:: FRECHET semantic presentation

Lemma1: for b1 being Nat holds
not ( b1 <> 0 & not 1 / b1 > 0 )
proof end;

Lemma2: for b1 being Real holds
not ( b1 > 0 & ( for b2 being Nat holds
not ( 1 / b2 < b1 & b2 > 0 ) ) )
proof end;

theorem Th1: :: FRECHET:1
for b1 being non empty 1-sorted
for b2 being sequence of b1 holds
rng b2 is Subset of b1 ;

theorem Th2: :: FRECHET:2
for b1 being non empty 1-sorted
for b2 being 1-sorted
for b3 being sequence of b1 holds
( rng b3 c= the carrier of b2 implies b3 is sequence of b2 )
proof end;

theorem Th3: :: FRECHET:3
for b1 being non empty TopSpace
for b2 being Point of b1
for b3 being Basis of b2 holds
b3 <> {}
proof end;

registration
let c1 be non empty TopSpace;
let c2 be Point of c1;
cluster -> non empty Basis of a2;
coherence
for b1 being Basis of c2 holds
not b1 is empty
by Th3;
end;

Lemma5: for b1 being TopStruct
for b2 being Subset of b1 holds
( b2 is open iff ([#] b1) \ b2 is closed )
proof end;

theorem Th4: :: FRECHET:4
for b1 being TopSpace
for b2, b3 being Subset of b1 holds
( b2 is open & b3 is closed implies b2 \ b3 is open )
proof end;

theorem Th5: :: FRECHET:5
for b1 being TopStruct holds
( {} b1 is closed & [#] b1 is closed & ( for b2, b3 being Subset of b1 holds
( b2 is closed & b3 is closed implies b2 \/ b3 is closed ) ) & ( for b2 being Subset-Family of b1 holds
( b2 is closed implies meet b2 is closed ) ) implies b1 is TopSpace )
proof end;

theorem Th6: :: FRECHET:6
for b1 being TopSpace
for b2 being non empty TopStruct
for b3 being Function of b1,b2 holds
( ( for b4 being Subset of b2 holds
( b4 is closed iff b3 " b4 is closed ) ) implies b2 is TopSpace )
proof end;

theorem Th7: :: FRECHET:7
for b1 being Point of RealSpace
for b2, b3 being Real holds
( b2 = b1 & b3 > 0 implies Ball b1,b3 = ].(b2 - b3),(b2 + b3).[ )
proof end;

theorem Th8: :: FRECHET:8
for b1 being Subset of R^1 holds
( b1 is open iff for b2 being Real holds
not ( b2 in b1 & ( for b3 being Real holds
not ( b3 > 0 & ].(b2 - b3),(b2 + b3).[ c= b1 ) ) ) )
proof end;

theorem Th9: :: FRECHET:9
for b1 being sequence of R^1 holds
( ( for b2 being Nat holds b1 . b2 in ].(b2 - (1 / 4)),(b2 + (1 / 4)).[ ) implies rng b1 is closed )
proof end;

theorem Th10: :: FRECHET:10
for b1 being Subset of R^1 holds
( b1 = NAT implies b1 is closed )
proof end;

theorem Th11: :: FRECHET:11
for b1 being non empty MetrSpace
for b2 being Point of (TopSpaceMetr b1)
for b3 being Point of b1 holds
not ( b2 = b3 & ( for b4 being Basis of b2 holds
not ( b4 = { (Ball b3,(1 / b5)) where B is Nat : b5 <> 0 } & b4 is countable & ex b5 being Function of NAT ,b4 st
for b6 being set holds
not ( b6 in NAT & ( for b7 being Nat holds
not ( b6 = b7 & b5 . b6 = Ball b3,(1 / (b7 + 1)) ) ) ) ) ) )
proof end;

theorem Th12: :: FRECHET:12
for b1, b2 being Function holds rng (b1 +* b2) = (b1 .: ((dom b1) \ (dom b2))) \/ (rng b2)
proof end;

theorem Th13: :: FRECHET:13
for b1, b2 being set holds
( b2 c= b1 implies (id b1) .: b2 = b2 )
proof end;

theorem Th14: :: FRECHET:14
canceled;

theorem Th15: :: FRECHET:15
for b1, b2, b3 being set holds dom ((id b1) +* (b2 --> b3)) = b1 \/ b2
proof end;

theorem Th16: :: FRECHET:16
for b1, b2, b3 being set holds
( b2 <> {} implies rng ((id b1) +* (b2 --> b3)) = (b1 \ b2) \/ {b3} )
proof end;

theorem Th17: :: FRECHET:17
for b1, b2, b3, b4 being set holds
( b3 c= b1 implies ((id b1) +* (b2 --> b4)) " (b3 \ {b4}) = (b3 \ b2) \ {b4} )
proof end;

theorem Th18: :: FRECHET:18
for b1, b2, b3 being set holds
( not b3 in b1 implies ((id b1) +* (b2 --> b3)) " {b3} = b2 )
proof end;

theorem Th19: :: FRECHET:19
for b1, b2, b3, b4 being set holds
( b3 c= b1 & not b4 in b1 implies ((id b1) +* (b2 --> b4)) " (b3 \/ {b4}) = b3 \/ b2 )
proof end;

theorem Th20: :: FRECHET:20
for b1, b2, b3, b4 being set holds
( b3 c= b1 & not b4 in b1 implies ((id b1) +* (b2 --> b4)) " (b3 \ {b4}) = b3 \ b2 )
proof end;

Lemma21: for b1, b2, b3, b4 being set holds
( not b4 in b1 implies ((id b1) +* (b2 --> b4)) " ((b1 \ b3) \ {b4}) = (b1 \ b3) \ b2 )
by Th20;

definition
let c1 be non empty TopStruct ;
attr a1 is first-countable means :Def1: :: FRECHET:def 1
for b1 being Point of a1 holds
ex b2 being Basis of b1 st b2 is countable;
end;

:: deftheorem Def1 defines first-countable FRECHET:def 1 :
for b1 being non empty TopStruct holds
( b1 is first-countable iff for b2 being Point of b1 holds
ex b3 being Basis of b2 st b3 is countable );

theorem Th21: :: FRECHET:21
for b1 being non empty MetrSpace holds TopSpaceMetr b1 is first-countable
proof end;

theorem Th22: :: FRECHET:22
R^1 is first-countable by Th21, TOPMETR:def 7;

registration
cluster R^1 -> first-countable ;
coherence
R^1 is first-countable
by Th21, TOPMETR:def 7;
end;

definition
let c1 be TopStruct ;
let c2 be sequence of c1;
let c3 be Point of c1;
pred c2 is_convergent_to c3 means :Def2: :: FRECHET:def 2
for b1 being Subset of a1 holds
not ( b1 is open & a3 in b1 & ( for b2 being Nat holds
ex b3 being Nat st
( b2 <= b3 & not a2 . b3 in b1 ) ) );
end;

:: deftheorem Def2 defines is_convergent_to FRECHET:def 2 :
for b1 being TopStruct
for b2 being sequence of b1
for b3 being Point of b1 holds
( b2 is_convergent_to b3 iff for b4 being Subset of b1 holds
not ( b4 is open & b3 in b4 & ( for b5 being Nat holds
ex b6 being Nat st
( b5 <= b6 & not b2 . b6 in b4 ) ) ) );

theorem Th23: :: FRECHET:23
for b1 being non empty TopStruct
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 = NAT --> b2 implies b3 is_convergent_to b2 )
proof end;

definition
let c1 be TopStruct ;
let c2 be sequence of c1;
attr a2 is convergent means :Def3: :: FRECHET:def 3
ex b1 being Point of a1 st a2 is_convergent_to b1;
end;

:: deftheorem Def3 defines convergent FRECHET:def 3 :
for b1 being TopStruct
for b2 being sequence of b1 holds
( b2 is convergent iff ex b3 being Point of b1 st b2 is_convergent_to b3 );

definition
let c1 be non empty TopStruct ;
let c2 be sequence of c1;
func Lim c2 -> Subset of a1 means :Def4: :: FRECHET:def 4
for b1 being Point of a1 holds
( b1 in a3 iff a2 is_convergent_to b1 );
existence
ex b1 being Subset of c1 st
for b2 being Point of c1 holds
( b2 in b1 iff c2 is_convergent_to b2 )
proof end;
uniqueness
for b1, b2 being Subset of c1 holds
( ( for b3 being Point of c1 holds
( b3 in b1 iff c2 is_convergent_to b3 ) ) & ( for b3 being Point of c1 holds
( b3 in b2 iff c2 is_convergent_to b3 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def4 defines Lim FRECHET:def 4 :
for b1 being non empty TopStruct
for b2 being sequence of b1
for b3 being Subset of b1 holds
( b3 = Lim b2 iff for b4 being Point of b1 holds
( b4 in b3 iff b2 is_convergent_to b4 ) );

definition
let c1 be non empty TopStruct ;
attr a1 is Frechet means :Def5: :: FRECHET:def 5
for b1 being Subset of a1
for b2 being Point of a1 holds
not ( b2 in Cl b1 & ( for b3 being sequence of a1 holds
not ( rng b3 c= b1 & b2 in Lim b3 ) ) );
end;

:: deftheorem Def5 defines Frechet FRECHET:def 5 :
for b1 being non empty TopStruct holds
( b1 is Frechet iff for b2 being Subset of b1
for b3 being Point of b1 holds
not ( b3 in Cl b2 & ( for b4 being sequence of b1 holds
not ( rng b4 c= b2 & b3 in Lim b4 ) ) ) );

definition
let c1 be non empty TopStruct ;
attr a1 is sequential means :: FRECHET:def 6
for b1 being Subset of a1 holds
( b1 is closed iff for b2 being sequence of a1 holds
( b2 is convergent & rng b2 c= b1 implies Lim b2 c= b1 ) );
end;

:: deftheorem Def6 defines sequential FRECHET:def 6 :
for b1 being non empty TopStruct holds
( b1 is sequential iff for b2 being Subset of b1 holds
( b2 is closed iff for b3 being sequence of b1 holds
( b3 is convergent & rng b3 c= b2 implies Lim b3 c= b2 ) ) );

theorem Th24: :: FRECHET:24
for b1 being non empty TopSpace holds
( b1 is first-countable implies b1 is Frechet )
proof end;

registration
cluster non empty first-countable -> non empty Frechet TopStruct ;
coherence
for b1 being non empty TopSpace holds
( b1 is first-countable implies b1 is Frechet )
by Th24;
end;

theorem Th25: :: FRECHET:25
canceled;

theorem Th26: :: FRECHET:26
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is closed implies for b3 being sequence of b1 holds
( b3 is convergent & rng b3 c= b2 implies Lim b3 c= b2 ) )
proof end;

theorem Th27: :: FRECHET:27
for b1 being non empty TopSpace holds
( ( for b2 being Subset of b1 holds
( ( for b3 being sequence of b1 holds
( b3 is convergent & rng b3 c= b2 implies Lim b3 c= b2 ) ) implies b2 is closed ) ) implies b1 is sequential )
proof end;

theorem Th28: :: FRECHET:28
for b1 being non empty TopSpace holds
( b1 is Frechet implies b1 is sequential )
proof end;

registration
cluster non empty Frechet -> non empty sequential TopStruct ;
coherence
for b1 being non empty TopSpace holds
( b1 is Frechet implies b1 is sequential )
by Th28;
end;

definition
func REAL? -> non empty strict TopSpace means :Def7: :: FRECHET:def 7
( the carrier of a1 = (REAL \ NAT ) \/ {REAL } & ex b1 being Function of R^1 ,a1 st
( b1 = (id REAL ) +* (NAT --> REAL ) & ( for b2 being Subset of a1 holds
( b2 is closed iff b1 " b2 is closed ) ) ) );
existence
ex b1 being non empty strict TopSpace st
( the carrier of b1 = (REAL \ NAT ) \/ {REAL } & ex b2 being Function of R^1 ,b1 st
( b2 = (id REAL ) +* (NAT --> REAL ) & ( for b3 being Subset of b1 holds
( b3 is closed iff b2 " b3 is closed ) ) ) )
proof end;
uniqueness
for b1, b2 being non empty strict TopSpace holds
( the carrier of b1 = (REAL \ NAT ) \/ {REAL } & ex b3 being Function of R^1 ,b1 st
( b3 = (id REAL ) +* (NAT --> REAL ) & ( for b4 being Subset of b1 holds
( b4 is closed iff b3 " b4 is closed ) ) ) & the carrier of b2 = (REAL \ NAT ) \/ {REAL } & ex b3 being Function of R^1 ,b2 st
( b3 = (id REAL ) +* (NAT --> REAL ) & ( for b4 being Subset of b2 holds
( b4 is closed iff b3 " b4 is closed ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def7 defines REAL? FRECHET:def 7 :
for b1 being non empty strict TopSpace holds
( b1 = REAL? iff ( the carrier of b1 = (REAL \ NAT ) \/ {REAL } & ex b2 being Function of R^1 ,b1 st
( b2 = (id REAL ) +* (NAT --> REAL ) & ( for b3 being Subset of b1 holds
( b3 is closed iff b2 " b3 is closed ) ) ) ) );

Lemma34: {REAL } c= the carrier of REAL?
proof end;

theorem Th29: :: FRECHET:29
canceled;

theorem Th30: :: FRECHET:30
REAL is Point of REAL?
proof end;

theorem Th31: :: FRECHET:31
for b1 being Subset of REAL? holds
( ( b1 is open & REAL in b1 ) iff ex b2 being Subset of R^1 st
( b2 is open & NAT c= b2 & b1 = (b2 \ NAT ) \/ {REAL } ) )
proof end;

theorem Th32: :: FRECHET:32
for b1 being set holds
( ( b1 is Subset of REAL? & not REAL in b1 implies ( b1 is Subset of R^1 & NAT /\ b1 = {} ) ) & ( b1 is Subset of R^1 & NAT /\ b1 = {} implies ( b1 is Subset of REAL? & not REAL in b1 ) ) )
proof end;

theorem Th33: :: FRECHET:33
for b1 being Subset of R^1
for b2 being Subset of REAL? holds
( b1 = b2 implies ( ( NAT /\ b1 = {} & b1 is open implies ( not REAL in b2 & b2 is open ) ) & ( not REAL in b2 & b2 is open implies ( NAT /\ b1 = {} & b1 is open ) ) ) )
proof end;

theorem Th34: :: FRECHET:34
for b1 being Subset of REAL? holds
( b1 = {REAL } implies b1 is closed )
proof end;

theorem Th35: :: FRECHET:35
not REAL? is first-countable
proof end;

theorem Th36: :: FRECHET:36
REAL? is Frechet
proof end;

theorem Th37: :: FRECHET:37
ex b1 being non empty TopSpace st
( b1 is Frechet & not b1 is first-countable ) by Th35, Th36;

theorem Th38: :: FRECHET:38
canceled;

theorem Th39: :: FRECHET:39
for b1, b2 being Function holds
( b1 tolerates b2 implies rng (b1 +* b2) = (rng b1) \/ (rng b2) )
proof end;

theorem Th40: :: FRECHET:40
for b1 being Real holds
not ( b1 > 0 & ( for b2 being Nat holds
not ( 1 / b2 < b1 & b2 > 0 ) ) ) by Lemma2;