Article RCOMP_1, MML version 4.99.1005
:: RCOMP_1:sch 1
scheme RCOMP_1:sch 1
ex b1 being Function-like quasi_total Relation of NAT,REAL st
for b2 being Element of NAT holds
P1[b2, b1 . b2]
provided
for b1 being Element of NAT holds
ex b2 being real set st
P1[b1, b2];
:: RCOMP_1:th 1
theorem
for b1, b2 being Element of bool REAL
st for b3 being real set
st b3 in b1
holds b3 in b2
holds b1 c= b2;
:: RCOMP_1:th 3
theorem
for b1, b2 being Element of bool REAL
st b1 c= b2 & b2 is bounded_below
holds b1 is bounded_below;
:: RCOMP_1:th 4
theorem
for b1, b2 being Element of bool REAL
st b1 c= b2 & b2 is bounded_above
holds b1 is bounded_above;
:: RCOMP_1:th 5
theorem
for b1, b2 being Element of bool REAL
st b1 c= b2 & b2 is bounded
holds b1 is bounded;
:: RCOMP_1:funcnot 1 => RCOMP_1:func 1
definition
let a1, a2 be real set;
redefine func [.A1,A2.] -> Element of bool REAL equals
{b1 where b1 is Element of REAL: a1 <= b1 & b1 <= a2};
end;
:: RCOMP_1:def 1
theorem
for b1, b2 being real set holds
[.b1,b2.] = {b3 where b3 is Element of REAL: b1 <= b3 & b3 <= b2};
:: RCOMP_1:funcnot 2 => RCOMP_1:func 2
definition
let a1, a2 be ext-real set;
redefine func ].A1,A2.[ -> Element of bool REAL equals
{b1 where b1 is Element of REAL: a1 < b1 & b1 < a2};
end;
:: RCOMP_1:def 2
theorem
for b1, b2 being ext-real set holds
].b1,b2.[ = {b3 where b3 is Element of REAL: b1 < b3 & b3 < b2};
:: RCOMP_1:th 8
theorem
for b1, b2, b3 being real set holds
b1 in ].b2 - b3,b2 + b3.[
iff
abs (b1 - b2) < b3;
:: RCOMP_1:th 9
theorem
for b1, b2, b3 being real set holds
b1 in [.b2,b3.]
iff
abs ((b2 + b3) - (2 * b1)) <= b3 - b2;
:: RCOMP_1:th 10
theorem
for b1, b2, b3 being real set holds
b1 in ].b2,b3.[
iff
abs ((b2 + b3) - (2 * b1)) < b3 - b2;
:: RCOMP_1:th 11
theorem
for b1, b2 being real set
st b1 <= b2
holds [.b1,b2.] = ].b1,b2.[ \/ {b1,b2};
:: RCOMP_1:th 12
theorem
for b1, b2 being real set
st b1 <= b2
holds ].b2,b1.[ = {};
:: RCOMP_1:th 13
theorem
for b1, b2 being real set
st b1 < b2
holds [.b2,b1.] = {};
:: RCOMP_1:th 14
theorem
for b1 being real set holds
[.b1,b1.] = {b1};
:: RCOMP_1:th 15
theorem
for b1, b2 being real set holds
(b1 < b2 implies ].b1,b2.[ <> {}) &
(b1 <= b2 implies b1 in [.b1,b2.] & b2 in [.b1,b2.]) &
].b1,b2.[ c= [.b1,b2.];
:: RCOMP_1:th 16
theorem
for b1, b2, b3, b4 being real set
st b1 in [.b2,b3.] & b4 in [.b2,b3.]
holds [.b1,b4.] c= [.b2,b3.];
:: RCOMP_1:th 17
theorem
for b1, b2, b3, b4 being real set
st b1 in ].b2,b3.[ & b4 in ].b2,b3.[
holds [.b1,b4.] c= ].b2,b3.[;
:: RCOMP_1:th 18
theorem
for b1, b2 being real set
st b1 <= b2
holds [.b1,b2.] = [.b1,b2.] \/ [.b2,b1.];
:: RCOMP_1:attrnot 1 => RCOMP_1:attr 1
definition
let a1 be Element of bool REAL;
attr a1 is compact means
for b1 being Function-like quasi_total Relation of NAT,REAL
st proj2 b1 c= a1
holds ex b2 being Function-like quasi_total Relation of NAT,REAL st
b2 is subsequence of b1 & b2 is convergent & lim b2 in a1;
end;
:: RCOMP_1:dfs 3
definiens
let a1 be Element of bool REAL;
To prove
a1 is compact
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of NAT,REAL
st proj2 b1 c= a1
holds ex b2 being Function-like quasi_total Relation of NAT,REAL st
b2 is subsequence of b1 & b2 is convergent & lim b2 in a1;
:: RCOMP_1:def 3
theorem
for b1 being Element of bool REAL holds
b1 is compact
iff
for b2 being Function-like quasi_total Relation of NAT,REAL
st proj2 b2 c= b1
holds ex b3 being Function-like quasi_total Relation of NAT,REAL st
b3 is subsequence of b2 & b3 is convergent & lim b3 in b1;
:: RCOMP_1:attrnot 2 => RCOMP_1:attr 2
definition
let a1 be Element of bool REAL;
attr a1 is closed means
for b1 being Function-like quasi_total Relation of NAT,REAL
st proj2 b1 c= a1 & b1 is convergent
holds lim b1 in a1;
end;
:: RCOMP_1:dfs 4
definiens
let a1 be Element of bool REAL;
To prove
a1 is closed
it is sufficient to prove
thus for b1 being Function-like quasi_total Relation of NAT,REAL
st proj2 b1 c= a1 & b1 is convergent
holds lim b1 in a1;
:: RCOMP_1:def 4
theorem
for b1 being Element of bool REAL holds
b1 is closed
iff
for b2 being Function-like quasi_total Relation of NAT,REAL
st proj2 b2 c= b1 & b2 is convergent
holds lim b2 in b1;
:: RCOMP_1:attrnot 3 => RCOMP_1:attr 3
definition
let a1 be Element of bool REAL;
attr a1 is open means
a1 ` is closed;
end;
:: RCOMP_1:dfs 5
definiens
let a1 be Element of bool REAL;
To prove
a1 is open
it is sufficient to prove
thus a1 ` is closed;
:: RCOMP_1:def 5
theorem
for b1 being Element of bool REAL holds
b1 is open
iff
b1 ` is closed;
:: RCOMP_1:th 22
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
st proj2 b3 c= [.b1,b2.]
holds b3 is bounded;
:: RCOMP_1:th 23
theorem
for b1, b2 being real set holds
[.b1,b2.] is closed;
:: RCOMP_1:th 24
theorem
for b1, b2 being real set holds
[.b1,b2.] is compact;
:: RCOMP_1:th 25
theorem
for b1, b2 being real set holds
].b1,b2.[ is open;
:: RCOMP_1:funcreg 1
registration
let a1, a2 be real set;
cluster ].a1,a2.[ -> open;
end;
:: RCOMP_1:funcreg 2
registration
let a1, a2 be real set;
cluster [.a1,a2.] -> closed;
end;
:: RCOMP_1:th 26
theorem
for b1 being Element of bool REAL
st b1 is compact
holds b1 is closed;
:: RCOMP_1:condreg 1
registration
cluster compact -> closed (Element of bool REAL);
end;
:: RCOMP_1:th 27
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of bool REAL
st for b3 being real set
st b3 in b2
holds ex b4 being real set st
ex b5 being Element of NAT st
0 < b4 &
(for b6 being Element of NAT
st b5 < b6
holds b4 < abs ((b1 . b6) - b3))
for b3 being Function-like quasi_total Relation of NAT,REAL
st b3 is subsequence of b1 & b3 is convergent
holds not lim b3 in b2;
:: RCOMP_1:th 28
theorem
for b1 being Element of bool REAL
st b1 is compact
holds b1 is bounded;
:: RCOMP_1:th 29
theorem
for b1 being Element of bool REAL
st b1 is bounded & b1 is closed
holds b1 is compact;
:: RCOMP_1:th 30
theorem
for b1 being Element of bool REAL
st b1 <> {} & b1 is closed & b1 is bounded_above
holds upper_bound b1 in b1;
:: RCOMP_1:th 31
theorem
for b1 being Element of bool REAL
st b1 <> {} & b1 is closed & b1 is bounded_below
holds lower_bound b1 in b1;
:: RCOMP_1:th 32
theorem
for b1 being Element of bool REAL
st b1 <> {} & b1 is compact
holds upper_bound b1 in b1 & lower_bound b1 in b1;
:: RCOMP_1:th 33
theorem
for b1 being Element of bool REAL
st b1 is compact &
(for b2, b3 being real set
st b2 in b1 & b3 in b1
holds [.b2,b3.] c= b1)
holds ex b2, b3 being real set st
b1 = [.b2,b3.];
:: RCOMP_1:exreg 1
registration
cluster complex-membered ext-real-membered real-membered open Element of bool REAL;
end;
:: RCOMP_1:modenot 1 => RCOMP_1:mode 1
definition
let a1 be real set;
mode Neighbourhood of A1 -> Element of bool REAL means
ex b1 being real set st
0 < b1 & it = ].a1 - b1,a1 + b1.[;
end;
:: RCOMP_1:dfs 6
definiens
let a1 be real set;
let a2 be Element of bool REAL;
To prove
a2 is Neighbourhood of a1
it is sufficient to prove
thus ex b1 being real set st
0 < b1 & a2 = ].a1 - b1,a1 + b1.[;
:: RCOMP_1:def 7
theorem
for b1 being real set
for b2 being Element of bool REAL holds
b2 is Neighbourhood of b1
iff
ex b3 being real set st
0 < b3 & b2 = ].b1 - b3,b1 + b3.[;
:: RCOMP_1:condreg 2
registration
let a1 be real set;
cluster -> open (Neighbourhood of a1);
end;
:: RCOMP_1:th 37
theorem
for b1 being real set
for b2 being Neighbourhood of b1 holds
b1 in b2;
:: RCOMP_1:th 38
theorem
for b1 being real set
for b2, b3 being Neighbourhood of b1 holds
ex b4 being Neighbourhood of b1 st
b4 c= b2 & b4 c= b3;
:: RCOMP_1:th 39
theorem
for b1 being open Element of bool REAL
for b2 being real set
st b2 in b1
holds ex b3 being Neighbourhood of b2 st
b3 c= b1;
:: RCOMP_1:th 40
theorem
for b1 being open Element of bool REAL
for b2 being real set
st b2 in b1
holds ex b3 being real set st
0 < b3 & ].b2 - b3,b2 + b3.[ c= b1;
:: RCOMP_1:th 41
theorem
for b1 being Element of bool REAL
st for b2 being real set
st b2 in b1
holds ex b3 being Neighbourhood of b2 st
b3 c= b1
holds b1 is open;
:: RCOMP_1:th 42
theorem
for b1 being Element of bool REAL holds
for b2 being real set
st b2 in b1
holds ex b3 being Neighbourhood of b2 st
b3 c= b1
iff
b1 is open;
:: RCOMP_1:th 43
theorem
for b1 being Element of bool REAL
st b1 is open & b1 is bounded_above
holds not upper_bound b1 in b1;
:: RCOMP_1:th 44
theorem
for b1 being Element of bool REAL
st b1 is open & b1 is bounded_below
holds not lower_bound b1 in b1;
:: RCOMP_1:th 45
theorem
for b1 being Element of bool REAL
st b1 is open &
b1 is bounded &
(for b2, b3 being real set
st b2 in b1 & b3 in b1
holds [.b2,b3.] c= b1)
holds ex b2, b3 being real set st
b1 = ].b2,b3.[;
:: RCOMP_1:th 46
theorem
for b1, b2 being real set holds
].b1,b2.[ misses {b1,b2};
:: RCOMP_1:th 47
theorem
for b1, b2, b3 being real set holds
b3 in ].b1,b2.[
iff
b1 < b3 & b3 < b2;
:: RCOMP_1:th 48
theorem
for b1, b2, b3 being real set holds
b3 in [.b1,b2.]
iff
b1 <= b3 & b3 <= b2;
:: RCOMP_1:th 49
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds [.b2,b3.] c= [.b1,b4.];
:: RCOMP_1:th 50
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4 & b2 <= b3
holds [.b1,b3.] \/ [.b2,b4.] = [.b1,b4.];
:: RCOMP_1:th 51
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4 & b2 <= b3
holds [.b1,b3.] /\ [.b2,b4.] = [.b2,b3.];
:: RCOMP_1:th 52
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds ].b2,b3.[ c= ].b1,b4.[;