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 Th57:
  p in rng f & p..f > k implies p in rng(f/^k)
proof
  assume that
A1: p in rng f and
A2: p..f > k;
A3: k+p..(f/^k) = p..f by A1,A2,Th56;
  then p..(f/^k) <> 0 by A2;
  then
A4: 1 <= p..(f/^k) by NAT_1:14;
  p..f <= len f by A1,FINSEQ_4:21;
  then k <= len f by A2,XXREAL_0:2;
  then len(f/^k) = len f - k by RFINSEQ:def 1;
  then
A5: len(f/^k) + k = len f;
  k + p..(f/^k) <= len f by A1,A3,FINSEQ_4:21;
  then p..(f/^k) <= len(f/^k) by A5,XREAL_1:6;
  then
A6: p..(f/^k) in dom(f/^k) by A4,FINSEQ_3:25;
  then (f/^k)/.(p..(f/^k)) in rng(f/^k) by PARTFUN2:2;
  then f/.(k+p..(f/^k)) in rng(f/^k) by A6,FINSEQ_5:27;
  hence thesis by A1,A3,FINSEQ_5:38;
end;
