reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
reconsider T = Table2 (m,e,f,n) as Element of NAT ;
defpred S₁[ Nat, set , set ] means ex i1, i2 being Nat st
( i1 = $2 & i2 = $3 & i2 = (((i1 |^ (Radix k)) mod f) * (Table2 (m,e,f,(n1 -' $1)))) mod f );
A2: for i being Nat st 1 <= i & i < n1 holds
for x being Element of NAT ex y being Element of NAT st S₁[i,x,y]

let i be Nat; :: thesis:
assume that
1 <= i and
i < n1 ; :: thesis:
let x be Element of NAT ; :: thesis:
reconsider x = x as Element of NAT ;
consider y being Element of NAT such that
A3: y = (((x |^ (Radix k)) mod f) * (Table2 (m,e,f,(n -' i)))) mod f ;
reconsider z = y as Element of NAT ;
take z ; :: thesis:
thus S₁[i,x,z] by A3; :: thesis:

end;

consider r being FinSequence of NAT such that
A4: ( len r = n1 & ( r . 1 = T or n1 = 0 ) ) and
A5: for i being Nat st 1 <= i & i < n1 holds
S₁[i,r . i,r . (i + 1)] from RECDEF_1:sch 4(A2);
reconsider r = r as Tuple of n, NAT by A4, CARD_1:def 7;
take r ; :: thesis:
thus r . 1 = Table2 (m,e,f,n) by A1, A4; :: thesis:
let i be Nat; :: thesis:
assume A6: ( 1 <= i & i <= n - 1 ) ; :: thesis:
thus ex i1, i2 being Nat st
( i1 = r . i & i2 = r . (i + 1) & i2 = (((i1 |^ (Radix k)) mod f) * (Table2 (m,e,f,(n -' i)))) mod f ) by A5, A6, XREAL_1:147; :: thesis: