Article COMPL_SP, MML version 4.99.1005
:: COMPL_SP:attrnot 1 => COMPL_SP:attr 1
definition
let a1 be non empty MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
attr a2 is bounded means
for b1 being natural set holds
a2 . b1 is bounded(a1);
end;
:: COMPL_SP:dfs 1
definiens
let a1 be non empty MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
To prove
a2 is bounded
it is sufficient to prove
thus for b1 being natural set holds
a2 . b1 is bounded(a1);
:: COMPL_SP:def 1
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
b2 is bounded(b1)
iff
for b3 being natural set holds
b2 . b3 is bounded(b1);
:: COMPL_SP:exreg 1
registration
let a1 be non empty Reflexive MetrStruct;
cluster Relation-like non-empty Function-like non empty quasi_total total bounded Relation of NAT,bool the carrier of a1;
end;
:: COMPL_SP:funcnot 1 => COMPL_SP:func 1
definition
let a1 be non empty Reflexive MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
func diameter A2 -> Function-like quasi_total Relation of NAT,REAL means
for b1 being natural set holds
it . b1 = diameter (a2 . b1);
end;
:: COMPL_SP:def 2
theorem
for b1 being non empty Reflexive MetrStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL holds
b3 = diameter b2
iff
for b4 being natural set holds
b3 . b4 = diameter (b2 . b4);
:: COMPL_SP:th 1
theorem
for b1 being non empty Reflexive MetrStruct
for b2 being Function-like quasi_total bounded Relation of NAT,bool the carrier of b1 holds
diameter b2 is bounded_below;
:: COMPL_SP:th 2
theorem
for b1 being non empty Reflexive MetrStruct
for b2 being Function-like quasi_total bounded Relation of NAT,bool the carrier of b1
st b2 is descending(the carrier of b1)
holds diameter b2 is bounded_above & diameter b2 is non-increasing;
:: COMPL_SP:th 3
theorem
for b1 being non empty Reflexive MetrStruct
for b2 being Function-like quasi_total bounded Relation of NAT,bool the carrier of b1
st b2 is ascending(the carrier of b1)
holds diameter b2 is non-decreasing;
:: COMPL_SP:th 4
theorem
for b1 being non empty Reflexive MetrStruct
for b2 being Function-like quasi_total bounded Relation of NAT,bool the carrier of b1
st b2 is descending(the carrier of b1) & lim diameter b2 = 0
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st for b4 being natural set holds
b3 . b4 in b2 . b4
holds b3 is Cauchy(b1);
:: COMPL_SP:th 5
theorem
for b1 being Element of REAL
for b2 being non empty Reflexive symmetric triangle MetrStruct
for b3 being Element of the carrier of b2
st 0 <= b1
holds diameter cl_Ball(b3,b1) <= 2 * b1;
:: COMPL_SP:attrnot 2 => COMPL_SP:attr 2
definition
let a1 be MetrStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is open means
a2 in Family_open_set a1;
end;
:: COMPL_SP:dfs 3
definiens
let a1 be MetrStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is open
it is sufficient to prove
thus a2 in Family_open_set a1;
:: COMPL_SP:def 3
theorem
for b1 being MetrStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open(b1)
iff
b2 in Family_open_set b1;
:: COMPL_SP:attrnot 3 => COMPL_SP:attr 3
definition
let a1 be MetrStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is closed means
a2 ` is open(a1);
end;
:: COMPL_SP:dfs 4
definiens
let a1 be MetrStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is closed
it is sufficient to prove
thus a2 ` is open(a1);
:: COMPL_SP:def 4
theorem
for b1 being MetrStruct
for b2 being Element of bool the carrier of b1 holds
b2 is closed(b1)
iff
b2 ` is open(b1);
:: COMPL_SP:exreg 2
registration
let a1 be MetrStruct;
cluster empty open Element of bool the carrier of a1;
end;
:: COMPL_SP:exreg 3
registration
let a1 be MetrStruct;
cluster empty closed Element of bool the carrier of a1;
end;
:: COMPL_SP:exreg 4
registration
let a1 be non empty MetrStruct;
cluster non empty open Element of bool the carrier of a1;
end;
:: COMPL_SP:exreg 5
registration
let a1 be non empty MetrStruct;
cluster non empty closed Element of bool the carrier of a1;
end;
:: COMPL_SP:th 6
theorem
for b1 being MetrStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of TopSpaceMetr b1
st b3 = b2
holds (b2 is open(b1) implies b3 is open(TopSpaceMetr b1)) & (b3 is open(TopSpaceMetr b1) implies b2 is open(b1)) & (b2 is closed(b1) implies b3 is closed(TopSpaceMetr b1)) & (b3 is closed(TopSpaceMetr b1) implies b2 is closed(b1));
:: COMPL_SP:attrnot 4 => COMPL_SP:attr 4
definition
let a1 be TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
attr a2 is open means
for b1 being natural set holds
a2 . b1 is open(a1);
end;
:: COMPL_SP:dfs 5
definiens
let a1 be TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
To prove
a2 is open
it is sufficient to prove
thus for b1 being natural set holds
a2 . b1 is open(a1);
:: COMPL_SP:def 5
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
b2 is open(b1)
iff
for b3 being natural set holds
b2 . b3 is open(b1);
:: COMPL_SP:attrnot 5 => COMPL_SP:attr 5
definition
let a1 be TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
attr a2 is closed means
for b1 being natural set holds
a2 . b1 is closed(a1);
end;
:: COMPL_SP:dfs 6
definiens
let a1 be TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
To prove
a2 is closed
it is sufficient to prove
thus for b1 being natural set holds
a2 . b1 is closed(a1);
:: COMPL_SP:def 6
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
b2 is closed(b1)
iff
for b3 being natural set holds
b2 . b3 is closed(b1);
:: COMPL_SP:exreg 6
registration
let a1 be TopSpace-like TopStruct;
cluster Relation-like Function-like non empty quasi_total total open Relation of NAT,bool the carrier of a1;
end;
:: COMPL_SP:exreg 7
registration
let a1 be TopSpace-like TopStruct;
cluster Relation-like Function-like non empty quasi_total total closed Relation of NAT,bool the carrier of a1;
end;
:: COMPL_SP:exreg 8
registration
let a1 be non empty TopSpace-like TopStruct;
cluster Relation-like non-empty Function-like non empty quasi_total total open Relation of NAT,bool the carrier of a1;
end;
:: COMPL_SP:exreg 9
registration
let a1 be non empty TopSpace-like TopStruct;
cluster Relation-like non-empty Function-like non empty quasi_total total closed Relation of NAT,bool the carrier of a1;
end;
:: COMPL_SP:attrnot 6 => COMPL_SP:attr 6
definition
let a1 be MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
attr a2 is open means
for b1 being natural set holds
a2 . b1 is open(a1);
end;
:: COMPL_SP:dfs 7
definiens
let a1 be MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
To prove
a2 is open
it is sufficient to prove
thus for b1 being natural set holds
a2 . b1 is open(a1);
:: COMPL_SP:def 7
theorem
for b1 being MetrStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
b2 is open(b1)
iff
for b3 being natural set holds
b2 . b3 is open(b1);
:: COMPL_SP:attrnot 7 => COMPL_SP:attr 7
definition
let a1 be MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
attr a2 is closed means
for b1 being natural set holds
a2 . b1 is closed(a1);
end;
:: COMPL_SP:dfs 8
definiens
let a1 be MetrStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
To prove
a2 is closed
it is sufficient to prove
thus for b1 being natural set holds
a2 . b1 is closed(a1);
:: COMPL_SP:def 8
theorem
for b1 being MetrStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
b2 is closed(b1)
iff
for b3 being natural set holds
b2 . b3 is closed(b1);
:: COMPL_SP:exreg 10
registration
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
cluster Relation-like non-empty Function-like non empty quasi_total total bounded open Relation of NAT,bool the carrier of a1;
end;
:: COMPL_SP:exreg 11
registration
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
cluster Relation-like non-empty Function-like non empty quasi_total total bounded closed Relation of NAT,bool the carrier of a1;
end;
:: COMPL_SP:th 7
theorem
for b1 being MetrStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,bool the carrier of TopSpaceMetr b1
st b3 = b2
holds (b2 is open(b1) implies b3 is open(TopSpaceMetr b1)) & (b3 is open(TopSpaceMetr b1) implies b2 is open(b1)) & (b2 is closed(b1) implies b3 is closed(TopSpaceMetr b1)) & (b3 is closed(TopSpaceMetr b1) implies b2 is closed(b1));
:: COMPL_SP:th 8
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is bounded(b1)
for b4 being Element of bool the carrier of TopSpaceMetr b1
st b2 = b4 & b3 = Cl b4
holds b3 is bounded(b1) & diameter b2 = diameter b3;
:: COMPL_SP:th 9
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
ex b3 being non-empty Function-like quasi_total closed Relation of NAT,bool the carrier of b1 st
b3 is descending(the carrier of b1) &
(b2 is Cauchy(b1) implies b3 is bounded(b1) & lim diameter b3 = 0) &
(for b4 being natural set holds
ex b5 being Element of bool the carrier of TopSpaceMetr b1 st
b5 = {b2 . b6 where b6 is Element of NAT: b4 <= b6} &
b3 . b4 = Cl b5);
:: COMPL_SP:th 10
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct holds
b1 is complete
iff
for b2 being non-empty Function-like quasi_total bounded closed Relation of NAT,bool the carrier of b1
st b2 is descending(the carrier of b1) & lim diameter b2 = 0
holds meet b2 is not empty;
:: COMPL_SP:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non-empty Function-like quasi_total Relation of NAT,bool the carrier of b1
st b2 is descending(the carrier of b1)
for b3 being Element of bool bool the carrier of b1
st b3 = rng b2
holds b3 is centered;
:: COMPL_SP:th 12
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Element of bool bool the carrier of TopSpaceMetr b1
st b3 = rng b2
holds (b2 is open(b1) implies b3 is open(TopSpaceMetr b1)) & (b2 is closed(b1) implies b3 is closed(TopSpaceMetr b1));
:: COMPL_SP:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
st rng b3 c= b2
holds ex b4 being Function-like quasi_total Relation of NAT,bool the carrier of b1 st
b4 is descending(the carrier of b1) &
(b2 is centered implies b4 is non-empty) &
(b2 is open(b1) implies b4 is open(b1)) &
(b2 is closed(b1) implies b4 is closed(b1)) &
(for b5 being natural set holds
b4 . b5 = meet {b3 . b6 where b6 is Element of NAT: b6 <= b5});
:: COMPL_SP:th 14
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct holds
b1 is complete
iff
for b2 being Element of bool bool the carrier of TopSpaceMetr b1
st b2 is closed(TopSpaceMetr b1) &
b2 is centered &
(for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of bool the carrier of b1 st
b4 in b2 & b4 is bounded(b1) & diameter b4 < b3)
holds meet b2 is not empty;
:: COMPL_SP:th 15
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b1 | b2
st b3 = b4
holds b4 is bounded(b1 | b2)
iff
b3 is bounded(b1);
:: COMPL_SP:th 16
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b1 | b2
st b3 = b4 & b3 is bounded(b1)
holds diameter b4 <= diameter b3;
:: COMPL_SP:th 17
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 | b2 holds
b3 is Function-like quasi_total Relation of NAT,the carrier of b1;
:: COMPL_SP:th 18
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 | b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 = b4
holds b4 is Cauchy(b1)
iff
b3 is Cauchy(b1 | b2);
:: COMPL_SP:th 19
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
st b1 is complete
for b2 being non empty Element of bool the carrier of b1
for b3 being Element of bool the carrier of TopSpaceMetr b1
st b2 = b3
holds b1 | b2 is complete
iff
b3 is closed(TopSpaceMetr b1);
:: COMPL_SP:attrnot 8 => COMPL_SP:attr 8
definition
let a1 be TopStruct;
attr a1 is countably_compact means
for b1 being Element of bool bool the carrier of a1
st b1 is_a_cover_of a1 & b1 is open(a1) & b1 is countable
holds ex b2 being Element of bool bool the carrier of a1 st
b2 c= b1 & b2 is_a_cover_of a1 & b2 is finite;
end;
:: COMPL_SP:dfs 9
definiens
let a1 be TopStruct;
To prove
a1 is countably_compact
it is sufficient to prove
thus for b1 being Element of bool bool the carrier of a1
st b1 is_a_cover_of a1 & b1 is open(a1) & b1 is countable
holds ex b2 being Element of bool bool the carrier of a1 st
b2 c= b1 & b2 is_a_cover_of a1 & b2 is finite;
:: COMPL_SP:def 9
theorem
for b1 being TopStruct holds
b1 is countably_compact
iff
for b2 being Element of bool bool the carrier of b1
st b2 is_a_cover_of b1 & b2 is open(b1) & b2 is countable
holds ex b3 being Element of bool bool the carrier of b1 st
b3 c= b2 & b3 is_a_cover_of b1 & b3 is finite;
:: COMPL_SP:th 20
theorem
for b1 being TopStruct
st b1 is compact
holds b1 is countably_compact;
:: COMPL_SP:th 21
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is countably_compact
iff
for b2 being Element of bool bool the carrier of b1
st b2 is centered & b2 is closed(b1) & b2 is countable
holds meet b2 <> {};
:: COMPL_SP:th 22
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is countably_compact
iff
for b2 being non-empty Function-like quasi_total closed Relation of NAT,bool the carrier of b1
st b2 is descending(the carrier of b1)
holds meet b2 <> {};
:: COMPL_SP:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
st rng b3 c= b2 & b3 is non-empty
holds ex b4 being non-empty Function-like quasi_total closed Relation of NAT,bool the carrier of b1 st
b4 is descending(the carrier of b1) &
(b2 is locally_finite(b1) & b3 is one-to-one implies meet b4 = {}) &
(for b5 being natural set holds
ex b6 being Element of bool bool the carrier of b1 st
b4 . b5 = Cl union b6 &
b6 = {b3 . b7 where b7 is Element of NAT: b5 <= b7});
:: COMPL_SP:th 24
theorem
for b1 being Relation-like Function-like set
st proj1 b1 is infinite & proj2 b1 is finite
holds ex b2 being set st
b2 in proj2 b1 & b1 " {b2} is infinite;
:: COMPL_SP:th 25
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is descending(b1)
for b3 being Function-like quasi_total Relation of NAT,b1
st (for b4 being natural set holds
b3 . b4 in b2 . b4) &
rng b3 is finite
holds meet b2 is not empty;
:: COMPL_SP:th 26
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is countably_compact
iff
for b2 being Element of bool bool the carrier of b1
st b2 is locally_finite(b1) & b2 is with_non-empty_elements
holds b2 is finite;
:: COMPL_SP:th 27
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is countably_compact
iff
for b2 being Element of bool bool the carrier of b1
st b2 is locally_finite(b1) &
(for b3 being Element of bool the carrier of b1
st b3 in b2
holds Card b3 = 1)
holds b2 is finite;
:: COMPL_SP:th 28
theorem
for b1 being non empty TopSpace-like being_T1 TopStruct holds
b1 is countably_compact
iff
for b2 being Element of bool the carrier of b1
st b2 is infinite
holds Der b2 is not empty;
:: COMPL_SP:th 29
theorem
for b1 being non empty TopSpace-like being_T1 TopStruct holds
b1 is countably_compact
iff
for b2 being Element of bool the carrier of b1
st b2 is infinite & b2 is countable
holds Der b2 is not empty;
:: COMPL_SP:sch 1
scheme COMPL_SP:sch 1
{F1 -> non empty set}:
ex b1 being Element of bool F1() st
(for b2, b3 being Element of F1()
st b2 in b1 & b3 in b1 & b2 <> b3
holds P1[b2, b3]) &
(for b2 being Element of F1() holds
ex b3 being Element of F1() st
b3 in b1 & not (P1[b2, b3]))
provided
for b1, b2 being Element of F1() holds
P1[b1, b2]
iff
P1[b2, b1]
and
for b1 being Element of F1() holds
not (P1[b1, b1]);
:: COMPL_SP:th 30
theorem
for b1 being non empty Reflexive symmetric MetrStruct
for b2 being Element of REAL
st 0 < b2
holds ex b3 being Element of bool the carrier of b1 st
(for b4, b5 being Element of the carrier of b1
st b4 <> b5 & b4 in b3 & b5 in b3
holds b2 <= dist(b4,b5)) &
(for b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
b5 in b3 & b4 in Ball(b5,b2));
:: COMPL_SP:th 31
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct holds
b1 is totally_bounded
iff
for b2 being Element of REAL
for b3 being Element of bool the carrier of b1
st 0 < b2 &
(for b4, b5 being Element of the carrier of b1
st b4 <> b5 & b4 in b3 & b5 in b3
holds b2 <= dist(b4,b5))
holds b3 is finite;
:: COMPL_SP:th 32
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
st TopSpaceMetr b1 is countably_compact
holds b1 is totally_bounded;
:: COMPL_SP:th 33
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
st b1 is totally_bounded
holds TopSpaceMetr b1 is second-countable;
:: COMPL_SP:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is second-countable
for b2 being Element of bool bool the carrier of b1
st b2 is_a_cover_of b1 & b2 is open(b1)
holds ex b3 being Element of bool bool the carrier of b1 st
b3 c= b2 & b3 is_a_cover_of b1 & b3 is countable;
:: COMPL_SP:th 35
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct holds
TopSpaceMetr b1 is compact
iff
TopSpaceMetr b1 is countably_compact;
:: COMPL_SP:th 36
theorem
for b1 being set
for b2 being Element of bool bool b1
st b2 is finite
for b3 being Element of bool b1
st b3 is infinite & b3 c= union b2
holds ex b4 being Element of bool b1 st
b4 in b2 & b4 /\ b3 is infinite;
:: COMPL_SP:th 37
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct holds
TopSpaceMetr b1 is compact
iff
b1 is totally_bounded & b1 is complete;
:: COMPL_SP:funcnot 2 => COMPL_SP:func 2
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,a1;
let a3 be natural set;
redefine func a2 . a3 -> Element of a1;
end;
:: COMPL_SP:th 38
theorem
for b1 being set
for b2 being MetrStruct
for b3 being Element of the carrier of b2
for b4 being set holds
b4 in [:b1,(the carrier of b2) \ {b3}:] \/ {[b1,b3]}
iff
ex b5 being set st
ex b6 being Element of the carrier of b2 st
b4 = [b5,b6] &
(b5 in b1 & b6 <> b3 or b5 = b1 & b6 = b3);
:: COMPL_SP:funcnot 3 => COMPL_SP:func 3
definition
let a1 be MetrStruct;
let a2 be Element of the carrier of a1;
let a3 be set;
func well_dist(A2,A3) -> Function-like quasi_total Relation of [:[:a3,(the carrier of a1) \ {a2}:] \/ {[a3,a2]},[:a3,(the carrier of a1) \ {a2}:] \/ {[a3,a2]}:],REAL means
for b1, b2 being Element of [:a3,(the carrier of a1) \ {a2}:] \/ {[a3,a2]}
for b3, b4 being set
for b5, b6 being Element of the carrier of a1
st b1 = [b3,b5] & b2 = [b4,b6]
holds (b3 = b4 implies it .(b1,b2) = dist(b5,b6)) &
(b3 = b4 or it .(b1,b2) = (dist(b5,a2)) + dist(a2,b6));
end;
:: COMPL_SP:def 10
theorem
for b1 being MetrStruct
for b2 being Element of the carrier of b1
for b3 being set
for b4 being Function-like quasi_total Relation of [:[:b3,(the carrier of b1) \ {b2}:] \/ {[b3,b2]},[:b3,(the carrier of b1) \ {b2}:] \/ {[b3,b2]}:],REAL holds
b4 = well_dist(b2,b3)
iff
for b5, b6 being Element of [:b3,(the carrier of b1) \ {b2}:] \/ {[b3,b2]}
for b7, b8 being set
for b9, b10 being Element of the carrier of b1
st b5 = [b7,b9] & b6 = [b8,b10]
holds (b7 = b8 implies b4 .(b5,b6) = dist(b9,b10)) &
(b7 = b8 or b4 .(b5,b6) = (dist(b9,b2)) + dist(b2,b10));
:: COMPL_SP:th 39
theorem
for b1 being MetrStruct
for b2 being Element of the carrier of b1
for b3 being non empty set holds
(well_dist(b2,b3) is Reflexive([:b3,(the carrier of b1) \ {b2}:] \/ {[b3,b2]}) implies b1 is Reflexive) &
(well_dist(b2,b3) is symmetric([:b3,(the carrier of b1) \ {b2}:] \/ {[b3,b2]}) implies b1 is symmetric) &
(well_dist(b2,b3) is triangle([:b3,(the carrier of b1) \ {b2}:] \/ {[b3,b2]}) &
well_dist(b2,b3) is Reflexive([:b3,(the carrier of b1) \ {b2}:] \/ {[b3,b2]}) implies b1 is triangle) &
(well_dist(b2,b3) is discerning([:b3,(the carrier of b1) \ {b2}:] \/ {[b3,b2]}) &
well_dist(b2,b3) is Reflexive([:b3,(the carrier of b1) \ {b2}:] \/ {[b3,b2]}) implies b1 is discerning);
:: COMPL_SP:funcnot 4 => COMPL_SP:func 4
definition
let a1 be MetrStruct;
let a2 be Element of the carrier of a1;
let a3 be set;
func WellSpace(A2,A3) -> strict MetrStruct equals
MetrStruct(#[:a3,(the carrier of a1) \ {a2}:] \/ {[a3,a2]},well_dist(a2,a3)#);
end;
:: COMPL_SP:def 11
theorem
for b1 being MetrStruct
for b2 being Element of the carrier of b1
for b3 being set holds
WellSpace(b2,b3) = MetrStruct(#[:b3,(the carrier of b1) \ {b2}:] \/ {[b3,b2]},well_dist(b2,b3)#);
:: COMPL_SP:funcreg 1
registration
let a1 be MetrStruct;
let a2 be Element of the carrier of a1;
let a3 be set;
cluster WellSpace(a2,a3) -> non empty strict;
end;
:: COMPL_SP:funcreg 2
registration
let a1 be Reflexive MetrStruct;
let a2 be Element of the carrier of a1;
let a3 be set;
cluster WellSpace(a2,a3) -> strict Reflexive;
end;
:: COMPL_SP:funcreg 3
registration
let a1 be symmetric MetrStruct;
let a2 be Element of the carrier of a1;
let a3 be set;
cluster WellSpace(a2,a3) -> strict symmetric;
end;
:: COMPL_SP:funcreg 4
registration
let a1 be Reflexive symmetric triangle MetrStruct;
let a2 be Element of the carrier of a1;
let a3 be set;
cluster WellSpace(a2,a3) -> strict triangle;
end;
:: COMPL_SP:funcreg 5
registration
let a1 be Reflexive discerning symmetric triangle MetrStruct;
let a2 be Element of the carrier of a1;
let a3 be set;
cluster WellSpace(a2,a3) -> strict discerning;
end;
:: COMPL_SP:th 40
theorem
for b1 being non empty Reflexive triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being non empty set
st WellSpace(b2,b3) is complete
holds b1 is complete;
:: COMPL_SP:th 41
theorem
for b1 being set
for b2 being non empty Reflexive symmetric triangle MetrStruct
for b3 being Element of the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of WellSpace(b3,b1)
st b4 is Cauchy(WellSpace(b3,b1)) &
(ex b5 being Element of the carrier of WellSpace(b3,b1) st
b5 = [b1,b3] &
(ex b6 being Element of REAL st
0 < b6 &
(for b7 being natural set holds
ex b8 being natural set st
b7 <= b8 & b6 <= dist(b4 . b8,b5))))
holds ex b5 being natural set st
ex b6 being set st
for b7 being natural set
st b5 <= b7
holds ex b8 being Element of the carrier of b2 st
b4 . b7 = [b6,b8];
:: COMPL_SP:th 42
theorem
for b1 being set
for b2 being non empty Reflexive symmetric triangle MetrStruct
for b3 being Element of the carrier of b2
st b2 is complete
holds WellSpace(b3,b1) is complete;
:: COMPL_SP:th 43
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
st b1 is complete
for b2 being Element of the carrier of b1
st ex b3 being Element of the carrier of b1 st
dist(b2,b3) <> 0
for b3 being infinite set holds
WellSpace(b2,b3) is complete &
(ex b4 being non-empty Function-like quasi_total bounded Relation of NAT,bool the carrier of WellSpace(b2,b3) st
b4 is closed(WellSpace(b2,b3)) & b4 is descending(the carrier of WellSpace(b2,b3)) & meet b4 is empty);
:: COMPL_SP:th 44
theorem
ex b1 being non empty Reflexive discerning symmetric triangle MetrStruct st
b1 is complete &
(ex b2 being non-empty Function-like quasi_total bounded Relation of NAT,bool the carrier of b1 st
b2 is closed(b1) & b2 is descending(the carrier of b1) & meet b2 is empty);