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 Th51:
  {v,v1} c= V & {a,a1} c= A implies ND_ex_3(v,v1,a,a1) in NDSS(V,A)
  proof
    assume that
A1: {v,v1} c= V and
A2: {a,a1} c= A;
    per cases;
    suppose v = v1;
      then
A3:   ND_ex_3(v,v1,a,a1) = ND_ex_1(v1,a1) by FUNCT_4:81;
      v1 in V & a1 in A by A1,A2,ZFMISC_1:32;
      hence thesis by A3,Th45;
    end;
    suppose v <> v1;
      then {v} misses {v1} by ZFMISC_1:11;
      then
A4:   ND_ex_1(v,a) tolerates ND_ex_1(v1,a1) by FUNCOP_1:87;
      v in V & v1 in V & a in A & a1 in A by A1,A2,ZFMISC_1:32;
      then
A5:   ND_ex_1(v,a) in NDSS(V,A) & ND_ex_1(v1,a1) in NDSS(V,A) by Th45;
      ND_ex_3(v,v1,a,a1) = ND_ex_1(v,a) \/ ND_ex_1(v1,a1) by A4,FUNCT_4:30;
      hence thesis by A4,A5,Th8;
    end;
  end;
