consider f being Function such that
A1: dom f = NAT and
A2: ( f . 0 = F2() & f . 1 = F3() & f . 2 = F4() ) and
A3: for n being Nat holds f . (n + 3) = F5(n,(f . n),(f . (n + 1)),(f . (n + 2))) from rng f c= F1()
proof
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng f or y in F1() )
assume y in rng f ; :: thesis: y in F1()
then consider n being object such that
A4: n in dom f and
A5: f . n = y by FUNCT_1:def 3;
reconsider n = n as Nat by A1, A4;
per cases ( n <= 2 or n > 2 ) ;
suppose n <= 2 ; :: thesis: y in F1()
then not not n = 0 & ... & not n = 2 ;
hence y in F1() by A2, A5; :: thesis: verum
end;
suppose n > 2 ; :: thesis: y in F1()
then 2 + 1 <= n by NAT_1:13;
then n - 3 in NAT by INT_1:5;
then f . ((n - 3) + 3) = F5((n - 3),(f . (n - 3)),(f . ((n - 3) + 1)),(f . ((n - 3) + 2))) by A3;
hence y in F1() by A5; :: thesis: verum
end;
end;
end;
then f is sequence of F1() by ;
hence ex f being sequence of F1() st
( f . 0 = F2() & f . 1 = F3() & f . 2 = F4() & ( for n being Nat holds f . (n + 3) = F5(n,(f . n),(f . (n + 1)),(f . (n + 2))) ) ) by A2, A3; :: thesis: verum