:: JCT_MISC semantic presentation
theorem Th1: :: JCT_MISC:1
canceled;
theorem Th2: :: JCT_MISC:2
canceled;
theorem Th3: :: JCT_MISC:3
canceled;
theorem Th4: :: JCT_MISC:4
canceled;
theorem Th5: :: JCT_MISC:5
for
b1,
b2 being
set for
b3 being
Function of
b1,
b2for
b4 being
Subset of
b2 st (
b2 = {} implies
b1 = {} ) holds
(b3 " b4) ` = b3 " (b4 ` )
theorem Th6: :: JCT_MISC:6
theorem Th7: :: JCT_MISC:7
for
b1,
b2 being
Nat st
b1 <= b2 holds
b2 -' (b2 -' b1) = b1
theorem Th8: :: JCT_MISC:8
canceled;
theorem Th9: :: JCT_MISC:9
Lemma3:
for b1 being increasing Seq_of_Nat
for b2, b3 being Nat st b2 <= b3 holds
b1 . b2 <= b1 . b3
by SEQM_3:12;
theorem Th10: :: JCT_MISC:10
canceled;
theorem Th11: :: JCT_MISC:11
theorem Th12: :: JCT_MISC:12
theorem Th13: :: JCT_MISC:13
theorem Th14: :: JCT_MISC:14
:: deftheorem Def1 defines connected JCT_MISC:def 1 :
theorem Th15: :: JCT_MISC:15
definition
let c1,
c2 be
Subset of
REAL ;
func dist c1,
c2 -> real number means :
Def2:
:: JCT_MISC:def 2
ex
b1 being
Subset of
REAL st
(
b1 = { (abs (b2 - b3)) where B is Element of REAL , B is Element of REAL : ( b2 in a1 & b3 in a2 ) } &
a3 = lower_bound b1 );
existence
ex b1 being real number ex b2 being Subset of REAL st
( b2 = { (abs (b3 - b4)) where B is Element of REAL , B is Element of REAL : ( b3 in c1 & b4 in c2 ) } & b1 = lower_bound b2 )
uniqueness
for b1, b2 being real number st ex b3 being Subset of REAL st
( b3 = { (abs (b4 - b5)) where B is Element of REAL , B is Element of REAL : ( b4 in c1 & b5 in c2 ) } & b1 = lower_bound b3 ) & ex b3 being Subset of REAL st
( b3 = { (abs (b4 - b5)) where B is Element of REAL , B is Element of REAL : ( b4 in c1 & b5 in c2 ) } & b2 = lower_bound b3 ) holds
b1 = b2
;
commutativity
for b1 being real number
for b2, b3 being Subset of REAL st ex b4 being Subset of REAL st
( b4 = { (abs (b5 - b6)) where B is Element of REAL , B is Element of REAL : ( b5 in b2 & b6 in b3 ) } & b1 = lower_bound b4 ) holds
ex b4 being Subset of REAL st
( b4 = { (abs (b5 - b6)) where B is Element of REAL , B is Element of REAL : ( b5 in b3 & b6 in b2 ) } & b1 = lower_bound b4 )
end;
:: deftheorem Def2 defines dist JCT_MISC:def 2 :
theorem Th16: :: JCT_MISC:16
theorem Th17: :: JCT_MISC:17
theorem Th18: :: JCT_MISC:18
theorem Th19: :: JCT_MISC:19
theorem Th20: :: JCT_MISC:20
theorem Th21: :: JCT_MISC:21
theorem Th22: :: JCT_MISC:22
theorem Th23: :: JCT_MISC:23
theorem Th24: :: JCT_MISC:24