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 implies f |-- p = f/^(p..f)
proof
  assume
A1: p in rng f;
  then
A2: len (f|--p) = len f - p..f by FINSEQ_4:def 6;
A3: p..f <= len f by A1,FINSEQ_4:21;
  then
A4: len (f/^(p..f)) = len f - p..f by RFINSEQ:def 1;
A5: Seg len (f|--p) = dom (f|--p) & Seg len (f/^(p..f)) = dom (f/^(p..f)) by
FINSEQ_1:def 3;
  now
    let i be Nat;
    assume
A6: i in dom (f|--p);
    hence (f|--p).i = f.(i + p..f) by A1,FINSEQ_4:def 6
      .= (f/^(p..f)).i by A2,A3,A4,A5,A6,RFINSEQ:def 1;
  end;
  hence thesis by A2,A4,FINSEQ_2:9;
end;
