let k be Nat; :: thesis: ex seq being sequence of F₁() st
for n being Nat holds
( ( n <= k implies seq . n = F₂(k,n) ) & ( n > k implies seq . n = 0. F₁() ) )
defpred S₁[ object , object ] means ex n being Nat st
( n = $1 & ( n <= k implies $2 = F₂(k,n) ) & ( n > k implies $2 = 0. F₁() ) );

A1: now :: thesis: for x being object st x in NAT holds
ex y being object st S₁[x,y]

let x be object ; :: thesis: ( x in NAT implies ex y being object st S₁[x,y] )
assume x in NAT ; :: thesis: ex y being object st S₁[x,y]
then consider n being Nat such that
A2: n = x ;
reconsider y = (CHK (n,k)) * F₂(k,n) as object ;
A3: ( n > k implies (CHK (n,k)) * F₂(k,n) = 0. F₁() ) by CLVECT_1:1, SIN_COS:def 1;
take y = y; :: thesis: S₁[x,y]

now :: thesis: ( n <= k implies (CHK (n,k)) * F₂(k,n) = F₂(k,n) )

assume n <= k ; :: thesis: (CHK (n,k)) * F₂(k,n) = F₂(k,n)
hence (CHK (n,k)) * F₂(k,n) = 1r * F₂(k,n) by COMPLEX1:def 4, SIN_COS:def 1
.= F₂(k,n) by CLVECT_1:def 5 ;
:: thesis: verum

end;

hence S₁[x,y] by A2, A3; :: thesis: verum

end;

consider f being Function such that
A4: dom f = NAT and
A5: for x being object st x in NAT holds
S₁[x,f . x] from CLASSES1:sch 1(A1);

now :: thesis: for x being object st x in NAT holds
f . x in the carrier of F₁()

let x be object ; :: thesis: ( x in NAT implies f . x in the carrier of F₁() )
assume x in NAT ; :: thesis: f . x in the carrier of F₁()
then ex n being Nat st
( n = x & ( n <= k implies f . x = F₂(k,n) ) & ( n > k implies f . x = 0. F₁() ) ) by A5;
hence f . x in the carrier of F₁() ; :: thesis: verum

end;

then reconsider f = f as sequence of F₁() by A4, FUNCT_2:3;
take seq = f; :: thesis: for n being Nat holds
( ( n <= k implies seq . n = F₂(k,n) ) & ( n > k implies seq . n = 0. F₁() ) )
let n be Nat; :: thesis: ( ( n <= k implies seq . n = F₂(k,n) ) & ( n > k implies seq . n = 0. F₁() ) )
n in NAT by ORDINAL1:def 12;
then ex l being Nat st
( l = n & ( l <= k implies seq . n = F₂(k,l) ) & ( l > k implies seq . n = 0. F₁() ) ) by A5;
hence ( ( n <= k implies seq . n = F₂(k,n) ) & ( n > k implies seq . n = 0. F₁() ) ) ; :: thesis: verum