reserve n, k, r, m, i, j for Nat;

theorem Th19:
  for q being FinSubsequence, k, n being Element of NAT st dom q
  c= Seg k & n > k ex p being FinSequence st q c= p & dom p = Seg n
proof
  let q be FinSubsequence, k, n be Element of NAT;
  assume that
A1: dom q c= Seg k and
A2: n > k;
  reconsider IK = id Seg n as Function;
  set IS = IK +* q;
A3: Seg k c= Seg n by A2,FINSEQ_1:5;
A4: dom IS = dom IK \/ dom q by FUNCT_4:def 1
    .= Seg n \/ dom q
    .= Seg n by A1,A3,XBOOLE_1:1,12;
  then reconsider IS as FinSequence by FINSEQ_1:def 2;
  q c= IS by FUNCT_4:25;
  hence thesis by A4;
end;
