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 Th32:
  for d being set st d c= D holds d is NonatomicND of V,A
  proof
    let d be set;
    assume
A1: d c= D;
    then reconsider d as Function;
    consider S being FinSequence such that
A2: S IsNDRankSeq V,A and
A3: D in Union S by Def5;
    consider x such that
A4: x in dom S and
A5: D in S.x by A3,CARD_5:2;
    reconsider x as Element of NAT by A4;
    now
      1 <= x by A4,FINSEQ_3:25;
      then per cases by XXREAL_0:1;
      suppose
A6:     x = 1;
        then D is TypeSSNominativeData of V,A by A2,A5,Th4;
        then d is TypeSSNominativeData of V,A by A1,RELSET_1:1;
        then d in NDSS(V,A);
        hence d in Union S by A2,A4,A6,CARD_5:2;
      end;
      suppose
A7:     x > 1;
        then reconsider n = x-1 as Element of NAT by INT_1:5;
A8:     S.(n+1) = NDSS(V,A\/S.n) by A2,A4,A7,CGAMES_1:20;
        then D is TypeSSNominativeData of V,A\/S.n by A5,Th4;
        then d is TypeSSNominativeData of V,A\/S.n by A1,RELSET_1:1;
        then d in NDSS(V,A\/S.n);
        hence d in Union S by A4,A8,CARD_5:2;
      end;
    end;
    hence thesis by A2,Def5;
  end;
