let i, k be Nat; :: thesis: for D being non empty set
for f being FinSequence of D st k < i & i in dom f holds
f /. i in rng (f /^ k)
let D be non empty set ; :: thesis: for f being FinSequence of D st k < i & i in dom f holds
f /. i in rng (f /^ k)
let f be FinSequence of D; :: thesis: ( k < i & i in dom f implies f /. i in rng (f /^ k) )
assume that
A1: k < i and
A2: i in dom f ; :: thesis: f /. i in rng (f /^ k)
reconsider j = i - k as Element of NAT by A1, INT_1:5;
j > 0 by A1, XREAL_1:50;
then A3: 1 <= j by NAT_1:14;
A4: i = j + k ;
A5: i <= len f by A2, FINSEQ_3:25;
then k <= len f by A1, XXREAL_0:2;
then len (f /^ k) = (len f) - k by RFINSEQ:def 1;
then (len (f /^ k)) + k = len f ;
then j <= len (f /^ k) by A4, A5, XREAL_1:6;
then A6: j in dom (f /^ k) by A3, FINSEQ_3:25;
then f /. i = (f /^ k) /. j by A4, FINSEQ_5:27;
hence f /. i in rng (f /^ k) by A6, PARTFUN2:2; :: thesis: verum