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 Th48:
  {v,v1} c= V & a1 in A implies ND_ex_2(v,v1,a1) in NDSS(V,A\/NDSS(V,A))
  proof
    assume that
A1: {v,v1} c= V and
A2: a1 in A;
    reconsider V1 = V, A1 = A as non empty set by A1,A2;
    reconsider v,v1 as Element of V1 by A1,ZFMISC_1:32;
    reconsider a1 as Element of A1 by A2;
    set d = ND_ex_2(v,v1,a1);
A3: dom(v.-->(v1.-->a1)) c= V;
A4: rng d = {v1.-->a1} by FUNCOP_1:88;
    v1.-->a1 in NDSS(V,A);
    then v1.-->a1 in A\/NDSS(V,A) by XBOOLE_0:def 3;
    then rng d c= A\/NDSS(V,A) by A4,ZFMISC_1:31;
    then d is PartFunc of V,A\/NDSS(V,A) by A3,RELSET_1:4;
    hence thesis;
  end;
