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 Th19:
  S IsNDRankSeq V,A & n in dom S implies S.n = FNDSC(V,A).n
  proof
    assume
A1: S IsNDRankSeq V,A;
    set F = FNDSC(V,A);
    defpred P[Nat] means $1 in dom S implies S.$1 = F.$1;
A2: P[0] by FINSEQ_3:24;
A3: for n st P[n] holds P[n+1]
    proof
      let n such that
A4:   P[n] and
A5:   n+1 in dom S;
      per cases;
      suppose n = 0;
        hence thesis by A1,Th9;
      end;
      suppose n <> 0;
        then
A6:     1 <= n by NAT_1:14;
A7:     n+1 <= len S by A5,FINSEQ_3:25;
        n <= n+1 by NAT_1:11;
        then
A8:     n <= len S by A7,XXREAL_0:2;
        then n in dom S by A6,FINSEQ_3:25;
        hence S.(n+1) = NDSS(V,A\/S.n) by A1,A5
        .= F.(n+1) by A4,A6,A8,Def3,FINSEQ_3:25;
      end;
    end;
    for n holds P[n] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
