Article SCHEME1, MML version 4.99.1005
:: SCHEME1:th 1
theorem
for b1 being Element of NAT holds
ex b2 being Element of NAT st
(b1 = 2 * b2 or b1 = (2 * b2) + 1);
:: SCHEME1:th 2
theorem
for b1 being Element of NAT holds
ex b2 being Element of NAT st
(b1 <> 3 * b2 & b1 <> (3 * b2) + 1 implies b1 = (3 * b2) + 2);
:: SCHEME1:th 3
theorem
for b1 being Element of NAT holds
ex b2 being Element of NAT st
(b1 <> 4 * b2 & b1 <> (4 * b2) + 1 & b1 <> (4 * b2) + 2 implies b1 = (4 * b2) + 3);
:: SCHEME1:th 4
theorem
for b1 being Element of NAT holds
ex b2 being Element of NAT st
(b1 <> 5 * b2 & b1 <> (5 * b2) + 1 & b1 <> (5 * b2) + 2 & b1 <> (5 * b2) + 3 implies b1 = (5 * b2) + 4);
:: SCHEME1:sch 1
scheme SCHEME1:sch 1
{F1 -> Function-like quasi_total Relation of NAT,REAL}:
ex b1 being Function-like quasi_total Relation of NAT,REAL st
b1 is subsequence of F1() &
(for b2 being Element of NAT holds
P1[b1 . b2]) &
(for b2 being Element of NAT
st for b3 being Element of REAL
st b3 = F1() . b2
holds P1[b3]
holds ex b3 being Element of NAT st
F1() . b2 = b1 . b3)
provided
for b1 being Element of NAT holds
ex b2 being Element of NAT st
b1 <= b2 & P1[F1() . b2];
:: SCHEME1:sch 2
scheme SCHEME1:sch 2
{F1 -> Element of REAL,
F2 -> Element of REAL}:
ex b1 being Function-like quasi_total Relation of NAT,REAL st
for b2 being Element of NAT holds
b1 . (2 * b2) = F1(b2) & b1 . ((2 * b2) + 1) = F2(b2)
:: SCHEME1:sch 3
scheme SCHEME1:sch 3
{F1 -> Element of REAL,
F2 -> Element of REAL,
F3 -> Element of REAL}:
ex b1 being Function-like quasi_total Relation of NAT,REAL st
for b2 being Element of NAT holds
b1 . (3 * b2) = F1(b2) & b1 . ((3 * b2) + 1) = F2(b2) & b1 . ((3 * b2) + 2) = F3(b2)
:: SCHEME1:sch 4
scheme SCHEME1:sch 4
{F1 -> Element of REAL,
F2 -> Element of REAL,
F3 -> Element of REAL,
F4 -> Element of REAL}:
ex b1 being Function-like quasi_total Relation of NAT,REAL st
for b2 being Element of NAT holds
b1 . (4 * b2) = F1(b2) & b1 . ((4 * b2) + 1) = F2(b2) & b1 . ((4 * b2) + 2) = F3(b2) & b1 . ((4 * b2) + 3) = F4(b2)
:: SCHEME1:sch 5
scheme SCHEME1:sch 5
{F1 -> Element of REAL,
F2 -> Element of REAL,
F3 -> Element of REAL,
F4 -> Element of REAL,
F5 -> Element of REAL}:
ex b1 being Function-like quasi_total Relation of NAT,REAL st
for b2 being Element of NAT holds
b1 . (5 * b2) = F1(b2) & b1 . ((5 * b2) + 1) = F2(b2) & b1 . ((5 * b2) + 2) = F3(b2) & b1 . ((5 * b2) + 3) = F4(b2) & b1 . ((5 * b2) + 4) = F5(b2)
:: SCHEME1:sch 6
scheme SCHEME1:sch 6
{F1 -> non empty set,
F2 -> non empty set,
F3 -> Element of F2(),
F4 -> Element of F2()}:
ex b1 being Function-like Relation of F1(),F2() st
(for b2 being Element of F1() holds
b2 in dom b1
iff
(P1[b2] or P2[b2])) &
(for b2 being Element of F1()
st b2 in dom b1
holds (P1[b2] implies b1 . b2 = F3(b2)) & (P2[b2] implies b1 . b2 = F4(b2)))
provided
for b1 being Element of F1()
st P1[b1]
holds not (P2[b1]);
:: SCHEME1:sch 7
scheme SCHEME1:sch 7
{F1 -> non empty set,
F2 -> non empty set,
F3 -> Element of F2(),
F4 -> Element of F2()}:
ex b1 being Function-like Relation of F1(),F2() st
(for b2 being Element of F1() holds
b2 in dom b1
iff
(P1[b2] or P2[b2])) &
(for b2 being Element of F1()
st b2 in dom b1
holds (P1[b2] implies b1 . b2 = F3(b2)) & (P2[b2] implies b1 . b2 = F4(b2)))
provided
for b1 being Element of F1()
st P1[b1] & P2[b1]
holds F3(b1) = F4(b1);
:: SCHEME1:sch 8
scheme SCHEME1:sch 8
{F1 -> non empty set,
F2 -> non empty set,
F3 -> Element of F2(),
F4 -> Element of F2()}:
ex b1 being Function-like Relation of F1(),F2() st
b1 is total(F1(), F2()) &
(for b2 being Element of F1()
st b2 in dom b1
holds (P1[b2] implies b1 . b2 = F3(b2)) & (P1[b2] or b1 . b2 = F4(b2)))
:: SCHEME1:sch 9
scheme SCHEME1:sch 9
{F1 -> non empty set,
F2 -> non empty set,
F3 -> Element of F2(),
F4 -> Element of F2(),
F5 -> Element of F2()}:
ex b1 being Function-like Relation of F1(),F2() st
(for b2 being Element of F1() holds
b2 in dom b1
iff
(not (P1[b2]) & not (P2[b2]) implies P3[b2])) &
(for b2 being Element of F1()
st b2 in dom b1
holds (P1[b2] implies b1 . b2 = F3(b2)) & (P2[b2] implies b1 . b2 = F4(b2)) & (P3[b2] implies b1 . b2 = F5(b2)))
provided
for b1 being Element of F1() holds
(P1[b1] implies not (P2[b1])) & (P1[b1] implies not (P3[b1])) & (P2[b1] implies not (P3[b1]));
:: SCHEME1:sch 10
scheme SCHEME1:sch 10
{F1 -> non empty set,
F2 -> non empty set,
F3 -> Element of F2(),
F4 -> Element of F2(),
F5 -> Element of F2()}:
ex b1 being Function-like Relation of F1(),F2() st
(for b2 being Element of F1() holds
b2 in dom b1
iff
(not (P1[b2]) & not (P2[b2]) implies P3[b2])) &
(for b2 being Element of F1()
st b2 in dom b1
holds (P1[b2] implies b1 . b2 = F3(b2)) & (P2[b2] implies b1 . b2 = F4(b2)) & (P3[b2] implies b1 . b2 = F5(b2)))
provided
for b1 being Element of F1() holds
(P1[b1] & P2[b1] implies F3(b1) = F4(b1)) & (P1[b1] & P3[b1] implies F3(b1) = F5(b1)) & (P2[b1] & P3[b1] implies F4(b1) = F5(b1));
:: SCHEME1:sch 11
scheme SCHEME1:sch 11
{F1 -> non empty set,
F2 -> non empty set,
F3 -> Element of F2(),
F4 -> Element of F2(),
F5 -> Element of F2(),
F6 -> Element of F2()}:
ex b1 being Function-like Relation of F1(),F2() st
(for b2 being Element of F1() holds
b2 in dom b1
iff
(not (P1[b2]) & not (P2[b2]) & not (P3[b2]) implies P4[b2])) &
(for b2 being Element of F1()
st b2 in dom b1
holds (P1[b2] implies b1 . b2 = F3(b2)) & (P2[b2] implies b1 . b2 = F4(b2)) & (P3[b2] implies b1 . b2 = F5(b2)) & (P4[b2] implies b1 . b2 = F6(b2)))
provided
for b1 being Element of F1() holds
(P1[b1] implies not (P2[b1])) & (P1[b1] implies not (P3[b1])) & (P1[b1] implies not (P4[b1])) & (P2[b1] implies not (P3[b1])) & (P2[b1] implies not (P4[b1])) & (P3[b1] implies not (P4[b1]));
:: SCHEME1:sch 12
scheme SCHEME1:sch 12
{F1 -> set,
F2 -> set,
F3 -> set,
F4 -> set}:
ex b1 being Function-like Relation of F1(),F2() st
(for b2 being set holds
b2 in dom b1
iff
b2 in F1() & (P1[b2] or P2[b2])) &
(for b2 being set
st b2 in dom b1
holds (P1[b2] implies b1 . b2 = F3(b2)) & (P2[b2] implies b1 . b2 = F4(b2)))
provided
for b1 being set
st b1 in F1() & P1[b1]
holds not (P2[b1])
and
for b1 being set
st b1 in F1() & P1[b1]
holds F3(b1) in F2()
and
for b1 being set
st b1 in F1() & P2[b1]
holds F4(b1) in F2();
:: SCHEME1:sch 13
scheme SCHEME1:sch 13
{F1 -> set,
F2 -> set,
F3 -> set,
F4 -> set,
F5 -> set}:
ex b1 being Function-like Relation of F1(),F2() st
(for b2 being set holds
b2 in dom b1
iff
b2 in F1() & (not (P1[b2]) & not (P2[b2]) implies P3[b2])) &
(for b2 being set
st b2 in dom b1
holds (P1[b2] implies b1 . b2 = F3(b2)) & (P2[b2] implies b1 . b2 = F4(b2)) & (P3[b2] implies b1 . b2 = F5(b2)))
provided
for b1 being set
st b1 in F1()
holds (P1[b1] implies not (P2[b1])) & (P1[b1] implies not (P3[b1])) & (P2[b1] implies not (P3[b1]))
and
for b1 being set
st b1 in F1() & P1[b1]
holds F3(b1) in F2()
and
for b1 being set
st b1 in F1() & P2[b1]
holds F4(b1) in F2()
and
for b1 being set
st b1 in F1() & P3[b1]
holds F5(b1) in F2();
:: SCHEME1:sch 14
scheme SCHEME1:sch 14
{F1 -> set,
F2 -> set,
F3 -> set,
F4 -> set,
F5 -> set,
F6 -> set}:
ex b1 being Function-like Relation of F1(),F2() st
(for b2 being set holds
b2 in dom b1
iff
b2 in F1() & (not (P1[b2]) & not (P2[b2]) & not (P3[b2]) implies P4[b2])) &
(for b2 being set
st b2 in dom b1
holds (P1[b2] implies b1 . b2 = F3(b2)) & (P2[b2] implies b1 . b2 = F4(b2)) & (P3[b2] implies b1 . b2 = F5(b2)) & (P4[b2] implies b1 . b2 = F6(b2)))
provided
for b1 being set
st b1 in F1()
holds (P1[b1] implies not (P2[b1])) & (P1[b1] implies not (P3[b1])) & (P1[b1] implies not (P4[b1])) & (P2[b1] implies not (P3[b1])) & (P2[b1] implies not (P4[b1])) & (P3[b1] implies not (P4[b1]))
and
for b1 being set
st b1 in F1() & P1[b1]
holds F3(b1) in F2()
and
for b1 being set
st b1 in F1() & P2[b1]
holds F4(b1) in F2()
and
for b1 being set
st b1 in F1() & P3[b1]
holds F5(b1) in F2()
and
for b1 being set
st b1 in F1() & P4[b1]
holds F6(b1) in F2();
:: SCHEME1:sch 15
scheme SCHEME1:sch 15
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> Element of F3(),
F5 -> Element of F3()}:
ex b1 being Function-like Relation of [:F1(),F2():],F3() st
(for b2 being Element of F1()
for b3 being Element of F2() holds
[b2,b3] in dom b1
iff
(P1[b2, b3] or P2[b2, b3])) &
(for b2 being Element of F1()
for b3 being Element of F2()
st [b2,b3] in dom b1
holds (P1[b2, b3] implies b1 . [b2,b3] = F4(b2, b3)) & (P2[b2, b3] implies b1 . [b2,b3] = F5(b2, b3)))
provided
for b1 being Element of F1()
for b2 being Element of F2()
st P1[b1, b2]
holds not (P2[b1, b2]);
:: SCHEME1:sch 16
scheme SCHEME1:sch 16
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> Element of F3(),
F5 -> Element of F3(),
F6 -> Element of F3()}:
ex b1 being Function-like Relation of [:F1(),F2():],F3() st
(for b2 being Element of F1()
for b3 being Element of F2() holds
[b2,b3] in dom b1
iff
(not (P1[b2, b3]) & not (P2[b2, b3]) implies P3[b2, b3])) &
(for b2 being Element of F1()
for b3 being Element of F2()
st [b2,b3] in dom b1
holds (P1[b2, b3] implies b1 . [b2,b3] = F4(b2, b3)) & (P2[b2, b3] implies b1 . [b2,b3] = F5(b2, b3)) & (P3[b2, b3] implies b1 . [b2,b3] = F6(b2, b3)))
provided
for b1 being Element of F1()
for b2 being Element of F2() holds
(P1[b1, b2] implies not (P2[b1, b2])) & (P1[b1, b2] implies not (P3[b1, b2])) & (P2[b1, b2] implies not (P3[b1, b2]));
:: SCHEME1:sch 17
scheme SCHEME1:sch 17
{F1 -> set,
F2 -> set,
F3 -> set,
F4 -> set,
F5 -> set}:
ex b1 being Function-like Relation of [:F1(),F2():],F3() st
(for b2, b3 being set holds
[b2,b3] in dom b1
iff
b2 in F1() & b3 in F2() & (P1[b2, b3] or P2[b2, b3])) &
(for b2, b3 being set
st [b2,b3] in dom b1
holds (P1[b2, b3] implies b1 . [b2,b3] = F4(b2, b3)) & (P2[b2, b3] implies b1 . [b2,b3] = F5(b2, b3)))
provided
for b1, b2 being set
st b1 in F1() & b2 in F2() & P1[b1, b2]
holds not (P2[b1, b2])
and
for b1, b2 being set
st b1 in F1() & b2 in F2() & P1[b1, b2]
holds F4(b1, b2) in F3()
and
for b1, b2 being set
st b1 in F1() & b2 in F2() & P2[b1, b2]
holds F5(b1, b2) in F3();
:: SCHEME1:sch 18
scheme SCHEME1:sch 18
{F1 -> set,
F2 -> set,
F3 -> set,
F4 -> set,
F5 -> set,
F6 -> set}:
ex b1 being Function-like Relation of [:F1(),F2():],F3() st
(for b2, b3 being set holds
[b2,b3] in dom b1
iff
b2 in F1() & b3 in F2() & (not (P1[b2, b3]) & not (P2[b2, b3]) implies P3[b2, b3])) &
(for b2, b3 being set
st [b2,b3] in dom b1
holds (P1[b2, b3] implies b1 . [b2,b3] = F4(b2, b3)) & (P2[b2, b3] implies b1 . [b2,b3] = F5(b2, b3)) & (P3[b2, b3] implies b1 . [b2,b3] = F6(b2, b3)))
provided
for b1, b2 being set
st b1 in F1() & b2 in F2()
holds (P1[b1, b2] implies not (P2[b1, b2])) & (P1[b1, b2] implies not (P3[b1, b2])) & (P2[b1, b2] implies not (P3[b1, b2]))
and
for b1, b2 being set
st b1 in F1() & b2 in F2() & P1[b1, b2]
holds F4(b1, b2) in F3()
and
for b1, b2 being set
st b1 in F1() & b2 in F2() & P2[b1, b2]
holds F5(b1, b2) in F3()
and
for b1, b2 being set
st b1 in F1() & b2 in F2() & P3[b1, b2]
holds F6(b1, b2) in F3();
:: SCHEME1:sch 19
scheme SCHEME1:sch 19
{F1 -> non empty set,
F2 -> non empty set,
F3 -> Element of F2(),
F4 -> Element of F2(),
F5 -> Element of F2()}:
ex b1 being Function-like quasi_total Relation of F1(),F2() st
for b2 being Element of F1() holds
(P1[b2] implies b1 . b2 = F3(b2)) & (P2[b2] implies b1 . b2 = F4(b2)) & (P3[b2] implies b1 . b2 = F5(b2))
provided
for b1 being Element of F1() holds
(P1[b1] implies not (P2[b1])) & (P1[b1] implies not (P3[b1])) & (P2[b1] implies not (P3[b1]))
and
for b1 being Element of F1()
st not (P1[b1]) & not (P2[b1])
holds P3[b1];
:: SCHEME1:sch 20
scheme SCHEME1:sch 20
{F1 -> non empty set,
F2 -> non empty set,
F3 -> Element of F2(),
F4 -> Element of F2(),
F5 -> Element of F2(),
F6 -> Element of F2()}:
ex b1 being Function-like quasi_total Relation of F1(),F2() st
for b2 being Element of F1() holds
(P1[b2] implies b1 . b2 = F3(b2)) & (P2[b2] implies b1 . b2 = F4(b2)) & (P3[b2] implies b1 . b2 = F5(b2)) & (P4[b2] implies b1 . b2 = F6(b2))
provided
for b1 being Element of F1() holds
(P1[b1] implies not (P2[b1])) & (P1[b1] implies not (P3[b1])) & (P1[b1] implies not (P4[b1])) & (P2[b1] implies not (P3[b1])) & (P2[b1] implies not (P4[b1])) & (P3[b1] implies not (P4[b1]))
and
for b1 being Element of F1()
st not (P1[b1]) & not (P2[b1]) & not (P3[b1])
holds P4[b1];
:: SCHEME1:sch 21
scheme SCHEME1:sch 21
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> Element of F3(),
F5 -> Element of F3()}:
ex b1 being Function-like quasi_total Relation of [:F1(),F2():],F3() st
for b2 being Element of F1()
for b3 being Element of F2()
st [b2,b3] in dom b1
holds (P1[b2, b3] implies b1 . [b2,b3] = F4(b2, b3)) & (P1[b2, b3] or b1 . [b2,b3] = F5(b2, b3))
:: SCHEME1:sch 22
scheme SCHEME1:sch 22
{F1 -> non empty set,
F2 -> non empty set,
F3 -> non empty set,
F4 -> Element of F3(),
F5 -> Element of F3(),
F6 -> Element of F3()}:
ex b1 being Function-like quasi_total Relation of [:F1(),F2():],F3() st
(for b2 being Element of F1()
for b3 being Element of F2() holds
[b2,b3] in dom b1
iff
(not (P1[b2, b3]) & not (P2[b2, b3]) implies P3[b2, b3])) &
(for b2 being Element of F1()
for b3 being Element of F2()
st [b2,b3] in dom b1
holds (P1[b2, b3] implies b1 . [b2,b3] = F4(b2, b3)) & (P2[b2, b3] implies b1 . [b2,b3] = F5(b2, b3)) & (P3[b2, b3] implies b1 . [b2,b3] = F6(b2, b3)))
provided
for b1 being Element of F1()
for b2 being Element of F2() holds
(P1[b1, b2] implies not (P2[b1, b2])) & (P1[b1, b2] implies not (P3[b1, b2])) & (P2[b1, b2] implies not (P3[b1, b2]))
and
for b1 being Element of F1()
for b2 being Element of F2()
st not (P1[b1, b2]) & not (P2[b1, b2])
holds P3[b1, b2];