reserve v,x for object;
reserve D,V,A for set;
reserve n for Nat;
reserve p,q for PartialPredicate of D;
reserve f,g for BinominativeFunction of D;
reserve D for non empty set;
reserve d for Element of D;
reserve f,g for BinominativeFunction of D;
reserve p,q,r,s for PartialPredicate of D;
reserve p,q for SCPartialNominativePredicate of V,A;
reserve f,g for SCBinominativeFunction of V,A;
reserve E for (V,A)-FPrg-yielding FinSequence;
reserve e for Element of product E;
reserve d for TypeSCNominativeData of V,A;

theorem
  product E <> {} implies
  <*SC_Psuperpos(p,e,E),SC_Fsuperpos(PPid(ND(V,A)),e,E),p*> is SFHT of ND(V,A)
  proof
    assume
A1: product E <> {};
    set I = PPid(ND(V,A));
    set P = SC_Psuperpos(p,e,E);
    set F = SC_Fsuperpos(I,e,E);
    for d holds d in dom P & P.d = TRUE & d in dom F & F.d in dom p implies
     p.(F.d) = TRUE
    proof
      let d such that
A2:   d in dom P and
A3:   P.d = TRUE and
A4:   d in dom F and
      F.d in dom p;
      set X = E;
      set o = global_overlapping(V,A,d,NDentry(E,X,d));
      o in ND(V,A);
      then o = I.o by FUNCT_1:18
      .= F.d by A1,A4,NOMIN_2:37;
      hence p.(F.d) = TRUE by A1,A2,A3,NOMIN_2:34;
    end;
    hence thesis by Th27;
  end;
