reserve a,b,c,v,v1,x,y for object;
reserve V,A for set;
reserve d for TypeSCNominativeData of V,A;

theorem Th6:
  for D being finite Function st dom D c= V & rng D c= ND(V,A) holds
  D is NonatomicND of V,A
  proof
    let D be finite Function such that
A1: dom D c= V and
A2: rng D c= ND(V,A);
    defpred P[set] means $1 is NonatomicND of V,A;
A3: D is finite;
A4: P[{}] by NOMIN_1:30;
A5: for x,B being set st x in D & B c= D & P[B] holds P[B \/ {x}]
    proof
      let x,B be set such that
A6:   x in D and
A7:   B c= D;
      assume P[B];
      then reconsider B as NonatomicND of V,A;
      consider a,b such that
A8:   x = [a,b] by A6,RELAT_1:def 1;
A9:   a in dom D by A6,A8,XTUPLE_0:def 12;
      b in rng D by A6,A8,XTUPLE_0:def 13;
      then b is TypeSCNominativeData of V,A by A2,NOMIN_1:39;
      then
A10:  {[a,b]} is NonatomicND of V,A by A1,A9,Th5;
      {x} c= D by A6,ZFMISC_1:31;
      then B \/ {[a,b]} is Function by A7,A8,GRFUNC_1:14;
      hence thesis by A8,A10,NOMIN_1:36,PARTFUN1:51;
    end;
    P[D] from FINSET_1:sch 2(A3,A4,A5);
    hence thesis;
  end;
