:: JORDAN24 semantic presentation
:: deftheorem Def1 defines realize-max-dist-in JORDAN24:def 1 :
set c1 = - 1;
set c2 = 1;
set c3 = |[(- 1),0]|;
set c4 = |[1,0]|;
theorem Th1: :: JORDAN24:1
:: deftheorem Def2 defines isometric JORDAN24:def 2 :
definition
let c
5 be
Real;
func Rotate c
1 -> Function of
(TOP-REAL 2),
(TOP-REAL 2) means :
Def3:
:: JORDAN24:def 3
for b
1 being
Point of
(TOP-REAL 2) holds a
2 . b
1 = |[(Re (Rotate ((b1 `1 ) + ((b1 `2 ) * <i> )),a1)),(Im (Rotate ((b1 `1 ) + ((b1 `2 ) * <i> )),a1))]|;
existence
ex b1 being Function of (TOP-REAL 2),(TOP-REAL 2) st
for b2 being Point of (TOP-REAL 2) holds b1 . b2 = |[(Re (Rotate ((b2 `1 ) + ((b2 `2 ) * <i> )),c5)),(Im (Rotate ((b2 `1 ) + ((b2 `2 ) * <i> )),c5))]|
uniqueness
for b1, b2 being Function of (TOP-REAL 2),(TOP-REAL 2) holds
( ( for b3 being Point of (TOP-REAL 2) holds b1 . b3 = |[(Re (Rotate ((b3 `1 ) + ((b3 `2 ) * <i> )),c5)),(Im (Rotate ((b3 `1 ) + ((b3 `2 ) * <i> )),c5))]| ) & ( for b3 being Point of (TOP-REAL 2) holds b2 . b3 = |[(Re (Rotate ((b3 `1 ) + ((b3 `2 ) * <i> )),c5)),(Im (Rotate ((b3 `1 ) + ((b3 `2 ) * <i> )),c5))]| ) implies b1 = b2 )
end;
:: deftheorem Def3 defines Rotate JORDAN24:def 3 :
theorem Th2: :: JORDAN24:2
theorem Th3: :: JORDAN24:3
for b
1, b
2 being
Point of
(TOP-REAL 2)for b
3 being
Subset of
(TOP-REAL 2)for b
4, b
5, b
6 being
real number holds
( b
1,b
2 realize-max-dist-in b
3 implies
(AffineMap b4,b5,b4,b6) . b
1,
(AffineMap b4,b5,b4,b6) . b
2 realize-max-dist-in (AffineMap b4,b5,b4,b6) .: b
3 )
theorem Th4: :: JORDAN24:4
theorem Th5: :: JORDAN24:5
theorem Th6: :: JORDAN24:6
theorem Th7: :: JORDAN24:7
:: deftheorem Def4 defines closed JORDAN24:def 4 :
theorem Th8: :: JORDAN24:8
theorem Th9: :: JORDAN24:9
for b
1 being
set for b
2 being
Subset of b
1 holds
( b
2 ` = {} iff b
2 = b
1 )
theorem Th10: :: JORDAN24:10
theorem Th11: :: JORDAN24:11
theorem Th12: :: JORDAN24:12
theorem Th13: :: JORDAN24:13
theorem Th14: :: JORDAN24:14
theorem Th15: :: JORDAN24:15
theorem Th16: :: JORDAN24:16