set p = f /^ n;

per cases ( n <= len f or len f < n ) ;

suppose A1: n <= len f ; :: thesis: f /^ n is FinSequence of D

then reconsider k = (len f) - n as Element of NAT by NAT_1:21;
A2: len (f /^ n) = k by A1, Def1;
rng (f /^ n) c= rng f

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (f /^ n) or x in rng f )
assume x in rng (f /^ n) ; :: thesis: x in rng f
then consider m being Nat such that
A3: m in dom (f /^ n) and
A4: (f /^ n) . m = x by FINSEQ_2:10;
m <= len (f /^ n) by A3, FINSEQ_3:25;
then A5: m + n <= len f by A2, XREAL_1:19;
A6: m <= m + n by NAT_1:11;
1 <= m by A3, FINSEQ_3:25;
then 1 <= m + n by A6, XXREAL_0:2;
then A7: m + n in dom f by A5, FINSEQ_3:25;
(f /^ n) . m = f . (m + n) by A1, A3, Def1;
hence x in rng f by A4, A7, FUNCT_1:def 3; :: thesis: verum

end;

then rng (f /^ n) c= D by XBOOLE_1:1;
hence f /^ n is FinSequence of D by FINSEQ_1:def 4; :: thesis: verum

end;

suppose len f < n ; :: thesis: f /^ n is FinSequence of D

then f /^ n = <*> D by Def1;
hence f /^ n is FinSequence of D ; :: thesis: verum

end;

end;