reconsider T = Table1 q,c,f,n as Element of INT by INT_1:def 2;
reconsider n1 = n as Element of NAT by ORDINAL1:def 13;
defpred S1[ Element of NAT , set , set ] means ex i1, i2 being Integer st
( i1 = $2 & i2 = $3 & i2 = (((Radix k) * i1) + (Table1 q,c,f,(n -' $1))) mod f );
A2:
for i being Element of NAT st 1 <= i & i < n1 holds
for x being Element of INT ex y being Element of INT st S1[i,x,y]
proof
let i be
Element of
NAT ;
:: thesis: ( 1 <= i & i < n1 implies for x being Element of INT ex y being Element of INT st S1[i,x,y] )
assume
( 1
<= i &
i < n1 )
;
:: thesis: for x being Element of INT ex y being Element of INT st S1[i,x,y]
let x be
Element of
INT ;
:: thesis: ex y being Element of INT st S1[i,x,y]
reconsider x =
x as
Integer ;
consider y being
Integer such that A3:
y = (((Radix k) * x) + (Table1 q,c,f,(n -' i))) mod f
;
reconsider z =
y as
Element of
INT by INT_1:def 2;
take
z
;
:: thesis: S1[i,x,z]
thus
S1[
i,
x,
z]
by A3;
:: thesis: verum
end;
consider r being FinSequence of INT such that
A4:
( len r = n1 & ( r . 1 = T or n1 = 0 ) )
and
A5:
for i being Element of 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,INT by A4, FINSEQ_2:110;
take
r
; :: thesis: ( r . 1 = Table1 q,c,f,n & ( for i being Nat st 1 <= i & i <= n - 1 holds
ex I1, I2 being Integer st
( I1 = r . i & I2 = r . (i + 1) & I2 = (((Radix k) * I1) + (Table1 q,c,f,(n -' i))) mod f ) ) )
thus
r . 1 = Table1 q,c,f,n
by A1, A4; :: thesis: for i being Nat st 1 <= i & i <= n - 1 holds
ex I1, I2 being Integer st
( I1 = r . i & I2 = r . (i + 1) & I2 = (((Radix k) * I1) + (Table1 q,c,f,(n -' i))) mod f )
let i be Nat; :: thesis: ( 1 <= i & i <= n - 1 implies ex I1, I2 being Integer st
( I1 = r . i & I2 = r . (i + 1) & I2 = (((Radix k) * I1) + (Table1 q,c,f,(n -' i))) mod f ) )
A6:
i in NAT
by ORDINAL1:def 13;
assume
( 1 <= i & i <= n - 1 )
; :: thesis: ex I1, I2 being Integer st
( I1 = r . i & I2 = r . (i + 1) & I2 = (((Radix k) * I1) + (Table1 q,c,f,(n -' i))) mod f )
then
( 1 <= i & i < n )
by XREAL_1:149;
hence
ex I1, I2 being Integer st
( I1 = r . i & I2 = r . (i + 1) & I2 = (((Radix k) * I1) + (Table1 q,c,f,(n -' i))) mod f )
by A5, A6; :: thesis: verum