Article TMAP_1, MML version 4.99.1005
:: TMAP_1:funcreg 1
registration
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be non empty SubSpace of a1;
cluster a2 union a3 -> non empty strict TopSpace-like;
end;
:: TMAP_1:funcnot 1 => TMAP_1:func 1
definition
let a1, a2 be non empty set;
let a3, a4 be non empty Element of bool a1;
let a5 be Function-like quasi_total Relation of a3,a2;
let a6 be Function-like quasi_total Relation of a4,a2;
assume a5 | (a3 /\ a4) = a6 | (a3 /\ a4);
func A5 union A6 -> Function-like quasi_total Relation of a3 \/ a4,a2 means
it | a3 = a5 & it | a4 = a6;
end;
:: TMAP_1:def 1
theorem
for b1, b2 being non empty set
for b3, b4 being non empty Element of bool b1
for b5 being Function-like quasi_total Relation of b3,b2
for b6 being Function-like quasi_total Relation of b4,b2
st b5 | (b3 /\ b4) = b6 | (b3 /\ b4)
for b7 being Function-like quasi_total Relation of b3 \/ b4,b2 holds
b7 = b5 union b6
iff
b7 | b3 = b5 & b7 | b4 = b6;
:: TMAP_1:th 6
theorem
for b1, b2 being non empty set
for b3, b4 being non empty Element of bool b1
st b3 misses b4
for b5 being Function-like quasi_total Relation of b3,b2
for b6 being Function-like quasi_total Relation of b4,b2 holds
b5 | (b3 /\ b4) = b6 | (b3 /\ b4) &
(b5 union b6) | b3 = b5 &
(b5 union b6) | b4 = b6;
:: TMAP_1:th 7
theorem
for b1, b2 being non empty set
for b3, b4 being non empty Element of bool b1
for b5 being Function-like quasi_total Relation of b3 \/ b4,b2
for b6 being Function-like quasi_total Relation of b3,b2
for b7 being Function-like quasi_total Relation of b4,b2
st b5 | b3 = b6 & b5 | b4 = b7
holds b5 = b6 union b7;
:: TMAP_1:th 8
theorem
for b1, b2 being non empty set
for b3, b4 being non empty Element of bool b1
for b5 being Function-like quasi_total Relation of b3,b2
for b6 being Function-like quasi_total Relation of b4,b2
st b5 | (b3 /\ b4) = b6 | (b3 /\ b4)
holds b5 union b6 = b6 union b5;
:: TMAP_1:th 9
theorem
for b1, b2 being non empty set
for b3, b4, b5, b6, b7 being non empty Element of bool b1
st b6 = b3 \/ b4 & b7 = b4 \/ b5
for b8 being Function-like quasi_total Relation of b3,b2
for b9 being Function-like quasi_total Relation of b4,b2
for b10 being Function-like quasi_total Relation of b5,b2
st b8 | (b3 /\ b4) = b9 | (b3 /\ b4) &
b9 | (b4 /\ b5) = b10 | (b4 /\ b5) &
b8 | (b3 /\ b5) = b10 | (b3 /\ b5)
for b11 being Function-like quasi_total Relation of b6,b2
for b12 being Function-like quasi_total Relation of b7,b2
st b11 = b8 union b9 & b12 = b9 union b10
holds b11 union b10 = b8 union b12;
:: TMAP_1:th 10
theorem
for b1, b2 being non empty set
for b3, b4 being non empty Element of bool b1
for b5 being Function-like quasi_total Relation of b3,b2
for b6 being Function-like quasi_total Relation of b4,b2
st b5 | (b3 /\ b4) = b6 | (b3 /\ b4)
holds (b3 is Element of bool b4 implies b5 union b6 = b6) & (b5 union b6 = b6 implies b3 is Element of bool b4) & (b4 is Element of bool b3 implies b5 union b6 = b5) & (b5 union b6 = b5 implies b4 is Element of bool b3);
:: TMAP_1:th 11
theorem
for b1 being TopStruct
for b2 being SubSpace of b1 holds
TopStruct(#the carrier of b2,the topology of b2#) is strict SubSpace of b1;
:: TMAP_1:th 12
theorem
for b1 being TopStruct
for b2, b3 being TopSpace-like TopStruct
st b2 = TopStruct(#the carrier of b3,the topology of b3#)
holds b2 is SubSpace of b1
iff
b3 is SubSpace of b1;
:: TMAP_1:th 13
theorem
for b1, b2, b3 being TopSpace-like TopStruct
st b3 = TopStruct(#the carrier of b2,the topology of b2#)
holds b2 is closed SubSpace of b1
iff
b3 is closed SubSpace of b1;
:: TMAP_1:th 14
theorem
for b1, b2, b3 being TopSpace-like TopStruct
st b3 = TopStruct(#the carrier of b2,the topology of b2#)
holds b2 is open SubSpace of b1
iff
b3 is open SubSpace of b1;
:: TMAP_1:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b2 is SubSpace of b3
for b4 being Element of the carrier of b2 holds
ex b5 being Element of the carrier of b3 st
b5 = b4;
:: TMAP_1:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
for b4 being Element of the carrier of b2 union b3
st for b5 being Element of the carrier of b2 holds
b5 <> b4
holds ex b5 being Element of the carrier of b3 st
b5 = b4;
:: TMAP_1:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b2 meets b3
for b4 being Element of the carrier of b2 meet b3 holds
(ex b5 being Element of the carrier of b2 st
b5 = b4) &
(ex b5 being Element of the carrier of b3 st
b5 = b4);
:: TMAP_1:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
for b4 being Element of the carrier of b2 union b3
for b5 being Element of bool the carrier of b2
for b6 being Element of bool the carrier of b3
st b5 is closed(b2) & b4 in b5 & b6 is closed(b3) & b4 in b6
holds ex b7 being Element of bool the carrier of b2 union b3 st
b7 is closed(b2 union b3) & b4 in b7 & b7 c= b5 \/ b6;
:: TMAP_1:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
for b4 being Element of the carrier of b2 union b3
for b5 being Element of bool the carrier of b2
for b6 being Element of bool the carrier of b3
st b5 is open(b2) & b4 in b5 & b6 is open(b3) & b4 in b6
holds ex b7 being Element of bool the carrier of b2 union b3 st
b7 is open(b2 union b3) & b4 in b7 & b7 c= b5 \/ b6;
:: TMAP_1:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
for b4 being Element of the carrier of b2 union b3
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b3
st b5 = b4 & b6 = b4
for b7 being a_neighborhood of b5
for b8 being a_neighborhood of b6 holds
ex b9 being Element of bool the carrier of b2 union b3 st
b9 is open(b2 union b3) & b4 in b9 & b9 c= b7 \/ b8;
:: TMAP_1:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
for b4 being Element of the carrier of b2 union b3
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b3
st b5 = b4 & b6 = b4
for b7 being a_neighborhood of b5
for b8 being a_neighborhood of b6 holds
ex b9 being a_neighborhood of b4 st
b9 c= b7 \/ b8;
:: TMAP_1:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b2 is SubSpace of b3
holds b2 meets b3 & b3 meets b2;
:: TMAP_1:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
st b2 is SubSpace of b3 & (b2 misses b4 implies b4 meets b2)
holds b3 meets b4 & b4 meets b3;
:: TMAP_1:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
st b2 is SubSpace of b3 & (b3 misses b4 or b4 misses b3)
holds b2 misses b4 & b4 misses b2;
:: TMAP_1:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
b2 union b2 = TopStruct(#the carrier of b2,the topology of b2#);
:: TMAP_1:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
b2 meet b2 = TopStruct(#the carrier of b2,the topology of b2#);
:: TMAP_1:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
st b2 is SubSpace of b3 & b4 is SubSpace of b5
holds b2 union b4 is SubSpace of b3 union b5;
:: TMAP_1:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
st b2 meets b3 & b2 is SubSpace of b4 & b3 is SubSpace of b5
holds b2 meet b3 is SubSpace of b4 meet b5;
:: TMAP_1:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
st b2 is SubSpace of b3 & b4 is SubSpace of b3
holds b2 union b4 is SubSpace of b3;
:: TMAP_1:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
st b2 meets b3 & b2 is SubSpace of b4 & b3 is SubSpace of b4
holds b2 meet b3 is SubSpace of b4;
:: TMAP_1:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1 holds
((b2 misses b3 or b3 misses b2) & (b4 misses b3 implies b3 meets b4) implies (b2 union b4) meet b3 = b4 meet b3 & b3 meet (b2 union b4) = b3 meet b4) &
((b2 misses b3 implies b3 meets b2) & (b4 misses b3 or b3 misses b4) implies (b2 union b4) meet b3 = b2 meet b3 & b3 meet (b2 union b4) = b3 meet b2);
:: TMAP_1:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
st b2 meets b3
holds (b2 is SubSpace of b4 implies b2 meet b3 is SubSpace of b4 meet b3) &
(b3 is SubSpace of b4 implies b2 meet b3 is SubSpace of b2 meet b4);
:: TMAP_1:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
st b2 is SubSpace of b3 & (b3 misses b4 or b4 misses b3)
holds b3 meet (b2 union b4) = TopStruct(#the carrier of b2,the topology of b2#) &
b3 meet (b4 union b2) = TopStruct(#the carrier of b2,the topology of b2#);
:: TMAP_1:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
st b2 meets b3
holds (b2 is SubSpace of b4 implies b4 meet b3 meets b2 & b3 meet b4 meets b2) &
(b3 is SubSpace of b4 implies b2 meet b4 meets b3 & b4 meet b2 meets b3);
:: TMAP_1:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
st b2 is SubSpace of b3 & b4 is SubSpace of b5 & (b3 misses b5 or b3 meet b5 misses b2 union b4)
holds b3 misses b4 & b5 misses b2;
:: TMAP_1:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
st b2 is not SubSpace of b3 & b3 is not SubSpace of b2 & b2 union b3 is SubSpace of b4 union b5 & b4 meet (b2 union b3) is SubSpace of b2 & b5 meet (b2 union b3) is SubSpace of b3
holds b4 meets b2 union b3 & b5 meets b2 union b3;
:: TMAP_1:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5, b6 being non empty SubSpace of b1
st b2 meets b3 &
b2 is not SubSpace of b3 &
b3 is not SubSpace of b2 &
TopStruct(#the carrier of b1,the topology of b1#) = (b4 union b5) union b6 &
b4 meet (b2 union b3) is SubSpace of b2 &
b5 meet (b2 union b3) is SubSpace of b3 &
b6 meet (b2 union b3) is SubSpace of b2 meet b3
holds b4 meets b2 union b3 & b5 meets b2 union b3;
:: TMAP_1:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5, b6 being non empty SubSpace of b1
st b2 meets b3 &
b2 is not SubSpace of b3 &
b3 is not SubSpace of b2 &
b2 union b3 is not SubSpace of b4 union b5 &
TopStruct(#the carrier of b1,the topology of b1#) = (b4 union b5) union b6 &
b4 meet (b2 union b3) is SubSpace of b2 &
b5 meet (b2 union b3) is SubSpace of b3 &
b6 meet (b2 union b3) is SubSpace of b2 meet b3
holds b4 union b5 meets b2 union b3 & b6 meets b2 union b3;
:: TMAP_1:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1 holds
(b2 union b3 meets b4 & b2 misses b4 implies b3 meets b4) &
((b2 misses b4 implies b3 meets b4) implies b2 union b3 meets b4) &
(b4 meets b2 union b3 & b4 misses b2 implies b4 meets b3) &
((b4 misses b2 implies b4 meets b3) implies b4 meets b2 union b3);
:: TMAP_1:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1 holds
(b2 union b3 misses b4 implies b2 misses b4 & b3 misses b4) &
(b2 misses b4 & b3 misses b4 implies b2 union b3 misses b4) &
(b4 misses b2 union b3 implies b4 misses b2 & b4 misses b3) &
(b4 misses b2 & b4 misses b3 implies b4 misses b2 union b3);
:: TMAP_1:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
st b2 meets b3
holds (b2 meet b3 misses b4 or b2 meets b4 & b3 meets b4) &
(b4 misses b2 meet b3 or b4 meets b2 & b4 meets b3);
:: TMAP_1:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
st b2 meets b3
holds (b2 meets b4 & b3 meets b4 or b2 meet b3 misses b4) &
(b4 meets b2 & b4 meets b3 or b4 misses b2 meet b3);
:: TMAP_1:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty closed SubSpace of b1
st b3 meets b2
holds b3 meet b2 is closed SubSpace of b2;
:: TMAP_1:th 44
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty open SubSpace of b1
st b3 meets b2
holds b3 meet b2 is open SubSpace of b2;
:: TMAP_1:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
for b4 being non empty closed SubSpace of b1
st b2 is SubSpace of b4 & b4 misses b3
holds b2 is closed SubSpace of b2 union b3 & b2 is closed SubSpace of b3 union b2;
:: TMAP_1:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
for b4 being non empty open SubSpace of b1
st b2 is SubSpace of b4 & b4 misses b3
holds b2 is open SubSpace of b2 union b3 & b2 is open SubSpace of b3 union b2;
:: TMAP_1:prednot 1 => TMAP_1:pred 1
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be Element of the carrier of a1;
pred A3 is_continuous_at A4 means
for b1 being a_neighborhood of a3 . a4 holds
ex b2 being a_neighborhood of a4 st
a3 .: b2 c= b1;
end;
:: TMAP_1:dfs 2
definiens
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be Element of the carrier of a1;
To prove
a3 is_continuous_at a4
it is sufficient to prove
thus for b1 being a_neighborhood of a3 . a4 holds
ex b2 being a_neighborhood of a4 st
a3 .: b2 c= b1;
:: TMAP_1:def 2
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_at b4
iff
for b5 being a_neighborhood of b3 . b4 holds
ex b6 being a_neighborhood of b4 st
b3 .: b6 c= b5;
:: TMAP_1:prednot 2 => not TMAP_1:pred 1
notation
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be Element of the carrier of a1;
antonym a3 is_not_continuous_at a4 for a3 is_continuous_at a4;
end;
:: TMAP_1:th 47
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2 holds
b3 is_continuous_at b4
iff
for b5 being a_neighborhood of b3 . b4 holds
b3 " b5 is a_neighborhood of b4;
:: TMAP_1:th 48
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
b3 is_continuous_at b4
iff
for b5 being Element of bool the carrier of b2
st b5 is open(b2) & b3 . b4 in b5
holds ex b6 being Element of bool the carrier of b1 st
b6 is open(b1) & b4 in b6 & b3 .: b6 c= b5;
:: TMAP_1:th 49
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 holds
b3 is continuous(b2, b1)
iff
for b4 being Element of the carrier of b2 holds
b3 is_continuous_at b4;
:: TMAP_1:th 50
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
st the carrier of b2 = the carrier of b3 & the topology of b3 c= the topology of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
st b4 = b5
for b6 being Element of the carrier of b1
st b4 is_continuous_at b6
holds b5 is_continuous_at b6;
:: TMAP_1:th 51
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
st the carrier of b1 = the carrier of b2 & the topology of b2 c= the topology of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
st b4 = b5
for b6 being Element of the carrier of b1
for b7 being Element of the carrier of b2
st b6 = b7 & b5 is_continuous_at b7
holds b4 is_continuous_at b6;
:: TMAP_1:th 52
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
for b6 being Element of the carrier of b2
for b7 being Element of the carrier of b3
st b7 = b4 . b6 & b4 is_continuous_at b6 & b5 is_continuous_at b7
holds b5 * b4 is_continuous_at b6;
:: TMAP_1:th 53
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b6 being Element of the carrier of b2
st b4 is continuous(b3, b2) & b5 is_continuous_at b6
for b7 being Element of the carrier of b3
st b7 in b4 " {b6}
holds b5 * b4 is_continuous_at b7;
:: TMAP_1:th 54
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Element of the carrier of b3
st b4 is_continuous_at b6 & b5 is continuous(b1, b2)
holds b5 * b4 is_continuous_at b6;
:: TMAP_1:th 55
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
iff
for b4 being Element of the carrier of b1 holds
b3 is_continuous_at b4;
:: TMAP_1:th 56
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
st the carrier of b2 = the carrier of b3 & the topology of b3 c= the topology of b2
for b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
b4 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b3;
:: TMAP_1:th 57
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
st the carrier of b1 = the carrier of b2 & the topology of b2 c= the topology of b1
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3 holds
b4 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b3;
:: TMAP_1:funcnot 2 => TMAP_1:func 2
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be SubSpace of a1;
let a4 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
func A4 | A3 -> Function-like quasi_total Relation of the carrier of a3,the carrier of a2 equals
a4 | the carrier of a3;
end;
:: TMAP_1:def 3
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being SubSpace of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b4 | b3 = b4 | the carrier of b3;
:: TMAP_1:th 59
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b2
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
st for b6 being Element of the carrier of b2
st b6 in the carrier of b3
holds b4 . b6 = b5 . b6
holds b4 | b3 = b5;
:: TMAP_1:th 60
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b2
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
st TopStruct(#the carrier of b3,the topology of b3#) = TopStruct(#the carrier of b2,the topology of b2#)
holds b4 = b4 | b3;
:: TMAP_1:th 61
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b2
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Element of bool the carrier of b2
st b5 c= the carrier of b3
holds b4 .: b5 = (b4 | b3) .: b5;
:: TMAP_1:th 62
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Element of bool the carrier of b2
st b4 " b5 c= the carrier of b3
holds b4 " b5 = (b4 | b3) " b5;
:: TMAP_1:th 63
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b2
for b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1 holds
ex b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 st
b5 | b3 = b4;
:: TMAP_1:th 64
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b4 being non empty SubSpace of b2
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b4
st b5 = b6 & b3 is_continuous_at b5
holds b3 | b4 is_continuous_at b6;
:: TMAP_1:th 65
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b4 being non empty SubSpace of b2
for b5 being Element of bool the carrier of b2
for b6 being Element of the carrier of b2
for b7 being Element of the carrier of b4
st b5 c= the carrier of b4 & b5 is a_neighborhood of b6 & b6 = b7
holds b3 is_continuous_at b6
iff
b3 | b4 is_continuous_at b7;
:: TMAP_1:th 66
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b4 being non empty SubSpace of b2
for b5 being Element of bool the carrier of b2
for b6 being Element of the carrier of b2
for b7 being Element of the carrier of b4
st b5 is open(b2) & b6 in b5 & b5 c= the carrier of b4 & b6 = b7
holds b3 is_continuous_at b6
iff
b3 | b4 is_continuous_at b7;
:: TMAP_1:th 67
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b4 being non empty open SubSpace of b2
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b4
st b5 = b6
holds b3 is_continuous_at b5
iff
b3 | b4 is_continuous_at b6;
:: TMAP_1:th 68
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b4 being non empty SubSpace of b1 holds
b3 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2;
:: TMAP_1:th 69
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3 holds
(b6 * b5) | b4 = b6 * (b5 | b4);
:: TMAP_1:th 70
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b5 is continuous(b2, b3) & b6 | b4 is continuous(b4, b2)
holds (b5 * b6) | b4 is continuous(b4, b3);
:: TMAP_1:th 71
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
for b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b3
for b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b6 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2
holds (b5 * b6) | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b3;
:: TMAP_1:funcnot 3 => TMAP_1:func 3
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3, a4 be SubSpace of a1;
let a5 be Function-like quasi_total Relation of the carrier of a3,the carrier of a2;
assume a4 is SubSpace of a3;
func A5 | A4 -> Function-like quasi_total Relation of the carrier of a4,the carrier of a2 equals
a5 | the carrier of a4;
end;
:: TMAP_1:def 4
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
st b4 is SubSpace of b3
holds b5 | b4 = b5 | the carrier of b4;
:: TMAP_1:th 72
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st b3 is SubSpace of b4
for b6 being Element of the carrier of b4
st b6 in the carrier of b3
holds b5 . b6 = (b5 | b3) . b6;
:: TMAP_1:th 73
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st b3 is SubSpace of b4
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
st for b7 being Element of the carrier of b4
st b7 in the carrier of b3
holds b5 . b7 = b6 . b7
holds b5 | b3 = b6;
:: TMAP_1:th 74
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b1
for b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2 holds
b4 = b4 | b3;
:: TMAP_1:th 75
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st b3 is SubSpace of b4
for b6 being Element of bool the carrier of b4
st b6 c= the carrier of b3
holds b5 .: b6 = (b5 | b3) .: b6;
:: TMAP_1:th 76
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st b3 is SubSpace of b4
for b6 being Element of bool the carrier of b2
st b5 " b6 c= the carrier of b3
holds b5 " b6 = (b5 | b3) " b6;
:: TMAP_1:th 77
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
st b6 = b5 | b3 & b4 is SubSpace of b3
holds b6 | b4 = b5 | b4;
:: TMAP_1:th 78
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is SubSpace of b4
holds (b5 | b4) | b3 = b5 | b3;
:: TMAP_1:th 79
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4, b5 being non empty SubSpace of b1
st b4 is SubSpace of b3 & b5 is SubSpace of b4
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2 holds
(b6 | b4) | b5 = b6 | b5;
:: TMAP_1:th 80
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b7 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st b4 = b1 & b5 = b7
holds b7 | b3 = b6
iff
b5 | b3 = b6;
:: TMAP_1:th 81
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b6 being Element of the carrier of b3
for b7 being Element of the carrier of b4
st b6 = b7 & b4 is SubSpace of b3 & b5 is_continuous_at b6
holds b5 | b4 is_continuous_at b7;
:: TMAP_1:th 82
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is SubSpace of b4
for b6 being Element of the carrier of b4
for b7 being Element of the carrier of b3
st b6 = b7 & b5 | b4 is_continuous_at b6
holds b5 | b3 is_continuous_at b7;
:: TMAP_1:th 83
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st b3 is SubSpace of b4
for b6 being Element of bool the carrier of b4
for b7 being Element of the carrier of b4
for b8 being Element of the carrier of b3
st b6 c= the carrier of b3 & b6 is a_neighborhood of b7 & b7 = b8
holds b5 is_continuous_at b7
iff
b5 | b3 is_continuous_at b8;
:: TMAP_1:th 84
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st b3 is SubSpace of b4
for b6 being Element of bool the carrier of b4
for b7 being Element of the carrier of b4
for b8 being Element of the carrier of b3
st b6 is open(b4) & b7 in b6 & b6 c= the carrier of b3 & b7 = b8
holds b5 is_continuous_at b7
iff
b5 | b3 is_continuous_at b8;
:: TMAP_1:th 85
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b2
for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b1
st b3 is SubSpace of b4
for b6 being Element of bool the carrier of b2
for b7 being Element of the carrier of b4
for b8 being Element of the carrier of b3
st b6 is open(b2) & b7 in b6 & b6 c= the carrier of b3 & b7 = b8
holds b5 is_continuous_at b7
iff
b5 | b3 is_continuous_at b8;
:: TMAP_1:th 86
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st b3 is open SubSpace of b4
for b6 being Element of the carrier of b4
for b7 being Element of the carrier of b3
st b6 = b7
holds b5 is_continuous_at b6
iff
b5 | b3 is_continuous_at b7;
:: TMAP_1:th 87
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b2
for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b1
st b3 is open SubSpace of b2 & b3 is SubSpace of b4
for b6 being Element of the carrier of b4
for b7 being Element of the carrier of b3
st b6 = b7
holds b5 is_continuous_at b6
iff
b5 | b3 is_continuous_at b7;
:: TMAP_1:th 88
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st TopStruct(#the carrier of b3,the topology of b3#) = b4
for b6 being Element of the carrier of b4
for b7 being Element of the carrier of b3
st b6 = b7 & b5 | b3 is_continuous_at b7
holds b5 is_continuous_at b6;
:: TMAP_1:th 89
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
st b4 is SubSpace of b3
holds b5 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2;
:: TMAP_1:th 90
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4, b5 being non empty SubSpace of b1
st b3 is SubSpace of b4 & b5 is SubSpace of b3
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st b6 | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
holds b6 | b5 is Function-like quasi_total continuous Relation of the carrier of b5,the carrier of b2;
:: TMAP_1:th 91
theorem
for b1 being non empty 1-sorted
for b2 being Element of the carrier of b1 holds
(id b1) . b2 = b2;
:: TMAP_1:th 92
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st for b3 being Element of the carrier of b1 holds
b2 . b3 = b3
holds b2 = id b1;
:: TMAP_1:th 93
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 * id b1 = b3 & (id b2) * b3 = b3;
:: TMAP_1:th 94
theorem
for b1 being non empty TopSpace-like TopStruct holds
id b1 is continuous(b1, b1);
:: TMAP_1:funcnot 4 => TMAP_1:func 4
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
func incl A2 -> Function-like quasi_total Relation of the carrier of a2,the carrier of a1 equals
(id a1) | a2;
end;
:: TMAP_1:def 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1 holds
incl b2 = (id b1) | b2;
:: TMAP_1:funcnot 5 => TMAP_1:func 4
notation
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
synonym a2 incl a1 for incl a2;
end;
:: TMAP_1:th 95
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being Element of the carrier of b1
st b3 in the carrier of b2
holds (incl b2) . b3 = b3;
:: TMAP_1:th 96
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
st for b4 being Element of the carrier of b1
st b4 in the carrier of b2
holds b4 = b3 . b4
holds incl b2 = b3;
:: TMAP_1:th 97
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b4 | b3 = b4 * incl b3;
:: TMAP_1:th 98
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
incl b2 is Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b1;
:: TMAP_1:funcnot 6 => TMAP_1:func 5
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
func A2 -extension_of_the_topology_of A1 -> Element of bool bool the carrier of a1 equals
{b1 \/ (b2 /\ a2) where b1 is Element of bool the carrier of a1, b2 is Element of bool the carrier of a1: b1 in the topology of a1 & b2 in the topology of a1};
end;
:: TMAP_1:def 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 -extension_of_the_topology_of b1 = {b3 \/ (b4 /\ b2) where b3 is Element of bool the carrier of b1, b4 is Element of bool the carrier of b1: b3 in the topology of b1 & b4 in the topology of b1};
:: TMAP_1:th 99
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
the topology of b1 c= b2 -extension_of_the_topology_of b1;
:: TMAP_1:th 100
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
{b3 /\ b2 where b3 is Element of bool the carrier of b1: b3 in the topology of b1} c= b2 -extension_of_the_topology_of b1;
:: TMAP_1:th 101
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
st b3 in the topology of b1 &
b4 in {b5 /\ b2 where b5 is Element of bool the carrier of b1: b5 in the topology of b1}
holds b3 \/ b4 in b2 -extension_of_the_topology_of b1 & b3 /\ b4 in b2 -extension_of_the_topology_of b1;
:: TMAP_1:th 102
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 in b2 -extension_of_the_topology_of b1;
:: TMAP_1:th 103
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 in the topology of b1
iff
the topology of b1 = b2 -extension_of_the_topology_of b1;
:: TMAP_1:funcnot 7 => TMAP_1:func 6
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
func A1 modified_with_respect_to A2 -> strict TopSpace-like TopStruct equals
TopStruct(#the carrier of a1,a2 -extension_of_the_topology_of a1#);
end;
:: TMAP_1:def 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b1 modified_with_respect_to b2 = TopStruct(#the carrier of b1,b2 -extension_of_the_topology_of b1#);
:: TMAP_1:funcreg 2
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
cluster a1 modified_with_respect_to a2 -> non empty strict TopSpace-like;
end;
:: TMAP_1:th 104
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
the carrier of b1 modified_with_respect_to b2 = the carrier of b1 & the topology of b1 modified_with_respect_to b2 = b2 -extension_of_the_topology_of b1;
:: TMAP_1:th 105
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 modified_with_respect_to b2
st b3 = b2
holds b3 is open(b1 modified_with_respect_to b2);
:: TMAP_1:th 106
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open(b1)
iff
TopStruct(#the carrier of b1,the topology of b1#) = b1 modified_with_respect_to b2;
:: TMAP_1:funcnot 8 => TMAP_1:func 7
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
func modid(A1,A2) -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 modified_with_respect_to a2 equals
id the carrier of a1;
end;
:: TMAP_1:def 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
modid(b1,b2) = id the carrier of b1;
:: TMAP_1:th 108
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st not b3 in b2
holds modid(b1,b2) is_continuous_at b3;
:: TMAP_1:th 109
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty SubSpace of b1
st the carrier of b3 misses b2
for b4 being Element of the carrier of b3 holds
(modid(b1,b2)) | b3 is_continuous_at b4;
:: TMAP_1:th 110
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty SubSpace of b1
st the carrier of b3 = b2
for b4 being Element of the carrier of b3 holds
(modid(b1,b2)) | b3 is_continuous_at b4;
:: TMAP_1:th 111
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty SubSpace of b1
st the carrier of b3 misses b2
holds (modid(b1,b2)) | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b1 modified_with_respect_to b2;
:: TMAP_1:th 112
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty SubSpace of b1
st the carrier of b3 = b2
holds (modid(b1,b2)) | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b1 modified_with_respect_to b2;
:: TMAP_1:th 113
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is open(b1)
iff
modid(b1,b2) is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b1 modified_with_respect_to b2;
:: TMAP_1:funcnot 9 => TMAP_1:func 8
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
func A1 modified_with_respect_to A2 -> strict TopSpace-like TopStruct means
for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds it = a1 modified_with_respect_to b1;
end;
:: TMAP_1:def 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being strict TopSpace-like TopStruct holds
b3 = b1 modified_with_respect_to b2
iff
for b4 being Element of bool the carrier of b1
st b4 = the carrier of b2
holds b3 = b1 modified_with_respect_to b4;
:: TMAP_1:funcreg 3
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
cluster a1 modified_with_respect_to a2 -> non empty strict TopSpace-like;
end;
:: TMAP_1:th 114
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
the carrier of b1 modified_with_respect_to b2 = the carrier of b1 &
(for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds the topology of b1 modified_with_respect_to b2 = b3 -extension_of_the_topology_of b1);
:: TMAP_1:th 115
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being SubSpace of b1 modified_with_respect_to b2
st the carrier of b3 = the carrier of b2
holds b3 is open SubSpace of b1 modified_with_respect_to b2;
:: TMAP_1:th 116
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
b2 is open SubSpace of b1
iff
TopStruct(#the carrier of b1,the topology of b1#) = b1 modified_with_respect_to b2;
:: TMAP_1:funcnot 10 => TMAP_1:func 9
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
func modid(A1,A2) -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 modified_with_respect_to a2 means
for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds it = modid(a1,b1);
end;
:: TMAP_1:def 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 modified_with_respect_to b2 holds
b3 = modid(b1,b2)
iff
for b4 being Element of bool the carrier of b1
st b4 = the carrier of b2
holds b3 = modid(b1,b4);
:: TMAP_1:th 117
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
modid(b1,b2) = id b1;
:: TMAP_1:th 118
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b2 misses b3
for b4 being Element of the carrier of b3 holds
(modid(b1,b2)) | b3 is_continuous_at b4;
:: TMAP_1:th 119
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being Element of the carrier of b2 holds
(modid(b1,b2)) | b2 is_continuous_at b3;
:: TMAP_1:th 120
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b2 misses b3
holds (modid(b1,b2)) | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b1 modified_with_respect_to b2;
:: TMAP_1:th 121
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
(modid(b1,b2)) | b2 is Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b1 modified_with_respect_to b2;
:: TMAP_1:th 122
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1 holds
b2 is open SubSpace of b1
iff
modid(b1,b2) is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b1 modified_with_respect_to b2;
:: TMAP_1:th 123
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3 union b4,the carrier of b2
for b6 being Element of the carrier of b3
for b7 being Element of the carrier of b4
for b8 being Element of the carrier of b3 union b4
st b8 = b6 & b8 = b7
holds b5 is_continuous_at b8
iff
b5 | b3 is_continuous_at b6 & b5 | b4 is_continuous_at b7;
:: TMAP_1:th 124
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4, b5 being non empty SubSpace of b1
for b6 being Element of the carrier of b4 union b5
for b7 being Element of the carrier of b4
for b8 being Element of the carrier of b5
st b6 = b7 & b6 = b8
holds b3 | (b4 union b5) is_continuous_at b6
iff
b3 | b4 is_continuous_at b7 & b3 | b5 is_continuous_at b8;
:: TMAP_1:th 125
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4, b5 being non empty SubSpace of b1
st b1 = b4 union b5
for b6 being Element of the carrier of b1
for b7 being Element of the carrier of b4
for b8 being Element of the carrier of b5
st b6 = b7 & b6 = b8
holds b3 is_continuous_at b6
iff
b3 | b4 is_continuous_at b7 & b3 | b5 is_continuous_at b8;
:: TMAP_1:th 126
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
st b3,b4 are_weakly_separated
for b5 being Function-like quasi_total Relation of the carrier of b3 union b4,the carrier of b2 holds
b5 is Function-like quasi_total continuous Relation of the carrier of b3 union b4,the carrier of b2
iff
b5 | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2 &
b5 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2;
:: TMAP_1:th 127
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty closed SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3 union b4,the carrier of b2 holds
b5 is Function-like quasi_total continuous Relation of the carrier of b3 union b4,the carrier of b2
iff
b5 | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2 &
b5 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2;
:: TMAP_1:th 128
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty open SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3 union b4,the carrier of b2 holds
b5 is Function-like quasi_total continuous Relation of the carrier of b3 union b4,the carrier of b2
iff
b5 | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2 &
b5 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2;
:: TMAP_1:th 129
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
st b3,b4 are_weakly_separated
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b5 | (b3 union b4) is Function-like quasi_total continuous Relation of the carrier of b3 union b4,the carrier of b2
iff
b5 | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2 &
b5 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2;
:: TMAP_1:th 130
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4, b5 being non empty closed SubSpace of b1 holds
b3 | (b4 union b5) is Function-like quasi_total continuous Relation of the carrier of b4 union b5,the carrier of b2
iff
b3 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2 &
b3 | b5 is Function-like quasi_total continuous Relation of the carrier of b5,the carrier of b2;
:: TMAP_1:th 131
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4, b5 being non empty open SubSpace of b1 holds
b3 | (b4 union b5) is Function-like quasi_total continuous Relation of the carrier of b4 union b5,the carrier of b2
iff
b3 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2 &
b3 | b5 is Function-like quasi_total continuous Relation of the carrier of b5,the carrier of b2;
:: TMAP_1:th 132
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4, b5 being non empty SubSpace of b1
st b1 = b4 union b5 & b4,b5 are_weakly_separated
holds b3 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
iff
b3 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2 &
b3 | b5 is Function-like quasi_total continuous Relation of the carrier of b5,the carrier of b2;
:: TMAP_1:th 133
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4, b5 being non empty closed SubSpace of b1
st b1 = b4 union b5
holds b3 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
iff
b3 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2 &
b3 | b5 is Function-like quasi_total continuous Relation of the carrier of b5,the carrier of b2;
:: TMAP_1:th 134
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4, b5 being non empty open SubSpace of b1
st b1 = b4 union b5
holds b3 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
iff
b3 | b4 is Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2 &
b3 | b5 is Function-like quasi_total continuous Relation of the carrier of b5,the carrier of b2;
:: TMAP_1:th 135
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
b2,b3 are_separated
iff
b2 misses b3 &
(for b4 being non empty TopSpace-like TopStruct
for b5 being Function-like quasi_total Relation of the carrier of b2 union b3,the carrier of b4
st b5 | b2 is Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b4 &
b5 | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b4
holds b5 is Function-like quasi_total continuous Relation of the carrier of b2 union b3,the carrier of b4);
:: TMAP_1:th 136
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
b2,b3 are_separated
iff
b2 misses b3 &
(for b4 being non empty TopSpace-like TopStruct
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b4
st b5 | b2 is Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b4 &
b5 | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b4
holds b5 | (b2 union b3) is Function-like quasi_total continuous Relation of the carrier of b2 union b3,the carrier of b4);
:: TMAP_1:th 137
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b1 = b2 union b3
holds b2,b3 are_separated
iff
b2 misses b3 &
(for b4 being non empty TopSpace-like TopStruct
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b4
st b5 | b2 is Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b4 &
b5 | b3 is Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b4
holds b5 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b4);
:: TMAP_1:funcnot 11 => TMAP_1:func 10
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3, a4 be non empty SubSpace of a1;
let a5 be Function-like quasi_total Relation of the carrier of a3,the carrier of a2;
let a6 be Function-like quasi_total Relation of the carrier of a4,the carrier of a2;
assume (a3 misses a4 or a5 | (a3 meet a4) = a6 | (a3 meet a4));
func A5 union A6 -> Function-like quasi_total Relation of the carrier of a3 union a4,the carrier of a2 means
it | a3 = a5 & it | a4 = a6;
end;
:: TMAP_1:def 12
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st (b3 misses b4 or b5 | (b3 meet b4) = b6 | (b3 meet b4))
for b7 being Function-like quasi_total Relation of the carrier of b3 union b4,the carrier of b2 holds
b7 = b5 union b6
iff
b7 | b3 = b5 & b7 | b4 = b6;
:: TMAP_1:th 138
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3 union b4,the carrier of b2 holds
b5 = (b5 | b3) union (b5 | b4);
:: TMAP_1:th 139
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
st b1 = b3 union b4
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b5 = (b5 | b3) union (b5 | b4);
:: TMAP_1:th 140
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
st b3 meets b4
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2 holds
(b5 union b6) | b3 = b5 & (b5 union b6) | b4 = b6
iff
b5 | (b3 meet b4) = b6 | (b3 meet b4);
:: TMAP_1:th 141
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st b5 | (b3 meet b4) = b6 | (b3 meet b4)
holds (b3 is SubSpace of b4 implies b5 union b6 = b6) & (b5 union b6 = b6 implies b3 is SubSpace of b4) & (b4 is SubSpace of b3 implies b5 union b6 = b5) & (b5 union b6 = b5 implies b4 is SubSpace of b3);
:: TMAP_1:th 142
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st (b3 misses b4 or b5 | (b3 meet b4) = b6 | (b3 meet b4))
holds b5 union b6 = b6 union b5;
:: TMAP_1:th 143
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4, b5 being non empty SubSpace of b1
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b7 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
for b8 being Function-like quasi_total Relation of the carrier of b5,the carrier of b2
st (b3 misses b4 or b6 | (b3 meet b4) = b7 | (b3 meet b4)) &
(b3 misses b5 or b6 | (b3 meet b5) = b8 | (b3 meet b5)) &
(b4 misses b5 or b7 | (b4 meet b5) = b8 | (b4 meet b5))
holds (b6 union b7) union b8 = b6 union (b7 union b8);
:: TMAP_1:th 144
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
st b3 meets b4
for b5 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2
st b5 | (b3 meet b4) = b6 | (b3 meet b4) &
b3,b4 are_weakly_separated
holds b5 union b6 is Function-like quasi_total continuous Relation of the carrier of b3 union b4,the carrier of b2;
:: TMAP_1:th 145
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
st b3 misses b4
for b5 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2
st b3,b4 are_weakly_separated
holds b5 union b6 is Function-like quasi_total continuous Relation of the carrier of b3 union b4,the carrier of b2;
:: TMAP_1:th 146
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty closed SubSpace of b1
st b3 meets b4
for b5 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2
st b5 | (b3 meet b4) = b6 | (b3 meet b4)
holds b5 union b6 is Function-like quasi_total continuous Relation of the carrier of b3 union b4,the carrier of b2;
:: TMAP_1:th 147
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty open SubSpace of b1
st b3 meets b4
for b5 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2
st b5 | (b3 meet b4) = b6 | (b3 meet b4)
holds b5 union b6 is Function-like quasi_total continuous Relation of the carrier of b3 union b4,the carrier of b2;
:: TMAP_1:th 148
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty closed SubSpace of b1
st b3 misses b4
for b5 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2 holds
b5 union b6 is Function-like quasi_total continuous Relation of the carrier of b3 union b4,the carrier of b2;
:: TMAP_1:th 149
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty open SubSpace of b1
st b3 misses b4
for b5 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2 holds
b5 union b6 is Function-like quasi_total continuous Relation of the carrier of b3 union b4,the carrier of b2;
:: TMAP_1:th 150
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
b2,b3 are_separated
iff
b2 misses b3 &
(for b4 being non empty TopSpace-like TopStruct
for b5 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b4
for b6 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b4 holds
b5 union b6 is Function-like quasi_total continuous Relation of the carrier of b2 union b3,the carrier of b4);
:: TMAP_1:th 151
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
st b1 = b3 union b4
for b5 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2
st (b5 union b6) | b3 = b5 & (b5 union b6) | b4 = b6 & b3,b4 are_weakly_separated
holds b5 union b6 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;
:: TMAP_1:th 152
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty closed SubSpace of b1
for b5 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2
st b1 = b3 union b4 & (b5 union b6) | b3 = b5 & (b5 union b6) | b4 = b6
holds b5 union b6 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;
:: TMAP_1:th 153
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty open SubSpace of b1
for b5 being Function-like quasi_total continuous Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total continuous Relation of the carrier of b4,the carrier of b2
st b1 = b3 union b4 & (b5 union b6) | b3 = b5 & (b5 union b6) | b4 = b6
holds b5 union b6 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;