let x1, x2, x3, x4, x5, x6, x7, x8, x9 be set ; :: thesis: for p being FinSequence st p = (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) ^ <*x9*> holds
( len p = 9 & p . 1 = x1 & p . 2 = x2 & p . 3 = x3 & p . 4 = x4 & p . 5 = x5 & p . 6 = x6 & p . 7 = x7 & p . 8 = x8 & p . 9 = x9 )
let p be FinSequence; :: thesis: ( p = (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) ^ <*x9*> implies ( len p = 9 & p . 1 = x1 & p . 2 = x2 & p . 3 = x3 & p . 4 = x4 & p . 5 = x5 & p . 6 = x6 & p . 7 = x7 & p . 8 = x8 & p . 9 = x9 ) )
assume A1: p = (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) ^ <*x9*> ; :: thesis: ( len p = 9 & p . 1 = x1 & p . 2 = x2 & p . 3 = x3 & p . 4 = x4 & p . 5 = x5 & p . 6 = x6 & p . 7 = x7 & p . 8 = x8 & p . 9 = x9 )
set p17 = ((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>;
A2: len (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) = 8 by Th70;
A3: ( (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) . 1 = x1 & (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) . 2 = x2 ) by Th70;
A4: ( (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) . 3 = x3 & (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) . 4 = x4 ) by Th70;
A5: ( (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) . 5 = x5 & (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) . 6 = x6 ) by Th70;
A6: ( (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) . 7 = x7 & (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) . 8 = x8 ) by Th70;
thus len p = (len (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>)) + (len <*x9*>) by A1, Th22
.= 8 + 1 by A2, Th40
.= 9 ; :: thesis: ( p . 1 = x1 & p . 2 = x2 & p . 3 = x3 & p . 4 = x4 & p . 5 = x5 & p . 6 = x6 & p . 7 = x7 & p . 8 = x8 & p . 9 = x9 )
A7: dom (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>) = Seg 8 by A2, Def3;
1 in Seg 8 & ... & 8 in Seg 8 ;
hence ( p . 1 = x1 & p . 2 = x2 & p . 3 = x3 & p . 4 = x4 & p . 5 = x5 & p . 6 = x6 & p . 7 = x7 & p . 8 = x8 ) by A1, A3, A4, A5, A6, Def7, A7; :: thesis: p . 9 = x9
thus p . 9 = p . ((len (((((((<*x1*> ^ <*x2*>) ^ <*x3*>) ^ <*x4*>) ^ <*x5*>) ^ <*x6*>) ^ <*x7*>) ^ <*x8*>)) + 1) by A2
.= x9 by A1, Th42 ; :: thesis: verum