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 Th49:
  {v,v1} c= V & a1 in A implies ND_ex_2(v,v1,a1) is NonatomicND of V,A
  proof
    assume
A1: {v,v1} c= V & a1 in A;
    take S = <*NDSS(V,A),NDSS(V,A\/NDSS(V,A))*>;
    thus S IsNDRankSeq V,A by Th28;
A2: Union S = NDSS(V,A) \/ NDSS(V,A\/NDSS(V,A)) by FINSEQ_3:136;
    ND_ex_2(v,v1,a1) in NDSS(V,A\/NDSS(V,A)) by A1,Th48;
    hence thesis by A2,XBOOLE_0:def 3;
  end;
