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)/.(p..f) = p
proof
  assume
A1: p in rng f;
  then
A2: p..f in dom f by FINSEQ_4:20;
  1 <= p..f by A1,FINSEQ_4:21;
  then p..f in Seg(p..f);
  hence (f-:p)/.(p..f) = f/.(p..f) by A1,Th43
    .= f.(p..f) by A2,PARTFUN1:def 6
    .= p by A1,FINSEQ_4:19;
end;
