reserve D for non empty set;
reserve f1,f2,f3,f4 for BinominativeFunction of D;
reserve p,q,r,t,w for PartialPredicate of D;
reserve d,v,v1 for object;
reserve V,A for set;
reserve z for Element of V;
reserve d1 for NonatomicND of V,A;
reserve f for SCBinominativeFunction of V,A;
reserve T for TypeSCNominativeData of V,A;
reserve loc for V-valued Function;
reserve val for Function;
reserve n0 for Nat;

theorem Th9:
  V is non empty & A is complex-containing & A is_without_nonatomicND_wrt V &
  loc/.1, loc/.2, loc/.3, loc/.4 are_mutually_distinct
  implies
  <* factorial_inv(A,loc,n0),
     factorial_main_loop(A,loc),
     PP_and(Equality(A,loc/.1,loc/.3),factorial_inv(A,loc,n0)) *>
  is SFHT of ND(V,A)
  proof
    set i = loc/.1, j = loc/.2, n = loc/.3, s = loc/.4;
    set D = ND(V,A);
    set inv = factorial_inv(A,loc,n0);
    set B = factorial_loop_body(A,loc);
    set E = Equality(A,i,n);
    set N = PP_inversion(inv);
    assume V is non empty & A is complex-containing &
    A is_without_nonatomicND_wrt V & i,j,n,s are_mutually_distinct;
    then
A1: <*inv,B,inv*> is SFHT of D by Th7;
    PP_and(PP_not(E),inv) ||= inv by NOMIN_3:3;
    then
A2: <*PP_and(PP_not(E),inv),B,inv*> is SFHT of D by A1,NOMIN_3:15;
A3: <*N,B,inv*> is SFHT of D by NOMIN_3:19;
    PP_and(PP_not(E),N) ||= N by NOMIN_3:3;
    then <*PP_and(PP_not(E),N),B,inv*> is SFHT of D by A3,NOMIN_3:15;
    hence thesis by A2,NOMIN_3:26;
  end;
