deffunc H₁( Nat) -> Element of NAT = IFEQ ((x /. $1),FALSE,0,(2 to_power ($1 -' 1)));
consider z being FinSequence of NAT such that
A1: len z = n and
A2: for j being Nat st j in dom z holds
z . j = H₁(j) from FINSEQ_2:sch 1();
A3: dom z = Seg n by A1, FINSEQ_1:def 3;
reconsider z = z as Tuple of n, NAT by A1, CARD_1:def 7;
take z ; :: thesis: for i being Nat st i in Seg n holds
z /. i = IFEQ ((x /. i),FALSE,0,(2 to_power (i -' 1)))
let j be Nat; :: thesis: ( j in Seg n implies z /. j = IFEQ ((x /. j),FALSE,0,(2 to_power (j -' 1))) )
assume A4: j in Seg n ; :: thesis: z /. j = IFEQ ((x /. j),FALSE,0,(2 to_power (j -' 1)))
then j in dom z by A1, FINSEQ_1:def 3;
hence z /. j = z . j by PARTFUN1:def 6
.= IFEQ ((x /. j),FALSE,0,(2 to_power (j -' 1))) by A2, A3, A4 ;
:: thesis: verum