )
proof end;

theorem Th14: :: JORDAN1K:14
for b1 being Nat
for b2, b3 being Subset of (TOP-REAL b1) holds
not ( b2 is_a_component_of b3 ` & not b2 is_inside_component_of b3 & not b2 is_outside_component_of b3 )
proof end;

theorem Th15: :: JORDAN1K:15
for b1 being Nat holds
( b1 >= 1 implies BDD ({} (TOP-REAL b1)) = {} (TOP-REAL b1) )
proof end;

theorem Th16: :: JORDAN1K:16
for b1 being Nat holds BDD ([#] (TOP-REAL b1)) = {} (TOP-REAL b1)
proof end;

theorem Th17: :: JORDAN1K:17
for b1 being Nat holds
( b1 >= 1 implies UBD ({} (TOP-REAL b1)) = [#] (TOP-REAL b1) )
proof end;

theorem Th18: :: JORDAN1K:18
for b1 being Nat holds UBD ([#] (TOP-REAL b1)) = {} (TOP-REAL b1)
proof end;

theorem Th19: :: JORDAN1K:19
for b1 being Nat
for b2 being connected Subset of (TOP-REAL b1)
for b3 being Subset of (TOP-REAL b1) holds
not ( b2 misses b3 & not b2 c= UBD b3 & not b2 c= BDD b3 )
proof end;

theorem Th20: :: JORDAN1K:20
for b1 being Point of (TOP-REAL 2)
for b2 being Real holds dist |[0,0]|,(b2 * b1) = (abs b2) * (dist |[0,0]|,b1)
proof end;

theorem Th21: :: JORDAN1K:21
for b1, b2, b3 being Point of (TOP-REAL 2) holds dist (b1 + b2),(b3 + b2) = dist b1,b3
proof end;

theorem Th22: :: JORDAN1K:22
for b1, b2 being Point of (TOP-REAL 2) holds
not ( b1 <> b2 & not dist b1,b2 > 0 )
proof end;

theorem Th23: :: JORDAN1K:23
for b1, b2, b3 being Point of (TOP-REAL 2) holds dist (b1 - b2),(b3 - b2) = dist b1,b3
proof end;

theorem Th24: :: JORDAN1K:24
for b1, b2 being Point of (TOP-REAL 2) holds dist b1,b2 = dist (- b1),(- b2)
proof end;

theorem Th25: :: JORDAN1K:25
for b1, b2, b3 being Point of (TOP-REAL 2) holds dist (b1 - b2),(b1 - b3) = dist b2,b3
proof end;

theorem Th26: :: JORDAN1K:26
for b1, b2 being Point of (TOP-REAL 2)
for b3 being Real holds dist (b3 * b1),(b3 * b2) = (abs b3) * (dist b1,b2)
proof end;

theorem Th27: :: JORDAN1K:27
for b1, b2 being Point of (TOP-REAL 2)
for b3 being Real holds
( b3 <= 1 implies dist b1,((b3 * b1) + ((1 - b3) * b2)) = (1 - b3) * (dist b1,b2) )
proof end;

theorem Th28: :: JORDAN1K:28
for b1, b2 being Point of (TOP-REAL 2)
for b3 being Real holds
( 0 <= b3 implies dist b1,((b3 * b2) + ((1 - b3) * b1)) = b3 * (dist b2,b1) )
proof end;

theorem Th29: :: JORDAN1K:29
for b1, b2, b3 being Point of (TOP-REAL 2) holds
( b1 in LSeg b2,b3 implies (dist b2,b1) + (dist b1,b3) = dist b2,b3 )
proof end;

theorem Th30: :: JORDAN1K:30
for b1, b2, b3 being Point of (TOP-REAL 2) holds
not ( b1 in LSeg b2,b3 & b1 <> b2 & not dist b1,b3 < dist b2,b3 )
proof end;

theorem Th31: :: JORDAN1K:31
for b1 being Real
for b2 being Point of (Euclid 2) holds
( b2 = |[0,0]| implies Ball b2,b1 = { b3 where B is Point of (TOP-REAL 2) : |.b3.| < b1 } )
proof end;

theorem Th32: :: JORDAN1K:32
for b1 being Point of (TOP-REAL 2)
for b2, b3, b4 being Real holds (AffineMap b2,b3,b2,b4) . b1 = (b2 * b1) + |[b3,b4]|
proof end;

theorem Th33: :: JORDAN1K:33
for b1, b2 being Point of (TOP-REAL 2)
for b3 being Real holds (AffineMap b3,(b1 `1 ),b3,(b1 `2 )) . b2 = (b3 * b2) + b1
proof end;

theorem Th34: :: JORDAN1K:34
for b1, b2, b3, b4 being Real holds
( b1 > 0 & b2 > 0 implies (AffineMap b1,b3,b2,b4) * (AffineMap (1 / b1),(- (b3 / b1)),(1 / b2),(- (b4 / b2))) = id (REAL 2) )
proof end;

theorem Th35: :: JORDAN1K:35
for b1 being Point of (TOP-REAL 2)
for b2 being Real
for b3, b4 being Point of (Euclid 2) holds
( b3 = |[0,0]| & b4 = b1 & b2 > 0 implies (AffineMap b2,(b1 `1 ),b2,(b1 `2 )) .: (Ball b3,1) = Ball b4,b2 )
proof end;

theorem Th36: :: JORDAN1K:36
for b1, b2, b3, b4 being Real holds
( b1 > 0 & b3 > 0 implies AffineMap b1,b2,b3,b4 is onto )
proof end;

theorem Th37: :: JORDAN1K:37
for b1 being Real
for b2 being Point of (Euclid 2) holds
(Ball b2,b1) ` is connected Subset of (TOP-REAL 2)
proof end;

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 )
proof end;
uniqueness
for b1, b2 being Real holds
( 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 ) implies b1 = b2 )
;
end;

:: deftheorem Def1 defines dist_min JORDAN1K:def 1 :
for b1 being Nat
for b2, b3 being Subset of (TOP-REAL b1)
for b4 being Real holds
( b4 = dist_min b2,b3 iff ex b5, b6 being Subset of (TopSpaceMetr (Euclid b1)) st
( b2 = b5 & b3 = b6 & b4 = min_dist_min b5,b6 ) );

definition
let c1 be non empty MetrSpace;
let c2, c3 be non empty compact Subset of (TopSpaceMetr c1);
redefine func min_dist_min as min_dist_min c2,c3 -> Element of REAL ;
commutativity
for b1, b2 being non empty compact Subset of (TopSpaceMetr c1) holds min_dist_min b1,b2 = min_dist_min b2,b1
proof end;
redefine func max_dist_max as max_dist_max c2,c3 -> Element of REAL ;
commutativity
for b1, b2 being non empty compact Subset of (TopSpaceMetr c1) holds max_dist_max b1,b2 = max_dist_max b2,b1
proof end;
end;

definition
let c1 be Nat;
let c2, c3 be non empty compact Subset of (TOP-REAL c1);
redefine func dist_min as dist_min c2,c3 -> Real;
commutativity
for b1, b2 being non empty compact Subset of (TOP-REAL c1) holds dist_min b1,b2 = dist_min b2,b1
proof end;
end;

theorem Th38: :: JORDAN1K:38
for b1 being Nat
for b2, b3 being non empty Subset of (TOP-REAL b1) holds dist_min b2,b3 >= 0
proof end;

theorem Th39: :: JORDAN1K:39
for b1 being Nat
for b2, b3 being compact Subset of (TOP-REAL b1) holds
( b2 meets b3 implies dist_min b2,b3 = 0 )
proof end;

theorem Th40: :: JORDAN1K:40
for b1 being Nat
for b2 being Real
for b3, b4 being non empty Subset of (TOP-REAL b1) holds
( ( for b5, b6 being Point of (TOP-REAL b1) holds
( b5 in b3 & b6 in b4 implies dist b5,b6 >= b2 ) ) implies dist_min b3,b4 >= b2 )
proof end;

theorem Th41: :: JORDAN1K:41
for b1 being Nat
for b2 being Subset of (TOP-REAL b1)
for b3, b4 being non empty Subset of (TOP-REAL b1) holds
( b4 c= b2 implies dist_min b3,b2 <= dist_min b3,b4 )
proof end;

theorem Th42: :: JORDAN1K:42
for b1 being Nat
for b2, b3 being non empty compact Subset of (TOP-REAL b1) holds
ex b4, b5 being Point of (TOP-REAL b1) st
( b4 in b2 & b5 in b3 & dist_min b2,b3 = dist b4,b5 )
proof end;

theorem Th43: :: JORDAN1K:43
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1) holds dist_min {b2},{b3} = dist b2,b3
proof end;

definition
let c1 be Nat;
let c2 be Point of (TOP-REAL c1);
let c3 be Subset of (TOP-REAL c1);
func dist c2,c3 -> Real equals :: JORDAN1K:def 2
dist_min {a2},a3;
coherence
dist_min {c2},c3 is Real
;
end;

:: deftheorem Def2 defines dist JORDAN1K:def 2 :
for b1 being Nat
for b2 being Point of (TOP-REAL b1)
for b3 being Subset of (TOP-REAL b1) holds dist b2,b3 = dist_min {b2},b3;

theorem Th44: :: JORDAN1K:44
for b1 being Nat
for b2 being non empty Subset of (TOP-REAL b1)
for b3 being Point of (TOP-REAL b1) holds dist b3,b2 >= 0 by Th38;

theorem Th45: :: JORDAN1K:45
for b1 being Nat
for b2 being compact Subset of (TOP-REAL b1)
for b3 being Point of (TOP-REAL b1) holds
( b3 in b2 implies dist b3,b2 = 0 )
proof end;

theorem Th46: :: JORDAN1K:46
for b1 being Nat
for b2 being non empty compact Subset of (TOP-REAL b1)
for b3 being Point of (TOP-REAL b1) holds
ex b4 being Point of (TOP-REAL b1) st
( b4 in b2 & dist b3,b2 = dist b3,b4 )
proof end;

theorem Th47: :: JORDAN1K:47
for b1 being Nat
for b2 being non empty Subset of (TOP-REAL b1)
for b3 being Subset of (TOP-REAL b1) holds
( b2 c= b3 implies for b4 being Point of (TOP-REAL b1) holds dist b4,b3 <= dist b4,b2 ) by Th41;

theorem Th48: :: JORDAN1K:48
for b1 being Nat
for b2 being Real
for b3 being non empty Subset of (TOP-REAL b1)
for b4 being Point of (TOP-REAL b1) holds
( ( for b5 being Point of (TOP-REAL b1) holds
( b5 in b3 implies dist b4,b5 >= b2 ) ) implies dist b4,b3 >= b2 )
proof end;

theorem Th49: :: JORDAN1K:49
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1) holds dist b2,{b3} = dist b2,b3 by Th43;

theorem Th50: :: JORDAN1K:50
for b1 being Nat
for b2 being non empty Subset of (TOP-REAL b1)
for b3, b4 being Point of (TOP-REAL b1) holds
( b4 in b2 implies dist b3,b2 <= dist b3,b4 )
proof end;

theorem Th51: :: JORDAN1K:51
for b1 being non empty compact Subset of (TOP-REAL 2)
for b2 being open Subset of (TOP-REAL 2) holds
( b1 c= b2 implies for b3 being Point of (TOP-REAL 2) holds
not ( not b3 in b2 & not dist b3,b2 < dist b3,b1 ) )
proof end;