:: URYSOHN3 semantic presentation
Lemma1:
for b1 being non empty being_T4 TopSpace
for b2, b3 being closed Subset of b1 holds
not ( b2 <> {} & b2 misses b3 & ( for b4 being Function of dyadic 0, bool the carrier of b1 holds
not ( b2 c= b4 . 0 & b3 = ([#] b1) \ (b4 . 1) & ( for b5, b6 being Element of dyadic 0 holds
( b5 < b6 implies ( b4 . b5 is open & b4 . b6 is open & Cl (b4 . b5) c= b4 . b6 ) ) ) ) ) )
theorem Th1: :: URYSOHN3:1
:: deftheorem Def1 defines Drizzle URYSOHN3:def 1 :
theorem Th2: :: URYSOHN3:2
canceled;
theorem Th3: :: URYSOHN3:3
theorem Th4: :: URYSOHN3:4
theorem Th5: :: URYSOHN3:5
:: deftheorem Def2 defines Rain URYSOHN3:def 2 :
:: deftheorem Def3 defines inf_number_dyadic URYSOHN3:def 3 :
theorem Th6: :: URYSOHN3:6
theorem Th7: :: URYSOHN3:7
theorem Th8: :: URYSOHN3:8
theorem Th9: :: URYSOHN3:9
theorem Th10: :: URYSOHN3:10
theorem Th11: :: URYSOHN3:11
:: deftheorem Def4 defines Tempest URYSOHN3:def 4 :
theorem Th12: :: URYSOHN3:12
theorem Th13: :: URYSOHN3:13
theorem Th14: :: URYSOHN3:14
:: deftheorem Def5 defines Rainbow URYSOHN3:def 5 :
theorem Th15: :: URYSOHN3:15
theorem Th16: :: URYSOHN3:16
definition
let c
1 be non
empty TopSpace;
let c
2, c
3 be
Subset of c
1;
let c
4 be
Rain of c
2,c
3;
func Thunder c
4 -> Function of a
1,
R^1 means :
Def6:
:: URYSOHN3:def 6
for b
1 being
Point of a
1 holds
( (
Rainbow b
1,a
4 = {} implies a
5 . b
1 = 0 ) & ( for b
2 being non
empty Subset of
ExtREAL holds
( b
2 = Rainbow b
1,a
4 implies a
5 . b
1 = sup b
2 ) ) );
existence
ex b1 being Function of c1,R^1 st
for b2 being Point of c1 holds
( ( Rainbow b2,c4 = {} implies b1 . b2 = 0 ) & ( for b3 being non empty Subset of ExtREAL holds
( b3 = Rainbow b2,c4 implies b1 . b2 = sup b3 ) ) )
by Th16;
uniqueness
for b1, b2 being Function of c1,R^1 holds
( ( for b3 being Point of c1 holds
( ( Rainbow b3,c4 = {} implies b1 . b3 = 0 ) & ( for b4 being non empty Subset of ExtREAL holds
( b4 = Rainbow b3,c4 implies b1 . b3 = sup b4 ) ) ) ) & ( for b3 being Point of c1 holds
( ( Rainbow b3,c4 = {} implies b2 . b3 = 0 ) & ( for b4 being non empty Subset of ExtREAL holds
( b4 = Rainbow b3,c4 implies b2 . b3 = sup b4 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines Thunder URYSOHN3:def 6 :
theorem Th17: :: URYSOHN3:17
theorem Th18: :: URYSOHN3:18
theorem Th19: :: URYSOHN3:19
theorem Th20: :: URYSOHN3:20
theorem Th21: :: URYSOHN3:21
theorem Th22: :: URYSOHN3:22
theorem Th23: :: URYSOHN3:23
theorem Th24: :: URYSOHN3:24