theorem
  for m being Nat for D being set, p being FinSequence of D st
    a in dom(p|m) holds (p|m qua FinSequence of D)/.a = p/.a
proof
  let m be Nat;
  let D be set, p be FinSequence of D;
  assume
A1: a in dom(p|m);
  then a in dom p /\ (Seg m) by RELAT_1:61;
  then
A2: a in dom p by XBOOLE_0:def 4;
  thus (p|m)/.a = (p|Seg m).a by A1,PARTFUN1:def 6
    .= p.a by A1,FUNCT_1:47
    .= p/.a by A2,PARTFUN1:def 6;
end;
