:: FRECHET2 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

Lm1: for T being non empty TopSpace holds
( T is_T1 iff for p being Point of T holds Cl {p} = {p} )
proof end;

Lm2: for T being non empty TopSpace st not T is_T1 holds
ex x1, x2 being Point of T st
( x1 <> x2 & x2 in Cl {x1} )
proof end;

Lm3: for T being non empty TopSpace st not T is_T1 holds
ex x1, x2 being Point of T ex S being sequence of T st
( S = NAT --> x1 & x1 <> x2 & S is_convergent_to x2 )
proof end;

Lm4: for T being non empty TopSpace st T is_T2 holds
T is_T1
proof end;

Lm5: for T being non empty 1-sorted
for S being sequence of T
for f being Function of NAT , NAT holds S * f is sequence of T
;

definition
let T be non empty 1-sorted ;
let f be Function of NAT , NAT ;
let S be sequence of T;
:: original: *
redefine func S * f -> sequence of T;
coherence
f * S is sequence of T
by Lm5;
end;

theorem Th1: :: FRECHET2:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty 1-sorted
for S being sequence of T
for NS being increasing Seq_of_Nat holds S * NS is sequence of T
proof end;

Lm6: id NAT is Real_Sequence
proof end;

Lm7: for RS being Real_Sequence st RS = id NAT holds
RS is natural-yielding
proof end;

Lm8: for RS being Real_Sequence st RS = id NAT holds
RS is increasing
proof end;

theorem :: FRECHET2:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RS being Real_Sequence st RS = id NAT holds
RS is increasing Seq_of_Nat by Lm7, Lm8;

Lm9: for T being non empty 1-sorted
for S being sequence of T ex NS being increasing Seq_of_Nat st S = S * NS
proof end;

theorem Th3: :: FRECHET2:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty 1-sorted
for S being sequence of T holds S is subsequence of S
proof end;

theorem Th4: :: FRECHET2:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty 1-sorted
for S being sequence of T
for S1 being subsequence of S holds rng S1 c= rng S
proof end;

Lm10: for T being non empty 1-sorted
for S being sequence of T
for NS being increasing Seq_of_Nat holds S * NS is subsequence of S
proof end;

definition
let T be non empty 1-sorted ;
let NS be increasing Seq_of_Nat;
let S be sequence of T;
:: original: *
redefine func S * NS -> subsequence of S;
correctness
coherence
NS * S is subsequence of S
;
by Lm10;
end;

theorem Th5: :: FRECHET2:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty 1-sorted
for S1 being sequence of T
for S2 being subsequence of S1
for S3 being subsequence of S2 holds S3 is subsequence of S1
proof end;

scheme :: FRECHET2:sch 1
SubSeqChoice{ F1() -> non empty 1-sorted , F2() -> sequence of F1(), P1[ set ] } :
ex S1 being subsequence of F2() st
for n being Nat holds P1[S1 . n]
provided
A1: for n being Nat ex m being Nat ex x being Point of F1() st
( n <= m & x = F2() . m & P1[x] )
proof end;

scheme :: FRECHET2:sch 2
SubSeqChoice1{ F1() -> non empty TopStruct , F2() -> sequence of F1(), P1[ set ] } :
ex S1 being subsequence of F2() st
for n being Nat holds P1[S1 . n]
provided
A1: for n being Nat ex m being Nat ex x being Point of F1() st
( n <= m & x = F2() . m & P1[x] )
proof end;

theorem Th6: :: FRECHET2:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty 1-sorted
for S being sequence of T
for A being Subset of T st ( for S1 being subsequence of S holds not rng S1 c= A ) holds
ex n being Nat st
for m being Nat st n <= m holds
not S . m in A
proof end;

theorem Th7: :: FRECHET2:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty 1-sorted
for S being sequence of T
for A, B being Subset of T st rng S c= A \/ B holds
ex S1 being subsequence of S st
( rng S1 c= A or rng S1 c= B )
proof end;

Lm11: for T being non empty TopSpace st T is first-countable holds
for x being Point of T ex B being Basis of x ex S being Function st
( dom S = NAT & rng S = B & ( for n, m being Nat st m >= n holds
S . m c= S . n ) )
proof end;

theorem :: FRECHET2:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st ( for S being sequence of T
for x1, x2 being Point of T st x1 in Lim S & x2 in Lim S holds
x1 = x2 ) holds
T is_T1
proof end;

theorem Th9: :: FRECHET2:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st T is_T2 holds
for S being sequence of T
for x1, x2 being Point of T st x1 in Lim S & x2 in Lim S holds
x1 = x2
proof end;

theorem :: FRECHET2:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st T is first-countable holds
( T is_T2 iff for S being sequence of T
for x1, x2 being Point of T st x1 in Lim S & x2 in Lim S holds
x1 = x2 )
proof end;

theorem Th11: :: FRECHET2:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T st not S is convergent holds
Lim S = {}
proof end;

theorem Th12: :: FRECHET2:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace
for A being Subset of T st A is closed holds
for S being sequence of T st rng S c= A holds
Lim S c= A
proof end;

theorem :: FRECHET2:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T
for x being Point of T st not S is_convergent_to x holds
ex S1 being subsequence of S st
for S2 being subsequence of S1 holds not S2 is_convergent_to x
proof end;

theorem Th14: :: FRECHET2:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T1, T2 being non empty TopSpace
for f being Function of T1,T2 st f is continuous holds
for S1 being sequence of T1
for S2 being sequence of T2 st S2 = f * S1 holds
f .: (Lim S1) c= Lim S2
proof end;

theorem :: FRECHET2:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T1, T2 being non empty TopSpace
for f being Function of T1,T2 st T1 is sequential holds
( f is continuous iff for S1 being sequence of T1
for S2 being sequence of T2 st S2 = f * S1 holds
f .: (Lim S1) c= Lim S2 )
proof end;

definition
let T be non empty TopStruct ;
let A be Subset of T;
canceled;
func Cl_Seq A -> Subset of T means :Def2: :: FRECHET2:def 2
for x being Point of T holds
( x in it iff ex S being sequence of T st
( rng S c= A & x in Lim S ) );
existence
ex b1 being Subset of T st
for x being Point of T holds
( x in b1 iff ex S being sequence of T st
( rng S c= A & x in Lim S ) )
proof end;
uniqueness
for b1, b2 being Subset of T st ( for x being Point of T holds
( x in b1 iff ex S being sequence of T st
( rng S c= A & x in Lim S ) ) ) & ( for x being Point of T holds
( x in b2 iff ex S being sequence of T st
( rng S c= A & x in Lim S ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem FRECHET2:def 1 :
canceled;

:: deftheorem Def2 defines Cl_Seq FRECHET2:def 2 :
for T being non empty TopStruct
for A, b3 being Subset of T holds
( b3 = Cl_Seq A iff for x being Point of T holds
( x in b3 iff ex S being sequence of T st
( rng S c= A & x in Lim S ) ) );

theorem Th16: :: FRECHET2:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for A being Subset of T
for S being sequence of T
for x being Point of T st rng S c= A & x in Lim S holds
x in Cl A
proof end;

theorem Th17: :: FRECHET2:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for A being Subset of T holds Cl_Seq A c= Cl A
proof end;

theorem Th18: :: FRECHET2:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T
for S1 being subsequence of S
for x being Point of T st S is_convergent_to x holds
S1 is_convergent_to x
proof end;

theorem Th19: :: FRECHET2:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T
for S1 being subsequence of S holds Lim S c= Lim S1
proof end;

theorem Th20: :: FRECHET2:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct holds Cl_Seq ({} T) = {}
proof end;

theorem Th21: :: FRECHET2:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for A being Subset of T holds A c= Cl_Seq A
proof end;

theorem Th22: :: FRECHET2:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for A, B being Subset of T holds (Cl_Seq A) \/ (Cl_Seq B) = Cl_Seq (A \/ B)
proof end;

theorem Th23: :: FRECHET2:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct holds
( T is Frechet iff for A being Subset of T holds Cl A = Cl_Seq A )
proof end;

theorem Th24: :: FRECHET2:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st T is Frechet holds
for A, B being Subset of T holds
( Cl_Seq ({} T) = {} & A c= Cl_Seq A & Cl_Seq (A \/ B) = (Cl_Seq A) \/ (Cl_Seq B) & Cl_Seq (Cl_Seq A) = Cl_Seq A )
proof end;

theorem Th25: :: FRECHET2:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st T is sequential & ( for A being Subset of T holds Cl_Seq (Cl_Seq A) = Cl_Seq A ) holds
T is Frechet
proof end;

theorem :: FRECHET2:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st T is sequential holds
( T is Frechet iff for A, B being Subset of T holds
( Cl_Seq ({} T) = {} & A c= Cl_Seq A & Cl_Seq (A \/ B) = (Cl_Seq A) \/ (Cl_Seq B) & Cl_Seq (Cl_Seq A) = Cl_Seq A ) ) by Th24, Th25;

definition
let T be non empty TopSpace;
let S be sequence of T;
assume A1: ex x being Point of T st Lim S = {x} ;
func lim S -> Point of T means :Def3: :: FRECHET2:def 3
S is_convergent_to it;
existence
ex b1 being Point of T st S is_convergent_to b1
proof end;
uniqueness
for b1, b2 being Point of T st S is_convergent_to b1 & S is_convergent_to b2 holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines lim FRECHET2:def 3 :
for T being non empty TopSpace
for S being sequence of T st ex x being Point of T st Lim S = {x} holds
for b3 being Point of T holds
( b3 = lim S iff S is_convergent_to b3 );

theorem Th27: :: FRECHET2:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st T is_T2 holds
for S being sequence of T st S is convergent holds
ex x being Point of T st Lim S = {x}
proof end;

theorem Th28: :: FRECHET2:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st T is_T2 holds
for S being sequence of T
for x being Point of T holds
( S is_convergent_to x iff ( S is convergent & x = lim S ) )
proof end;

theorem :: FRECHET2:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being MetrStruct
for S being sequence of M holds S is sequence of (TopSpaceMetr M)
proof end;

theorem :: FRECHET2:30  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty MetrStruct
for S being sequence of (TopSpaceMetr M) holds S is sequence of M
proof end;

theorem Th31: :: FRECHET2:31  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty MetrSpace
for S being sequence of M
for x being Point of M
for S' being sequence of (TopSpaceMetr M)
for x' being Point of (TopSpaceMetr M) st S = S' & x = x' holds
( S is_convergent_in_metrspace_to x iff S' is_convergent_to x' )
proof end;

theorem :: FRECHET2:32  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty MetrSpace
for Sm being sequence of M
for St being sequence of (TopSpaceMetr M) st Sm = St holds
( Sm is convergent iff St is convergent )
proof end;

theorem :: FRECHET2:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty MetrSpace
for Sm being sequence of M
for St being sequence of (TopSpaceMetr M) st Sm = St & Sm is convergent holds
lim Sm = lim St
proof end;

definition
let T be TopStruct ;
let S be sequence of T;
let x be Point of T;
pred x is_a_cluster_point_of S means :Def4: :: FRECHET2:def 4
for O being Subset of T
for n being Nat st O is open & x in O holds
ex m being Nat st
( n <= m & S . m in O );
end;

:: deftheorem Def4 defines is_a_cluster_point_of FRECHET2:def 4 :
for T being TopStruct
for S being sequence of T
for x being Point of T holds
( x is_a_cluster_point_of S iff for O being Subset of T
for n being Nat st O is open & x in O holds
ex m being Nat st
( n <= m & S . m in O ) );

theorem Th34: :: FRECHET2:34  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T
for x being Point of T st ex S1 being subsequence of S st S1 is_convergent_to x holds
x is_a_cluster_point_of S
proof end;

theorem :: FRECHET2:35  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T
for x being Point of T st S is_convergent_to x holds
x is_a_cluster_point_of S
proof end;

theorem Th36: :: FRECHET2:36  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T
for x being Point of T
for Y being Subset of T st Y = { y where y is Point of T : x in Cl {y} } & rng S c= Y holds
S is_convergent_to x
proof end;

theorem Th37: :: FRECHET2:37  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T
for x, y being Point of T st ( for n being Nat holds
( S . n = y & S is_convergent_to x ) ) holds
x in Cl {y}
proof end;

theorem Th38: :: FRECHET2:38  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for x being Point of T
for Y being Subset of T
for S being sequence of T st Y = { y where y is Point of T : x in Cl {y} } & rng S misses Y & S is_convergent_to x holds
ex S1 being subsequence of S st S1 is one-to-one
proof end;

Lm12: for f being Function st not dom f is finite & f is one-to-one holds
not rng f is finite
proof end;

theorem Th39: :: FRECHET2:39  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S1, S2 being sequence of T st rng S2 c= rng S1 & S2 is one-to-one holds
ex P being Permutation of NAT st S2 * P is subsequence of S1
proof end;

scheme :: FRECHET2:sch 3
PermSeq{ F1() -> non empty 1-sorted , F2() -> sequence of F1(), F3() -> Permutation of NAT , P1[ set ] } :
ex n being Nat st
for m being Nat st n <= m holds
P1[(F2() * F3()) . m]
provided
A1: ex n being Nat st
for m being Nat
for x being Point of F1() st n <= m & x = F2() . m holds
P1[x]
proof end;

scheme :: FRECHET2:sch 4
PermSeq2{ F1() -> non empty TopStruct , F2() -> sequence of F1(), F3() -> Permutation of NAT , P1[ set ] } :
ex n being Nat st
for m being Nat st n <= m holds
P1[(F2() * F3()) . m]
provided
A1: ex n being Nat st
for m being Nat
for x being Point of F1() st n <= m & x = F2() . m holds
P1[x]
proof end;

theorem Th40: :: FRECHET2:40  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T
for P being Permutation of NAT
for x being Point of T st S is_convergent_to x holds
S * P is_convergent_to x
proof end;

theorem Th41: :: FRECHET2:41  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for n0 being Nat ex NS being increasing Seq_of_Nat st
for n being Nat holds NS . n = n + n0
proof end;

theorem Th42: :: FRECHET2:42  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty 1-sorted
for S being sequence of T
for n0 being Nat ex S1 being subsequence of S st
for n being Nat holds S1 . n = S . (n + n0)
proof end;

theorem Th43: :: FRECHET2:43  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T
for x being Point of T
for S1 being subsequence of S st x is_a_cluster_point_of S & ex n0 being Nat st
for n being Nat holds S1 . n = S . (n + n0) holds
x is_a_cluster_point_of S1
proof end;

theorem Th44: :: FRECHET2:44  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for S being sequence of T
for x being Point of T st x is_a_cluster_point_of S holds
x in Cl (rng S)
proof end;

theorem :: FRECHET2:45  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct st T is Frechet holds
for S being sequence of T
for x being Point of T st x is_a_cluster_point_of S holds
ex S1 being subsequence of S st S1 is_convergent_to x
proof end;

theorem :: FRECHET2:46  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st T is first-countable holds
for x being Point of T ex B being Basis of x ex S being Function st
( dom S = NAT & rng S = B & ( for n, m being Nat st m >= n holds
S . m c= S . n ) ) by Lm11;

theorem :: FRECHET2:47  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace holds
( T is_T1 iff for p being Point of T holds Cl {p} = {p} ) by Lm1;

theorem :: FRECHET2:48  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st T is_T2 holds
T is_T1 by Lm4;

theorem :: FRECHET2:49  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace st not T is_T1 holds
ex x1, x2 being Point of T ex S being sequence of T st
( S = NAT --> x1 & x1 <> x2 & S is_convergent_to x2 ) by Lm3;

theorem :: FRECHET2:50  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for f being Function st not dom f is finite & f is one-to-one holds
not rng f is finite by Lm12;