reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;

theorem Th58:
  k < i & i in dom f implies f/.i in rng(f/^k)
proof
  assume that
A1: k < i and
A2: i in dom f;
  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 thesis by A6,PARTFUN2:2;
end;
