theorem
  for C being 0-based arr_computation of R st R is finite array of O &
  for a st inversions (C.a) <> {} holds succ a in dom C
  holds C is complete
  proof
    let C be 0-based arr_computation of R;
    assume R is finite array of O; then
    C is finite by Th76; then
    reconsider d = dom C as non empty finite Ordinal;
    assume
A1: for a st inversions (C.a) <> {} holds succ a in dom C;
    set u = union d;
    consider n being Nat such that
A2: d = n+1 by NAT_1:6;
    d = Segm(n+1) by A2;
    then
A3: d = succ Segm n by NAT_1:38; then
A4: u = n by ORDINAL2:2;
    inversions (C.u) <> 0 implies d in d by A1,A3,A4;
    hence last C is ascending by Th48;
  end;
