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 Th68:
  {v1,v2} c= V & v1 <> v2 & not v1.-->a1 in A & not v2.-->a2 in A &
  a1 is TypeSCNominativeData of V,A & a2 is TypeSCNominativeData of V,A
  implies
  global_overlapping(V,A,v1.-->a1,v2.-->a2) = (v2,v1)-->(a2,a1)
  proof
    set D1 = v1.-->a1;
    set D2 = v2.-->a2;
    assume that
A1: {v1,v2} c= V and
A2: v1 <> v2 and
A3: not D1 in A & not D2 in A and
A4: a1 is TypeSCNominativeData of V,A &
    a2 is TypeSCNominativeData of V,A;
    v1 in V & v2 in V by A1,ZFMISC_1:32;
    then
A5: naming(V,A,v1,a1) = v1.-->a1 & naming(V,A,v2,a2) = v2.-->a2 by A4,Def13;
A6: {v1}\{v2} = {v1} by A2,ZFMISC_1:14;
    {v1} misses {v2} by A2,ZFMISC_1:11;
    hence (v2,v1)-->(a2,a1) = D2 \/ (D1|(dom(D1)\dom(D2)))
    by A6,FUNCT_4:30,PARTFUN1:56
    .= global_overlapping(V,A,D1,D2) by A3,A5,Th64;
  end;
