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;
reserve D,D1,D2 for NonatomicND of V,A;

theorem Th33:
  ex n being Nat st D is TypeSSNominativeData of V,A\/FNDSC(V,A).n
  proof
    set F = FNDSC(V,A);
    consider S being FinSequence such that
A1: S IsNDRankSeq V,A and
A2: D in Union S by Def5;
    consider x being object such that
A3: x in dom S and
A4: D in S.x by A2,CARD_5:2;
    reconsider x as Element of NAT by A3;
    reconsider n = x-1 as Element of NAT by A3,FINSEQ_3:25,INT_1:5;
    take n;
A5: F.(n+1) = S.(n+1) by A1,A3,Th19;
    F.(n+1) = NDSS(V,A\/F.n) by Def3;
    hence thesis by A4,A5,Th4;
  end;
