let p be XFinSequence; :: thesis: for x1, x2, x3, x4, x5, x6, x7 being set st p = (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) ^ <%x7%> holds
( len p = 7 & p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 & p . 4 = x5 & p . 5 = x6 & p . 6 = x7 )
let x1, x2, x3, x4, x5, x6, x7 be set ; :: thesis: ( p = (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) ^ <%x7%> implies ( len p = 7 & p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 & p . 4 = x5 & p . 5 = x6 & p . 6 = x7 ) )
assume A1: p = (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) ^ <%x7%> ; :: thesis: ( len p = 7 & p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 & p . 4 = x5 & p . 5 = x6 & p . 6 = x7 )
set p16 = ((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>;
A2: len (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) = 6 by Th51;
A3: ( (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) . 0 = x1 & (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) . 1 = x2 ) by Th51;
A4: ( (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) . 2 = x3 & (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) . 3 = x4 ) by Th51;
A5: ( (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) . 4 = x5 & (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) . 5 = x6 ) by Th51;
thus len p = (len (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>)) + (len <%x7%>) by A1, Def4
.= 6 + 1 by A2, Th36
.= 7 ; :: thesis: ( p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 & p . 4 = x5 & p . 5 = x6 & p . 6 = x7 )
( 0 in 6 & 1 in 6 & 2 in 6 & 3 in 6 & 4 in 6 & 5 in 6 ) by CARD_1:54, ENUMSET1:def 4;
hence ( p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 & p . 4 = x5 & p . 5 = x6 ) by A1, A3, A4, A5, Def4, A2; :: thesis: p . 6 = x7
thus p . 6 = p . (len (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>)) by Th51
.= x7 by A1, Th40 ; :: thesis: verum