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
  {v,v1,v2} c= V & v <> v1 & a2 in A &
  not v1.-->a1 in A & not v.-->(v2.-->a2) in A &
  a1 is TypeSCNominativeData of V,A implies
  local_overlapping(V,A,v1.-->a1,v2.-->a2,v) = (v,v1)-->(v2.-->a2,a1)
  proof
    set d1 = v1.-->a1;
    set d2 = v2.-->a2;
    assume that
A1: {v,v1,v2} c= V and
A2: v <> v1 and
A3: a2 in A and
A4: not d1 in A and
A5: not v.-->d2 in A and
A6: a1 is TypeSCNominativeData of V,A;
A7: v in {v,v1,v2} by ENUMSET1:def 1;
    v2 in {v,v1,v2} by ENUMSET1:def 1;
    then
A8: ND_ex_1(v2,a2) is TypeSCNominativeData of V,A by A1,A3,Th46;
    then
A9: naming(V,A,v,d2) = v.-->d2 by A1,A7,Def13;
    {v1,v} c= V
    proof
      let x;
      assume x in {v1,v};
      then x = v1 or x = v by TARSKI:def 2;
      then x in {v,v1,v2} by ENUMSET1:def 1;
      hence thesis by A1;
    end;
    hence thesis by A4,A5,A6,A2,A8,A9,Th68;
  end;
