deffunc H_{1}( Nat) -> Element of k -SD_Sub = SDSubAddDigit (x,y,$1,k);

consider z being FinSequence of k -SD_Sub such that

A1: len z = n and

A2: for j being Nat st j in dom z holds

z . j = H_{1}(j)
from FINSEQ_2:sch 1();

A3: dom z = Seg n by A1, FINSEQ_1:def 3;

z is Element of n -tuples_on (k -SD_Sub) by A1, FINSEQ_2:92;

then reconsider z = z as Tuple of n,k -SD_Sub ;

take z ; :: thesis: for i being Nat st i in Seg n holds

DigA_SDSub (z,i) = SDSubAddDigit (x,y,i,k)

let i be Nat; :: thesis: ( i in Seg n implies DigA_SDSub (z,i) = SDSubAddDigit (x,y,i,k) )

assume A4: i in Seg n ; :: thesis: DigA_SDSub (z,i) = SDSubAddDigit (x,y,i,k)

hence DigA_SDSub (z,i) = z . i by RADIX_3:def 5

.= SDSubAddDigit (x,y,i,k) by A2, A3, A4 ;

:: thesis: verum

consider z being FinSequence of k -SD_Sub such that

A1: len z = n and

A2: for j being Nat st j in dom z holds

z . j = H

A3: dom z = Seg n by A1, FINSEQ_1:def 3;

z is Element of n -tuples_on (k -SD_Sub) by A1, FINSEQ_2:92;

then reconsider z = z as Tuple of n,k -SD_Sub ;

take z ; :: thesis: for i being Nat st i in Seg n holds

DigA_SDSub (z,i) = SDSubAddDigit (x,y,i,k)

let i be Nat; :: thesis: ( i in Seg n implies DigA_SDSub (z,i) = SDSubAddDigit (x,y,i,k) )

assume A4: i in Seg n ; :: thesis: DigA_SDSub (z,i) = SDSubAddDigit (x,y,i,k)

hence DigA_SDSub (z,i) = z . i by RADIX_3:def 5

.= SDSubAddDigit (x,y,i,k) by A2, A3, A4 ;

:: thesis: verum