reserve D for non empty set;
reserve m,n,N for Nat;
reserve size for non zero Nat;
reserve f1,f2,f3,f4,f5,f6 for BinominativeFunction of D;
reserve p1,p2,p3,p4,p5,p6,p7 for PartialPredicate of D;
reserve d,v for object;
reserve V,A for set;
reserve z for Element of V;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve T for TypeSCNominativeData of V,A;
reserve n0 for Nat;

theorem Th19:
  for val being 6-element FinSequence holds
  V is non empty & A is complex-containing & A is_without_nonatomicND_wrt V &
   (for T holds loc/.1 is_a_value_on T & loc/.2 is_a_value_on T &
    loc/.4 is_a_value_on T & loc/.6 is_a_value_on T) &
   Seg 6 c= dom loc & loc|Seg 6 is one-to-one & loc,val are_different_wrt 6
  implies
  <* valid_Fibonacci_input(V,A,val,n0),
     Fibonacci_main_part(A,loc,val),
     PP_and(Equality(A,loc/.1,loc/.3),Fibonacci_inv(A,loc,n0)) *>
  is SFHT of ND(V,A)
  proof
    let val be 6-element FinSequence;
    set i = loc/.1, j = loc/.2, n = loc/.3, s = loc/.4, b = loc/.5, c = loc/.6;
    set i1 = val.1, j1 = val.2, n1 = val.3, s1 = val.4, b1 = val.5, c1 = val.6;
    set D = ND(V,A);
    set p = valid_Fibonacci_input(V,A,val,n0);
    set f = initial_assignments(A,loc,val,6);
    set g = Fibonacci_main_loop(A,loc);
    set q = Fibonacci_inv(A,loc,n0);
    set r = PP_and(Equality(A,i,n),Fibonacci_inv(A,loc,n0));
    assume that
A1: V is non empty & A is complex-containing & A is_without_nonatomicND_wrt V &
    (for T holds i is_a_value_on T & j is_a_value_on T &
    s is_a_value_on T & c is_a_value_on T) and
A2: Seg 6 c= dom loc and
A3: loc|Seg 6 is one-to-one and
A4: loc,val are_different_wrt 6;
A5: <*p,f,q*> is SFHT of D by A1,A2,A3,A4,Th16;
A6: <*q,g,r*> is SFHT of D by A1,A2,A3,Th18;
    <*PP_inversion(q),g,r*> is SFHT of D by NOMIN_3:19;
    hence thesis by A5,A6,NOMIN_3:25;
  end;
