reserve a,a1,a2,v,v1,v2,x for object;
reserve V,A for set;
reserve m,n for Nat;
reserve S,S1,S2 for FinSequence;

theorem Th15:
  S IsNDRankSeq V,A & S1 c= S & S1 <> {} implies S1 IsNDRankSeq V,A
  proof
    assume that
A1: S IsNDRankSeq V,A and
A2: S1 c= S and
A3: S1 <> {};
A4: dom S1 c= dom S by A2,XTUPLE_0:8;
    rng S1 <> {} by A3;
    then 1 in dom S1 by FINSEQ_3:32;
    hence S1.1 = NDSS(V,A) by A1,A2,GRFUNC_1:2;
    let n be Nat such that
A5: n in dom S1 and
A6: n+1 in dom S1;
    S1.n = S.n by A2,A5,GRFUNC_1:2;
    hence NDSS(V,A\/S1.n) = S.(n+1) by A1,A4,A5,A6
    .= S1.(n+1) by A2,A6,GRFUNC_1:2;
  end;
