reserve i,j,k,n for Nat;
reserve D for non empty set,
  p for Element of D,
  f,g for FinSequence of D;

theorem
  p in rng(f|i) implies p..(f|i) = p..f
proof
A1: dom(f|i) c= dom f by Th18;
  assume
A2: p in rng(f|i);
  then
A3: p..(f|i) in dom(f|i) by FINSEQ_4:20;
  then f/.(p..(f|i)) = (f|i)/.(p..(f|i)) by FINSEQ_4:70
    .= p by A2,Th38;
  then
A4: p..f <= p..(f|i) by A3,A1,Th39;
  p..(f|i) <= len(f|i) by A2,FINSEQ_4:21;
  then
A5: p..f <= len(f|i) by A4,XXREAL_0:2;
A6: rng(f|i) c= rng f by Th19;
  then 1 <= p..f by A2,FINSEQ_4:21;
  then
A7: p..f in dom(f|i) by A5,FINSEQ_3:25;
  then (f|i)/.(p..f) = f/.(p..f) by FINSEQ_4:70
    .= p by A2,A6,Th38;
  then p..(f|i) <= p..f by A7,Th39;
  hence thesis by A4,XXREAL_0:1;
end;
