let x1, x2, x3, x4, x5, x6, x7, x8 be set ; for p being FinSequence st p = ((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*> holds
( len p = 8 & p . 1 = x1 & p . 2 = x2 & p . 3 = x3 & p . 4 = x4 & p . 5 = x5 & p . 6 = x6 & p . 7 = x7 & p . 8 = x8 )
let p be FinSequence; ( p = ((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*> implies ( len p = 8 & p . 1 = x1 & p . 2 = x2 & p . 3 = x3 & p . 4 = x4 & p . 5 = x5 & p . 6 = x6 & p . 7 = x7 & p . 8 = x8 ) )
thus
( p = ((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*> implies ( len p = 8 & p . 1 = x1 & p . 2 = x2 & p . 3 = x3 & p . 4 = x4 & p . 5 = x5 & p . 6 = x6 & p . 7 = x7 & p . 8 = x8 ) )
verumproof
assume A1:
p = ((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>
;
( len p = 8 & p . 1 = x1 & p . 2 = x2 & p . 3 = x3 & p . 4 = x4 & p . 5 = x5 & p . 6 = x6 & p . 7 = x7 & p . 8 = x8 )
set p17 =
(((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>;
A2:
len ((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) = 7
by Th69;
A3:
(
((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) . 1
= x1 &
((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) . 2
= x2 )
by Th69;
A4:
(
((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) . 3
= x3 &
((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) . 4
= x4 )
by Th69;
A5:
(
((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) . 5
= x5 &
((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) . 6
= x6 )
by Th69;
A6:
((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) . 7
= x7
by Th69;
thus len p =
(len ((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>)) + (len <*x8*>)
by A1, Th22
.=
7
+ 1
by A2, Th40
.=
8
;
( p . 1 = x1 & p . 2 = x2 & p . 3 = x3 & p . 4 = x4 & p . 5 = x5 & p . 6 = x6 & p . 7 = x7 & p . 8 = x8 )
A7:
dom ((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) = Seg 7
by A2, Def3;
1
in Seg 7 & ... & 7
in Seg 7
;
hence
(
p . 1
= x1 &
p . 2
= x2 &
p . 3
= x3 &
p . 4
= x4 &
p . 5
= x5 &
p . 6
= x6 &
p . 7
= x7 )
by A1, A3, A4, A5, A6, Def7, A7;
p . 8 = x8
thus p . 8 =
p . ((len ((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>)) + 1)
by A2
.=
x8
by A1, Th42
;
verum
end;