reserve D for non empty set;
reserve f1,f2,f3,f4,f5 for BinominativeFunction of D;
reserve p,q,r,t,w,u for PartialPredicate of D;
reserve d,v,v1 for object;
reserve V,A for set;
reserve i,j,b,n,s,z for Element of V;
reserve i1,j1,b1,n1,s1 for object;
reserve d1,Li,Lj,Lb,Ln,Ls for NonatomicND of V,A;
reserve Di,Dj,Db,Dn,Ds for SCBinominativeFunction of V,A;

theorem Th2:
  V is non empty & A is_without_nonatomicND_wrt V &
  Di = denaming(V,A,i1) & Dj = denaming(V,A,j1) &
  Db = denaming(V,A,b1) & Dn = denaming(V,A,n1) &
  Ds = denaming(V,A,s1) &
  Li = local_overlapping(V,A,d1,Di.d1,i) &
  Lj = local_overlapping(V,A,Li,Dj.Li,j) &
  Lb = local_overlapping(V,A,Lj,Db.Lj,b) &
  Ln = local_overlapping(V,A,Lb,Dn.Lb,n) &
  Ls = local_overlapping(V,A,Ln,Ds.Ln,s) &
  j1 in dom d1 & b1 in dom d1 & n1 in dom d1 & s1 in dom d1 &
  d1 in dom Di & s <> n implies Ls.n = Ln.n
  proof
    assume that
A1: V is non empty and
A2: A is_without_nonatomicND_wrt V and
A3: Di = denaming(V,A,i1) & Dj = denaming(V,A,j1) &
    Db = denaming(V,A,b1) & Dn = denaming(V,A,n1) &
    Ds = denaming(V,A,s1) &
    Li = local_overlapping(V,A,d1,Di.d1,i) &
    Lj = local_overlapping(V,A,Li,Dj.Li,j) &
    Lb = local_overlapping(V,A,Lj,Db.Lj,b) &
    Ln = local_overlapping(V,A,Lb,Dn.Lb,n) &
    Ls = local_overlapping(V,A,Ln,Ds.Ln,s) and
A4: j1 in dom d1 and
A5: b1 in dom d1 and
A6: n1 in dom d1 and
A7: s1 in dom d1 and
A8: d1 in dom Di and
A9: s <> n;
A10: dom Dj = {d where d is NonatomicND of V,A: j1 in dom d}
     by A3,NOMIN_1:def 18;
A11:  dom Db = {d where d is NonatomicND of V,A: b1 in dom d}
     by A3,NOMIN_1:def 18;
A12: dom Dn = {d where d is NonatomicND of V,A: n1 in dom d}
     by A3,NOMIN_1:def 18;
A13: dom Ds = {d where d is NonatomicND of V,A: s1 in dom d}
     by A3,NOMIN_1:def 18;
     Di.d1 is TypeSCNominativeData of V,A by A8,PARTFUN1:4,NOMIN_1:39;
     then
A14: dom Li = {i} \/ dom d1 by A1,A2,A3,NOMIN_4:4;
     j1 in dom Li by A4,A14,XBOOLE_0:def 3;
     then Li in dom Dj by A10;
     then Dj.Li is TypeSCNominativeData of V,A by PARTFUN1:4,NOMIN_1:39;
     then
A15: dom Lj = {j} \/ dom Li by A1,A2,A3,NOMIN_4:4;
     b1 in dom Li by A5,A14,XBOOLE_0:def 3;
     then b1 in dom Lj by A15,XBOOLE_0:def 3;
     then Lj in dom Db by A11;
     then Db.Lj is TypeSCNominativeData of V,A by PARTFUN1:4,NOMIN_1:39;
     then
A16: dom Lb = {b} \/ dom Lj by A1,A2,A3,NOMIN_4:4;
     n1 in dom Li by A6,A14,XBOOLE_0:def 3;
     then n1 in dom Lj by A15,XBOOLE_0:def 3;
     then n1 in dom Lb by A16,XBOOLE_0:def 3;
     then Lb in dom Dn by A12;
     then Dn.Lb is TypeSCNominativeData of V,A by PARTFUN1:4, NOMIN_1:39;
     then
A17: dom Ln = {n} \/ dom Lb by A1,A2,A3,NOMIN_4:4;
     s1 in dom Li by A7,A14,XBOOLE_0:def 3;
     then s1 in dom Lj by A15,XBOOLE_0:def 3;
     then s1 in dom Lb by A16,XBOOLE_0:def 3;
     then s1 in dom Ln by A17,XBOOLE_0:def 3;
     then
A18: Ln in dom Ds by A13;
     n in {n} by TARSKI:def 1;
     then
A19: n in dom Ln by A17,XBOOLE_0:def 3;
     Ds.Ln is TypeSCNominativeData of V,A by A18,PARTFUN1:4,NOMIN_1:39;
     hence Ls.n = Ln.n by A1,A2,A9,A19,A3,NOMIN_5:3;
   end;
