reserve v for object;
reserve V,A for set;
reserve f for SCBinominativeFunction of V,A;
reserve d for TypeSCNominativeData of V,A;
reserve d1 for NonatomicND of V,A;
reserve a,b,c,z for Element of V;
reserve x,y for object;
reserve p,q,r,s for SCPartialNominativePredicate of V,A;
reserve x0,y0 for Nat;

theorem Th15:
  <*PP_inversion(SC_Psuperpos(p,denaming(V,A,x),a)),
    SC_assignment(denaming(V,A,x),a),
    p*>
  is SFHT of ND(V,A)
  proof
    set Dx = denaming(V,A,x);
    set F = SC_assignment(Dx,a);
    set P = SC_Psuperpos(p,Dx,a);
    set I = PP_inversion(P);
    for d st d in dom I & I.d = TRUE & d in dom F & F.d in dom p holds
     p.(F.d) = TRUE
    proof
      let d such that
A1:   d in dom I and
      I.d = TRUE and
A2:   d in dom F and
A3:   F.d in dom p;
      dom I = {d where d is Element of ND(V,A): not d in dom P}
      by PARTPR_2:def 17;
      then
A4:   ex d1 being Element of ND(V,A) st d1 = d & not d1 in dom P by A1;
      dom F = dom Dx by NOMIN_2:def 7;
      then P,d =~ p,local_overlapping(V,A,d,Dx.d,a) by A2,NOMIN_2:def 11;
      hence thesis by A2,A3,A4,NOMIN_2:def 7;
    end;
    hence thesis by NOMIN_3:28;
  end;
