Article FLANG_3, MML version 4.99.1005
:: FLANG_3:th 1
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
st b2 c= b3 *
holds b3 * ^^ b2 c= b3 * & b2 ^^ (b3 *) c= b3 *;
:: FLANG_3:funcnot 1 => FLANG_3:func 1
definition
let a1 be set;
let a2 be Element of bool (a1 ^omega);
let a3 be natural set;
func A2 |^.. A3 -> Element of bool (a1 ^omega) equals
union {b1 where b1 is Element of bool (a1 ^omega): ex b2 being natural set st
a3 <= b2 & b1 = a2 |^ b2};
end;
:: FLANG_3:def 1
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 |^.. b3 = union {b4 where b4 is Element of bool (b1 ^omega): ex b5 being natural set st
b3 <= b5 & b4 = b2 |^ b5};
:: FLANG_3:th 2
theorem
for b1, b2 being set
for b3 being Element of bool (b1 ^omega)
for b4 being natural set holds
b2 in b3 |^.. b4
iff
ex b5 being natural set st
b4 <= b5 & b2 in b3 |^ b5;
:: FLANG_3:th 3
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set
st b3 <= b4
holds b2 |^ b4 c= b2 |^.. b3;
:: FLANG_3:th 4
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 |^.. b3 = {}
iff
0 < b3 & b2 = {};
:: FLANG_3:th 5
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set
st b3 <= b4
holds b2 |^.. b4 c= b2 |^.. b3;
:: FLANG_3:th 6
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4, b5 being natural set
st b3 <= b4
holds b2 |^(b4,b5) c= b2 |^.. b3;
:: FLANG_3:th 7
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set
st b3 <= b4 + 1
holds (b2 |^(b3,b4)) \/ (b2 |^.. (b4 + 1)) = b2 |^.. b3;
:: FLANG_3:th 8
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
(b2 |^ b3) \/ (b2 |^.. (b3 + 1)) = b2 |^.. b3;
:: FLANG_3:th 9
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 |^.. b3 c= b2 *;
:: FLANG_3:th 10
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
<%> b1 in b2 |^.. b3
iff
(b3 = 0 or <%> b1 in b2);
:: FLANG_3:th 11
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 |^.. b3 = b2 *
iff
(<%> b1 in b2 or b3 = 0);
:: FLANG_3:th 12
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 * = (b2 |^(0,b3)) \/ (b2 |^.. (b3 + 1));
:: FLANG_3:th 13
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4 being natural set
st b2 c= b3
holds b2 |^.. b4 c= b3 |^.. b4;
:: FLANG_3:th 14
theorem
for b1, b2 being set
for b3 being Element of bool (b2 ^omega)
for b4 being natural set
st b1 in b3 & b1 <> <%> b2
holds b3 |^.. b4 <> {<%> b2};
:: FLANG_3:th 15
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 |^.. b3 = {<%> b1}
iff
(b2 = {<%> b1} or b3 = 0 & b2 = {});
:: FLANG_3:th 16
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 |^.. (b3 + 1) = (b2 |^.. b3) ^^ b2;
:: FLANG_3:th 17
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
(b2 |^.. b3) ^^ (b2 *) = b2 |^.. b3;
:: FLANG_3:th 18
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set holds
(b2 |^.. b3) ^^ (b2 |^.. b4) = b2 |^.. (b3 + b4);
:: FLANG_3:th 19
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set
st 0 < b3
holds (b2 |^.. b4) |^ b3 = b2 |^.. (b4 * b3);
:: FLANG_3:th 20
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
(b2 |^.. b3) * = (b2 |^.. b3) ?;
:: FLANG_3:th 21
theorem
for b1 being set
for b2, b3, b4 being Element of bool (b1 ^omega)
for b5, b6 being natural set
st b2 c= b3 |^.. b5 & b4 c= b3 |^.. b6
holds b2 ^^ b4 c= b3 |^.. (b5 + b6);
:: FLANG_3:th 22
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set holds
b2 |^.. (b3 + b4) = (b2 |^.. b3) ^^ (b2 |^ b4);
:: FLANG_3:th 23
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 ^^ (b2 |^.. b3) = (b2 |^.. b3) ^^ b2;
:: FLANG_3:th 24
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set holds
(b2 |^ b3) ^^ (b2 |^.. b4) = (b2 |^.. b4) ^^ (b2 |^ b3);
:: FLANG_3:th 25
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4, b5 being natural set holds
(b2 |^(b3,b4)) ^^ (b2 |^.. b5) = (b2 |^.. b5) ^^ (b2 |^(b3,b4));
:: FLANG_3:th 26
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4 being natural set
st <%> b1 in b2
holds b3 c= b3 ^^ (b2 |^.. b4) & b3 c= (b2 |^.. b4) ^^ b3;
:: FLANG_3:th 27
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set holds
(b2 |^.. b3) ^^ (b2 |^.. b4) = (b2 |^.. b4) ^^ (b2 |^.. b3);
:: FLANG_3:th 28
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4, b5 being natural set
st b2 c= b3 |^.. b4 & 0 < b5
holds b2 |^ b5 c= b3 |^.. b4;
:: FLANG_3:th 29
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4, b5 being natural set
st b2 c= b3 |^.. b4 & 0 < b5
holds b2 |^.. b5 c= b3 |^.. b4;
:: FLANG_3:th 30
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 * ^^ b2 = b2 |^.. 1;
:: FLANG_3:th 31
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 * ^^ (b2 |^ b3) = b2 |^.. b3;
:: FLANG_3:th 32
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
(b2 |^.. b3) ^^ (b2 *) = b2 * ^^ (b2 |^.. b3);
:: FLANG_3:th 33
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4, b5 being natural set
st b3 <= b4
holds (b2 |^.. b5) ^^ (b2 |^(b3,b4)) = b2 |^.. (b5 + b3);
:: FLANG_3:th 34
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set
st b3 <= b4
holds b2 * ^^ (b2 |^(b3,b4)) = b2 |^.. b3;
:: FLANG_3:th 35
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set holds
(b2 |^ b3) |^.. b4 c= b2 |^.. (b3 * b4);
:: FLANG_3:th 36
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set holds
(b2 |^ b3) |^.. b4 c= (b2 |^.. b4) |^ b3;
:: FLANG_3:th 37
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being Element of b1 ^omega
for b5, b6 being natural set
st b3 in b2 |^.. b5 & b4 in b2 |^.. b6
holds b3 ^ b4 in b2 |^.. (b5 + b6);
:: FLANG_3:th 38
theorem
for b1, b2 being set
for b3 being Element of bool (b2 ^omega)
for b4 being natural set
st b3 |^.. b4 = {b1}
holds b1 = <%> b2;
:: FLANG_3:th 39
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4 being natural set
st b2 c= b3 *
holds b2 |^.. b4 c= b3 *;
:: FLANG_3:th 40
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 ? c= b2 |^.. b3
iff
(b3 = 0 or <%> b1 in b2);
:: FLANG_3:th 41
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
(b2 |^.. b3) ^^ (b2 ?) = b2 |^.. b3;
:: FLANG_3:th 42
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
(b2 |^.. b3) ^^ (b2 ?) = b2 ? ^^ (b2 |^.. b3);
:: FLANG_3:th 43
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4 being natural set
st b2 c= b3 *
holds (b3 |^.. b4) ^^ b2 c= b3 |^.. b4 & b2 ^^ (b3 |^.. b4) c= b3 |^.. b4;
:: FLANG_3:th 44
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4 being natural set holds
(b2 /\ b3) |^.. b4 c= (b2 |^.. b4) /\ (b3 |^.. b4);
:: FLANG_3:th 45
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4 being natural set holds
(b2 |^.. b4) \/ (b3 |^.. b4) c= (b2 \/ b3) |^.. b4;
:: FLANG_3:th 46
theorem
for b1, b2 being set
for b3 being Element of bool (b2 ^omega)
for b4 being natural set holds
<%b1%> in b3 |^.. b4
iff
<%b1%> in b3 & (<%> b2 in b3 or b4 <= 1);
:: FLANG_3:th 47
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4 being natural set
st b2 c= b3 |^.. b4
holds b3 |^.. b4 = (b3 \/ b2) |^.. b4;
:: FLANG_3:funcnot 2 => FLANG_3:func 2
definition
let a1 be set;
let a2 be Element of bool (a1 ^omega);
func A2 + -> Element of bool (a1 ^omega) equals
union {b1 where b1 is Element of bool (a1 ^omega): ex b2 being natural set st
0 < b2 & b1 = a2 |^ b2};
end;
:: FLANG_3:def 2
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + = union {b3 where b3 is Element of bool (b1 ^omega): ex b4 being natural set st
0 < b4 & b3 = b2 |^ b4};
:: FLANG_3:th 48
theorem
for b1, b2 being set
for b3 being Element of bool (b1 ^omega) holds
b2 in b3 +
iff
ex b4 being natural set st
0 < b4 & b2 in b3 |^ b4;
:: FLANG_3:th 49
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set
st 0 < b3
holds b2 |^ b3 c= b2 +;
:: FLANG_3:th 50
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + = b2 |^.. 1;
:: FLANG_3:th 51
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + = {}
iff
b2 = {};
:: FLANG_3:th 52
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + = b2 * ^^ b2;
:: FLANG_3:th 53
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 * = {<%> b1} \/ (b2 +);
:: FLANG_3:th 54
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 + = (b2 |^(1,b3)) \/ (b2 |^.. (b3 + 1));
:: FLANG_3:th 55
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + c= b2 *;
:: FLANG_3:th 56
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
<%> b1 in b2 +
iff
<%> b1 in b2;
:: FLANG_3:th 57
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + = b2 *
iff
<%> b1 in b2;
:: FLANG_3:th 58
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
st b2 c= b3
holds b2 + c= b3 +;
:: FLANG_3:th 59
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 c= b2 +;
:: FLANG_3:th 60
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 * + = b2 * & b2 + * = b2 *;
:: FLANG_3:th 61
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
st b2 c= b3 *
holds b2 + c= b3 *;
:: FLANG_3:th 62
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + + = b2 +;
:: FLANG_3:th 63
theorem
for b1, b2 being set
for b3 being Element of bool (b2 ^omega)
st b1 in b3 & b1 <> <%> b2
holds b3 + <> {<%> b2};
:: FLANG_3:th 64
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + = {<%> b1}
iff
b2 = {<%> b1};
:: FLANG_3:th 65
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + ? = b2 * & b2 ? + = b2 *;
:: FLANG_3:th 66
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being Element of b1 ^omega
st b3 in b2 + & b4 in b2 +
holds b3 ^ b4 in b2 +;
:: FLANG_3:th 67
theorem
for b1 being set
for b2, b3, b4 being Element of bool (b1 ^omega)
st b2 c= b3 + & b4 c= b3 +
holds b2 ^^ b4 c= b3 +;
:: FLANG_3:th 68
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 ^^ b2 c= b2 +;
:: FLANG_3:th 69
theorem
for b1, b2 being set
for b3 being Element of bool (b2 ^omega)
st b3 + = {b1}
holds b1 = <%> b2;
:: FLANG_3:th 70
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 ^^ (b2 +) = b2 + ^^ b2;
:: FLANG_3:th 71
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
(b2 |^ b3) ^^ (b2 +) = b2 + ^^ (b2 |^ b3);
:: FLANG_3:th 72
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set holds
(b2 |^(b3,b4)) ^^ (b2 +) = b2 + ^^ (b2 |^(b3,b4));
:: FLANG_3:th 73
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
st <%> b1 in b2
holds b3 c= b3 ^^ (b2 +) & b3 c= b2 + ^^ b3;
:: FLANG_3:th 74
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + ^^ (b2 +) = b2 |^.. 2;
:: FLANG_3:th 75
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 + ^^ (b2 |^ b3) = b2 |^.. (b3 + 1);
:: FLANG_3:th 76
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + ^^ b2 = b2 |^.. 2;
:: FLANG_3:th 77
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set
st b3 <= b4
holds b2 + ^^ (b2 |^(b3,b4)) = b2 |^.. (b3 + 1);
:: FLANG_3:th 78
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4 being natural set
st b2 c= b3 + & 0 < b4
holds b2 |^ b4 c= b3 +;
:: FLANG_3:th 79
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + ^^ (b2 ?) = b2 ? ^^ (b2 +);
:: FLANG_3:th 80
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + ^^ (b2 ?) = b2 +;
:: FLANG_3:th 81
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 ? c= b2 +
iff
<%> b1 in b2;
:: FLANG_3:th 82
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
st b2 c= b3 +
holds b2 + c= b3 +;
:: FLANG_3:th 83
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
st b2 c= b3 +
holds b3 + = (b3 \/ b2) +;
:: FLANG_3:th 84
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set
st 0 < b3
holds b2 |^.. b3 c= b2 +;
:: FLANG_3:th 85
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set
st 0 < b3
holds b2 |^(b3,b4) c= b2 +;
:: FLANG_3:th 86
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 * ^^ (b2 +) = b2 + ^^ (b2 *);
:: FLANG_3:th 87
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 + |^ b3 c= b2 |^.. b3;
:: FLANG_3:th 88
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3, b4 being natural set holds
b2 + |^(b3,b4) c= b2 |^.. b3;
:: FLANG_3:th 89
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
for b4 being natural set
st b2 c= b3 + & 0 < b4
holds b2 |^.. b4 c= b3 +;
:: FLANG_3:th 90
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 + ^^ (b2 |^.. b3) = b2 |^.. (b3 + 1);
:: FLANG_3:th 91
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega)
for b3 being natural set holds
b2 + ^^ (b2 |^.. b3) = (b2 |^.. b3) ^^ (b2 +);
:: FLANG_3:th 92
theorem
for b1 being set
for b2 being Element of bool (b1 ^omega) holds
b2 + ^^ (b2 *) = b2 +;
:: FLANG_3:th 93
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega)
st b2 c= b3 *
holds b3 + ^^ b2 c= b3 + & b2 ^^ (b3 +) c= b3 +;
:: FLANG_3:th 94
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega) holds
(b2 /\ b3) + c= b2 + /\ (b3 +);
:: FLANG_3:th 95
theorem
for b1 being set
for b2, b3 being Element of bool (b1 ^omega) holds
b2 + \/ (b3 +) c= (b2 \/ b3) +;
:: FLANG_3:th 96
theorem
for b1, b2 being set
for b3 being Element of bool (b1 ^omega) holds
<%b2%> in b3 +
iff
<%b2%> in b3;