Article JCT_MISC, MML version 4.99.1005
:: JCT_MISC:sch 1
scheme JCT_MISC:sch 1
{F1 -> non empty set,
F2 -> set}:
{F2(b1) where b1 is Element of F1(): TRUE} is not empty
:: JCT_MISC:th 5
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of bool b2
st (b2 = {} implies b1 = {})
holds (b3 " b4) ` = b3 " (b4 `);
:: JCT_MISC:th 6
theorem
for b1 being 1-sorted
for b2 being non empty set
for b3 being Function-like quasi_total Relation of the carrier of b1,b2
for b4 being Element of bool b2 holds
(b3 " b4) ` = b3 " (b4 `);
:: JCT_MISC:th 7
theorem
for b1, b2 being Element of NAT
st b1 <= b2
holds b2 -' (b2 -' b1) = b1;
:: JCT_MISC:th 9
theorem
for b1, b2, b3, b4 being real set
st b1 in [.b3,b4.] & b2 in [.b3,b4.]
holds (b1 + b2) / 2 in [.b3,b4.];
:: JCT_MISC:th 11
theorem
for b1, b2, b3, b4 being real set holds
abs ((abs (b1 - b2)) - abs (b3 - b4)) <= (abs (b1 - b3)) + abs (b2 - b4);
:: JCT_MISC:th 12
theorem
for b1, b2, b3 being real set
st b1 in ].b2,b3.[
holds abs b1 < max(abs b2,abs b3);
:: JCT_MISC:sch 2
scheme JCT_MISC:sch 2
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set}:
ex b1 being Function-like quasi_total Relation of F1(),F2() st
ex b2 being Function-like quasi_total Relation of F1(),F3() st
for b3 being Element of F1() holds
P1[b3, b1 . b3, b2 . b3]
provided
for b1 being Element of F1() holds
ex b2 being Element of F2() st
ex b3 being Element of F3() st
P1[b1, b2, b3];
:: JCT_MISC:th 13
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:]
st for b4 being Element of the carrier of [:b1,b2:]
st b4 in b3
holds ex b5 being Element of bool the carrier of b1 st
ex b6 being Element of bool the carrier of b2 st
b5 is open(b1) & b6 is open(b2) & b4 in [:b5,b6:] & [:b5,b6:] c= b3
holds b3 is open([:b1,b2:]);
:: JCT_MISC:th 14
theorem
for b1, b2 being compact Element of bool REAL holds
b1 /\ b2 is compact;
:: JCT_MISC:attrnot 1 => JCT_MISC:attr 1
definition
let a1 be Element of bool REAL;
attr a1 is connected means
for b1, b2 being real set
st b1 in a1 & b2 in a1
holds [.b1,b2.] c= a1;
end;
:: JCT_MISC:dfs 1
definiens
let a1 be Element of bool REAL;
To prove
a1 is connected
it is sufficient to prove
thus for b1, b2 being real set
st b1 in a1 & b2 in a1
holds [.b1,b2.] c= a1;
:: JCT_MISC:def 1
theorem
for b1 being Element of bool REAL holds
b1 is connected
iff
for b2, b3 being real set
st b2 in b1 & b3 in b1
holds [.b2,b3.] c= b1;
:: JCT_MISC:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total continuous Relation of the carrier of b1,REAL
for b3 being Element of bool the carrier of b1
st b3 is connected(b1)
holds b2 .: b3 is connected;
:: JCT_MISC:funcnot 1 => JCT_MISC:func 1
definition
let a1, a2 be Element of bool REAL;
func dist(A1,A2) -> real set means
ex b1 being Element of bool REAL st
b1 = {abs (b2 - b3) where b2 is Element of REAL, b3 is Element of REAL: b2 in a1 & b3 in a2} &
it = lower_bound b1;
commutativity;
:: for a1, a2 being Element of bool REAL holds
:: dist(a1,a2) = dist(a2,a1);
end;
:: JCT_MISC:def 2
theorem
for b1, b2 being Element of bool REAL
for b3 being real set holds
b3 = dist(b1,b2)
iff
ex b4 being Element of bool REAL st
b4 = {abs (b5 - b6) where b5 is Element of REAL, b6 is Element of REAL: b5 in b1 & b6 in b2} &
b3 = lower_bound b4;
:: JCT_MISC:th 16
theorem
for b1, b2 being Element of bool REAL
for b3, b4 being real set
st b3 in b1 & b4 in b2
holds dist(b1,b2) <= abs (b3 - b4);
:: JCT_MISC:th 17
theorem
for b1, b2 being Element of bool REAL
for b3, b4 being non empty Element of bool REAL
st b3 c= b1 & b4 c= b2
holds dist(b1,b2) <= dist(b3,b4);
:: JCT_MISC:th 18
theorem
for b1, b2 being non empty compact Element of bool REAL holds
ex b3, b4 being real set st
b3 in b1 & b4 in b2 & dist(b1,b2) = abs (b3 - b4);
:: JCT_MISC:th 19
theorem
for b1, b2 being non empty compact Element of bool REAL holds
0 <= dist(b1,b2);
:: JCT_MISC:th 20
theorem
for b1, b2 being non empty compact Element of bool REAL
st b1 misses b2
holds 0 < dist(b1,b2);
:: JCT_MISC:th 21
theorem
for b1, b2 being real set
for b3, b4 being compact Element of bool REAL
st b3 misses b4 & b3 c= [.b1,b2.] & b4 c= [.b1,b2.]
for b5 being Function-like quasi_total Relation of NAT,bool REAL
st for b6 being Element of NAT holds
b5 . b6 is connected & b5 . b6 meets b3 & b5 . b6 meets b4
holds ex b6 being real set st
b6 in [.b1,b2.] &
not b6 in b3 \/ b4 &
(for b7 being Element of NAT holds
ex b8 being Element of NAT st
b7 <= b8 & b6 in b5 . b8);
:: JCT_MISC:th 22
theorem
for b1 being closed Element of bool the carrier of TOP-REAL 2
st b1 is Bounded(2)
holds proj2 .: b1 is closed;
:: JCT_MISC:th 23
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st b1 is Bounded(2)
holds proj2 .: b1 is bounded;
:: JCT_MISC:th 24
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2 holds
proj2 .: b1 is compact;