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 Th55:
  {v,v1,v2} c= V & {a,a1} c= A implies
  ND_ex_4(v,v1,v2,a,a1) is NonatomicND of V,A
  proof
    assume
A1: {v,v1,v2} c= V & {a,a1} c= A;
    set R = ND_ex_4(v,v1,v2,a,a1);
    set S1 = NDSS(V,A);
    set S2 = NDSS(V,A\/NDSS(V,A));
    R in S2 by A1,Th54;
    then
A2: R in S1\/S2 by XBOOLE_0:def 3;
    Union <*S1,S2*> = S1\/S2 by FINSEQ_3:136;
    hence thesis by A2,Def5,Th28;
  end;
