:: RECDEF_2 semantic presentation
definition
let c
1 be
set ;
given c
2, c
3, c
4 being
set such that E1:
c
1 = [c2,c3,c4]
;
func c
1 `1_3 -> set means :
Def1:
:: RECDEF_2:def 1
for b
1, b
2, b
3 being
set holds
( a
1 = [b1,b2,b3] implies a
2 = b
1 );
existence
ex b1 being set st
for b2, b3, b4 being set holds
( c1 = [b2,b3,b4] implies b1 = b2 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5 being set holds
( c1 = [b3,b4,b5] implies b1 = b3 ) ) & ( for b3, b4, b5 being set holds
( c1 = [b3,b4,b5] implies b2 = b3 ) ) implies b1 = b2 )
func c
1 `2_3 -> set means :
Def2:
:: RECDEF_2:def 2
for b
1, b
2, b
3 being
set holds
( a
1 = [b1,b2,b3] implies a
2 = b
2 );
existence
ex b1 being set st
for b2, b3, b4 being set holds
( c1 = [b2,b3,b4] implies b1 = b3 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5 being set holds
( c1 = [b3,b4,b5] implies b1 = b4 ) ) & ( for b3, b4, b5 being set holds
( c1 = [b3,b4,b5] implies b2 = b4 ) ) implies b1 = b2 )
func c
1 `3_3 -> set means :
Def3:
:: RECDEF_2:def 3
for b
1, b
2, b
3 being
set holds
( a
1 = [b1,b2,b3] implies a
2 = b
3 );
existence
ex b1 being set st
for b2, b3, b4 being set holds
( c1 = [b2,b3,b4] implies b1 = b4 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5 being set holds
( c1 = [b3,b4,b5] implies b1 = b5 ) ) & ( for b3, b4, b5 being set holds
( c1 = [b3,b4,b5] implies b2 = b5 ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines `1_3 RECDEF_2:def 1 :
for b
1 being
set holds
( ex b
2, b
3, b
4 being
set st b
1 = [b2,b3,b4] implies for b
2 being
set holds
( b
2 = b
1 `1_3 iff for b
3, b
4, b
5 being
set holds
( b
1 = [b3,b4,b5] implies b
2 = b
3 ) ) );
:: deftheorem Def2 defines `2_3 RECDEF_2:def 2 :
for b
1 being
set holds
( ex b
2, b
3, b
4 being
set st b
1 = [b2,b3,b4] implies for b
2 being
set holds
( b
2 = b
1 `2_3 iff for b
3, b
4, b
5 being
set holds
( b
1 = [b3,b4,b5] implies b
2 = b
4 ) ) );
:: deftheorem Def3 defines `3_3 RECDEF_2:def 3 :
for b
1 being
set holds
( ex b
2, b
3, b
4 being
set st b
1 = [b2,b3,b4] implies for b
2 being
set holds
( b
2 = b
1 `3_3 iff for b
3, b
4, b
5 being
set holds
( b
1 = [b3,b4,b5] implies b
2 = b
5 ) ) );
theorem Th1: :: RECDEF_2:1
theorem Th2: :: RECDEF_2:2
theorem Th3: :: RECDEF_2:3
definition
let c
1 be
set ;
given c
2, c
3, c
4, c
5 being
set such that E5:
c
1 = [c2,c3,c4,c5]
;
func c
1 `1_4 -> set means :
Def4:
:: RECDEF_2:def 4
for b
1, b
2, b
3, b
4 being
set holds
( a
1 = [b1,b2,b3,b4] implies a
2 = b
1 );
existence
ex b1 being set st
for b2, b3, b4, b5 being set holds
( c1 = [b2,b3,b4,b5] implies b1 = b2 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5, b6 being set holds
( c1 = [b3,b4,b5,b6] implies b1 = b3 ) ) & ( for b3, b4, b5, b6 being set holds
( c1 = [b3,b4,b5,b6] implies b2 = b3 ) ) implies b1 = b2 )
func c
1 `2_4 -> set means :
Def5:
:: RECDEF_2:def 5
for b
1, b
2, b
3, b
4 being
set holds
( a
1 = [b1,b2,b3,b4] implies a
2 = b
2 );
existence
ex b1 being set st
for b2, b3, b4, b5 being set holds
( c1 = [b2,b3,b4,b5] implies b1 = b3 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5, b6 being set holds
( c1 = [b3,b4,b5,b6] implies b1 = b4 ) ) & ( for b3, b4, b5, b6 being set holds
( c1 = [b3,b4,b5,b6] implies b2 = b4 ) ) implies b1 = b2 )
func c
1 `3_4 -> set means :
Def6:
:: RECDEF_2:def 6
for b
1, b
2, b
3, b
4 being
set holds
( a
1 = [b1,b2,b3,b4] implies a
2 = b
3 );
existence
ex b1 being set st
for b2, b3, b4, b5 being set holds
( c1 = [b2,b3,b4,b5] implies b1 = b4 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5, b6 being set holds
( c1 = [b3,b4,b5,b6] implies b1 = b5 ) ) & ( for b3, b4, b5, b6 being set holds
( c1 = [b3,b4,b5,b6] implies b2 = b5 ) ) implies b1 = b2 )
func c
1 `4_4 -> set means :
Def7:
:: RECDEF_2:def 7
for b
1, b
2, b
3, b
4 being
set holds
( a
1 = [b1,b2,b3,b4] implies a
2 = b
4 );
existence
ex b1 being set st
for b2, b3, b4, b5 being set holds
( c1 = [b2,b3,b4,b5] implies b1 = b5 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5, b6 being set holds
( c1 = [b3,b4,b5,b6] implies b1 = b6 ) ) & ( for b3, b4, b5, b6 being set holds
( c1 = [b3,b4,b5,b6] implies b2 = b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines `1_4 RECDEF_2:def 4 :
for b
1 being
set holds
( ex b
2, b
3, b
4, b
5 being
set st b
1 = [b2,b3,b4,b5] implies for b
2 being
set holds
( b
2 = b
1 `1_4 iff for b
3, b
4, b
5, b
6 being
set holds
( b
1 = [b3,b4,b5,b6] implies b
2 = b
3 ) ) );
:: deftheorem Def5 defines `2_4 RECDEF_2:def 5 :
for b
1 being
set holds
( ex b
2, b
3, b
4, b
5 being
set st b
1 = [b2,b3,b4,b5] implies for b
2 being
set holds
( b
2 = b
1 `2_4 iff for b
3, b
4, b
5, b
6 being
set holds
( b
1 = [b3,b4,b5,b6] implies b
2 = b
4 ) ) );
:: deftheorem Def6 defines `3_4 RECDEF_2:def 6 :
for b
1 being
set holds
( ex b
2, b
3, b
4, b
5 being
set st b
1 = [b2,b3,b4,b5] implies for b
2 being
set holds
( b
2 = b
1 `3_4 iff for b
3, b
4, b
5, b
6 being
set holds
( b
1 = [b3,b4,b5,b6] implies b
2 = b
5 ) ) );
:: deftheorem Def7 defines `4_4 RECDEF_2:def 7 :
for b
1 being
set holds
( ex b
2, b
3, b
4, b
5 being
set st b
1 = [b2,b3,b4,b5] implies for b
2 being
set holds
( b
2 = b
1 `4_4 iff for b
3, b
4, b
5, b
6 being
set holds
( b
1 = [b3,b4,b5,b6] implies b
2 = b
6 ) ) );
theorem Th4: :: RECDEF_2:4
theorem Th5: :: RECDEF_2:5
theorem Th6: :: RECDEF_2:6
for b
1, b
2, b
3, b
4, b
5 being
set holds
( b
1 in [:b2,b3,b4,b5:] implies b
1 = [(b1 `1_4 ),(b1 `2_4 ),(b1 `3_4 ),(b1 `4_4 )] )
definition
let c
1 be
set ;
given c
2, c
3, c
4, c
5, c
6 being
set such that E10:
c
1 = [c2,c3,c4,c5,c6]
;
func c
1 `1_5 -> set means :
Def8:
:: RECDEF_2:def 8
for b
1, b
2, b
3, b
4, b
5 being
set holds
( a
1 = [b1,b2,b3,b4,b5] implies a
2 = b
1 );
existence
ex b1 being set st
for b2, b3, b4, b5, b6 being set holds
( c1 = [b2,b3,b4,b5,b6] implies b1 = b2 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5, b6, b7 being set holds
( c1 = [b3,b4,b5,b6,b7] implies b1 = b3 ) ) & ( for b3, b4, b5, b6, b7 being set holds
( c1 = [b3,b4,b5,b6,b7] implies b2 = b3 ) ) implies b1 = b2 )
func c
1 `2_5 -> set means :
Def9:
:: RECDEF_2:def 9
for b
1, b
2, b
3, b
4, b
5 being
set holds
( a
1 = [b1,b2,b3,b4,b5] implies a
2 = b
2 );
existence
ex b1 being set st
for b2, b3, b4, b5, b6 being set holds
( c1 = [b2,b3,b4,b5,b6] implies b1 = b3 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5, b6, b7 being set holds
( c1 = [b3,b4,b5,b6,b7] implies b1 = b4 ) ) & ( for b3, b4, b5, b6, b7 being set holds
( c1 = [b3,b4,b5,b6,b7] implies b2 = b4 ) ) implies b1 = b2 )
func c
1 `3_5 -> set means :
Def10:
:: RECDEF_2:def 10
for b
1, b
2, b
3, b
4, b
5 being
set holds
( a
1 = [b1,b2,b3,b4,b5] implies a
2 = b
3 );
existence
ex b1 being set st
for b2, b3, b4, b5, b6 being set holds
( c1 = [b2,b3,b4,b5,b6] implies b1 = b4 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5, b6, b7 being set holds
( c1 = [b3,b4,b5,b6,b7] implies b1 = b5 ) ) & ( for b3, b4, b5, b6, b7 being set holds
( c1 = [b3,b4,b5,b6,b7] implies b2 = b5 ) ) implies b1 = b2 )
func c
1 `4_5 -> set means :
Def11:
:: RECDEF_2:def 11
for b
1, b
2, b
3, b
4, b
5 being
set holds
( a
1 = [b1,b2,b3,b4,b5] implies a
2 = b
4 );
existence
ex b1 being set st
for b2, b3, b4, b5, b6 being set holds
( c1 = [b2,b3,b4,b5,b6] implies b1 = b5 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5, b6, b7 being set holds
( c1 = [b3,b4,b5,b6,b7] implies b1 = b6 ) ) & ( for b3, b4, b5, b6, b7 being set holds
( c1 = [b3,b4,b5,b6,b7] implies b2 = b6 ) ) implies b1 = b2 )
func c
1 `5_5 -> set means :
Def12:
:: RECDEF_2:def 12
for b
1, b
2, b
3, b
4, b
5 being
set holds
( a
1 = [b1,b2,b3,b4,b5] implies a
2 = b
5 );
existence
ex b1 being set st
for b2, b3, b4, b5, b6 being set holds
( c1 = [b2,b3,b4,b5,b6] implies b1 = b6 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4, b5, b6, b7 being set holds
( c1 = [b3,b4,b5,b6,b7] implies b1 = b7 ) ) & ( for b3, b4, b5, b6, b7 being set holds
( c1 = [b3,b4,b5,b6,b7] implies b2 = b7 ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines `1_5 RECDEF_2:def 8 :
for b
1 being
set holds
( ex b
2, b
3, b
4, b
5, b
6 being
set st b
1 = [b2,b3,b4,b5,b6] implies for b
2 being
set holds
( b
2 = b
1 `1_5 iff for b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 = [b3,b4,b5,b6,b7] implies b
2 = b
3 ) ) );
:: deftheorem Def9 defines `2_5 RECDEF_2:def 9 :
for b
1 being
set holds
( ex b
2, b
3, b
4, b
5, b
6 being
set st b
1 = [b2,b3,b4,b5,b6] implies for b
2 being
set holds
( b
2 = b
1 `2_5 iff for b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 = [b3,b4,b5,b6,b7] implies b
2 = b
4 ) ) );
:: deftheorem Def10 defines `3_5 RECDEF_2:def 10 :
for b
1 being
set holds
( ex b
2, b
3, b
4, b
5, b
6 being
set st b
1 = [b2,b3,b4,b5,b6] implies for b
2 being
set holds
( b
2 = b
1 `3_5 iff for b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 = [b3,b4,b5,b6,b7] implies b
2 = b
5 ) ) );
:: deftheorem Def11 defines `4_5 RECDEF_2:def 11 :
for b
1 being
set holds
( ex b
2, b
3, b
4, b
5, b
6 being
set st b
1 = [b2,b3,b4,b5,b6] implies for b
2 being
set holds
( b
2 = b
1 `4_5 iff for b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 = [b3,b4,b5,b6,b7] implies b
2 = b
6 ) ) );
:: deftheorem Def12 defines `5_5 RECDEF_2:def 12 :
for b
1 being
set holds
( ex b
2, b
3, b
4, b
5, b
6 being
set st b
1 = [b2,b3,b4,b5,b6] implies for b
2 being
set holds
( b
2 = b
1 `5_5 iff for b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 = [b3,b4,b5,b6,b7] implies b
2 = b
7 ) ) );
theorem Th7: :: RECDEF_2:7
for b
1 being
set holds
( ex b
2, b
3, b
4, b
5, b
6 being
set st b
1 = [b2,b3,b4,b5,b6] implies b
1 = [(b1 `1_5 ),(b1 `2_5 ),(b1 `3_5 ),(b1 `4_5 ),(b1 `5_5 )] )
theorem Th8: :: RECDEF_2:8
theorem Th9: :: RECDEF_2:9
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( b
1 in [:b2,b3,b4,b5,b6:] implies b
1 = [(b1 `1_5 ),(b1 `2_5 ),(b1 `3_5 ),(b1 `4_5 ),(b1 `5_5 )] )
scheme :: RECDEF_2:sch 1
s1{ F
1()
-> set , P
1[
set ], P
2[
set ], P
3[
set ], F
2(
set )
-> set , F
3(
set )
-> set , F
4(
set )
-> set } :
ex b
1 being
Function st
(
dom b
1 = F
1() & ( for b
2 being
set holds
( b
2 in F
1() implies ( ( P
1[b
2] implies b
1 . b
2 = F
2(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
3[b
2] implies b
1 . b
2 = F
4(b
2) ) ) ) ) )
provided
E16:
for b
1 being
set holds
( b
1 in F
1() implies ( not ( P
1[b
1] & P
2[b
1] ) & not ( P
1[b
1] & P
3[b
1] ) & not ( P
2[b
1] & P
3[b
1] ) ) )
and
E17:
for b
1 being
set holds
not ( b
1 in F
1() & not P
1[b
1] & not P
2[b
1] & not P
3[b
1] )
scheme :: RECDEF_2:sch 2
s2{ F
1()
-> set , P
1[
set ], P
2[
set ], P
3[
set ], P
4[
set ], F
2(
set )
-> set , F
3(
set )
-> set , F
4(
set )
-> set , F
5(
set )
-> set } :
ex b
1 being
Function st
(
dom b
1 = F
1() & ( for b
2 being
set holds
( b
2 in F
1() implies ( ( P
1[b
2] implies b
1 . b
2 = F
2(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
3[b
2] implies b
1 . b
2 = F
4(b
2) ) & ( P
4[b
2] implies b
1 . b
2 = F
5(b
2) ) ) ) ) )
provided
E16:
for b
1 being
set holds
( b
1 in F
1() implies ( not ( P
1[b
1] & P
2[b
1] ) & not ( P
1[b
1] & P
3[b
1] ) & not ( P
1[b
1] & P
4[b
1] ) & not ( P
2[b
1] & P
3[b
1] ) & not ( P
2[b
1] & P
4[b
1] ) & not ( P
3[b
1] & P
4[b
1] ) ) )
and
E17:
for b
1 being
set holds
not ( b
1 in F
1() & not P
1[b
1] & not P
2[b
1] & not P
3[b
1] & not P
4[b
1] )
scheme :: RECDEF_2:sch 6
s6{ F
1()
-> set , F
2()
-> set , F
3()
-> Function, F
4()
-> Function, F
5(
set ,
set ,
set )
-> set } :
provided
E16:
dom F
3()
= NAT
and
E17:
( F
3()
. 0
= F
1() & F
3()
. 1
= F
2() )
and
E18:
for b
1 being
Nat holds F
3()
. (b1 + 2) = F
5(b
1,
(F3() . b1),
(F3() . (b1 + 1)))
and
E19:
dom F
4()
= NAT
and
E20:
( F
4()
. 0
= F
1() & F
4()
. 1
= F
2() )
and
E21:
for b
1 being
Nat holds F
4()
. (b1 + 2) = F
5(b
1,
(F4() . b1),
(F4() . (b1 + 1)))
scheme :: RECDEF_2:sch 7
s7{ F
1()
-> non
empty set , F
2()
-> Element of F
1(), F
3()
-> Element of F
1(), F
4()
-> Function of
NAT ,F
1(), F
5()
-> Function of
NAT ,F
1(), F
6(
set ,
set ,
set )
-> Element of F
1() } :
provided
E16:
( F
4()
. 0
= F
2() & F
4()
. 1
= F
3() )
and
E17:
for b
1 being
Nat holds F
4()
. (b1 + 2) = F
6(b
1,
(F4() . b1),
(F4() . (b1 + 1)))
and
E18:
( F
5()
. 0
= F
2() & F
5()
. 1
= F
3() )
and
E19:
for b
1 being
Nat holds F
5()
. (b1 + 2) = F
6(b
1,
(F5() . b1),
(F5() . (b1 + 1)))
scheme :: RECDEF_2:sch 9
s9{ F
1()
-> non
empty set , F
2()
-> Element of F
1(), F
3()
-> Element of F
1(), F
4()
-> Element of F
1(), F
5(
set ,
set ,
set ,
set )
-> Element of F
1() } :
ex b
1 being
Function of
NAT ,F
1() st
( b
1 . 0
= F
2() & b
1 . 1
= F
3() & b
1 . 2
= F
4() & ( for b
2 being
Nat holds b
1 . (b2 + 3) = F
5(b
2,
(b1 . b2),
(b1 . (b2 + 1)),
(b1 . (b2 + 2))) ) )
scheme :: RECDEF_2:sch 10
s10{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> Function, F
5()
-> Function, F
6(
set ,
set ,
set ,
set )
-> set } :
provided
E16:
dom F
4()
= NAT
and
E17:
( F
4()
. 0
= F
1() & F
4()
. 1
= F
2() & F
4()
. 2
= F
3() )
and
E18:
for b
1 being
Nat holds F
4()
. (b1 + 3) = F
6(b
1,
(F4() . b1),
(F4() . (b1 + 1)),
(F4() . (b1 + 2)))
and
E19:
dom F
5()
= NAT
and
E20:
( F
5()
. 0
= F
1() & F
5()
. 1
= F
2() & F
5()
. 2
= F
3() )
and
E21:
for b
1 being
Nat holds F
5()
. (b1 + 3) = F
6(b
1,
(F5() . b1),
(F5() . (b1 + 1)),
(F5() . (b1 + 2)))
scheme :: RECDEF_2:sch 11
s11{ F
1()
-> non
empty set , F
2()
-> Element of F
1(), F
3()
-> Element of F
1(), F
4()
-> Element of F
1(), F
5()
-> Function of
NAT ,F
1(), F
6()
-> Function of
NAT ,F
1(), F
7(
set ,
set ,
set ,
set )
-> Element of F
1() } :
provided
E16:
( F
5()
. 0
= F
2() & F
5()
. 1
= F
3() & F
5()
. 2
= F
4() )
and
E17:
for b
1 being
Nat holds F
5()
. (b1 + 3) = F
7(b
1,
(F5() . b1),
(F5() . (b1 + 1)),
(F5() . (b1 + 2)))
and
E18:
( F
6()
. 0
= F
2() & F
6()
. 1
= F
3() & F
6()
. 2
= F
4() )
and
E19:
for b
1 being
Nat holds F
6()
. (b1 + 3) = F
7(b
1,
(F6() . b1),
(F6() . (b1 + 1)),
(F6() . (b1 + 2)))
scheme :: RECDEF_2:sch 12
s12{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> set , F
5(
set ,
set ,
set ,
set ,
set )
-> set } :
ex b
1 being
Function st
(
dom b
1 = NAT & b
1 . 0
= F
1() & b
1 . 1
= F
2() & b
1 . 2
= F
3() & b
1 . 3
= F
4() & ( for b
2 being
Nat holds b
1 . (b2 + 4) = F
5(b
2,
(b1 . b2),
(b1 . (b2 + 1)),
(b1 . (b2 + 2)),
(b1 . (b2 + 3))) ) )
scheme :: RECDEF_2:sch 13
s13{ F
1()
-> non
empty set , F
2()
-> Element of F
1(), F
3()
-> Element of F
1(), F
4()
-> Element of F
1(), F
5()
-> Element of F
1(), F
6(
set ,
set ,
set ,
set ,
set )
-> Element of F
1() } :
ex b
1 being
Function of
NAT ,F
1() st
( b
1 . 0
= F
2() & b
1 . 1
= F
3() & b
1 . 2
= F
4() & b
1 . 3
= F
5() & ( for b
2 being
Nat holds b
1 . (b2 + 4) = F
6(b
2,
(b1 . b2),
(b1 . (b2 + 1)),
(b1 . (b2 + 2)),
(b1 . (b2 + 3))) ) )
scheme :: RECDEF_2:sch 14
s14{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> set , F
5()
-> Function, F
6()
-> Function, F
7(
set ,
set ,
set ,
set ,
set )
-> set } :
provided
E16:
dom F
5()
= NAT
and
E17:
( F
5()
. 0
= F
1() & F
5()
. 1
= F
2() & F
5()
. 2
= F
3() & F
5()
. 3
= F
4() )
and
E18:
for b
1 being
Nat holds F
5()
. (b1 + 4) = F
7(b
1,
(F5() . b1),
(F5() . (b1 + 1)),
(F5() . (b1 + 2)),
(F5() . (b1 + 3)))
and
E19:
dom F
6()
= NAT
and
E20:
( F
6()
. 0
= F
1() & F
6()
. 1
= F
2() & F
6()
. 2
= F
3() & F
6()
. 3
= F
4() )
and
E21:
for b
1 being
Nat holds F
6()
. (b1 + 4) = F
7(b
1,
(F6() . b1),
(F6() . (b1 + 1)),
(F6() . (b1 + 2)),
(F6() . (b1 + 3)))
scheme :: RECDEF_2:sch 15
s15{ F
1()
-> non
empty set , F
2()
-> Element of F
1(), F
3()
-> Element of F
1(), F
4()
-> Element of F
1(), F
5()
-> Element of F
1(), F
6()
-> Function of
NAT ,F
1(), F
7()
-> Function of
NAT ,F
1(), F
8(
set ,
set ,
set ,
set ,
set )
-> Element of F
1() } :
provided
E16:
( F
6()
. 0
= F
2() & F
6()
. 1
= F
3() & F
6()
. 2
= F
4() & F
6()
. 3
= F
5() )
and
E17:
for b
1 being
Nat holds F
6()
. (b1 + 4) = F
8(b
1,
(F6() . b1),
(F6() . (b1 + 1)),
(F6() . (b1 + 2)),
(F6() . (b1 + 3)))
and
E18:
( F
7()
. 0
= F
2() & F
7()
. 1
= F
3() & F
7()
. 2
= F
4() & F
7()
. 3
= F
5() )
and
E19:
for b
1 being
Nat holds F
7()
. (b1 + 4) = F
8(b
1,
(F7() . b1),
(F7() . (b1 + 1)),
(F7() . (b1 + 2)),
(F7() . (b1 + 3)))