reserve x,y,P,Q for Integer;
reserve a,b,n for Nat;
reserve V,A for set;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve p for SCPartialNominativePredicate of V,A;
reserve d for object;
reserve z for Element of V;
reserve T for TypeSCNominativeData of V,A;
reserve size for non zero Nat;
reserve x0, y0, p0, q0 for Integer;
reserve n0 for Nat;

theorem
  for V being non empty set
  for loc being V-valued 10-element FinSequence
  for val being 10-element FinSequence
  for z being Element of V holds
  A is complex-containing & A is_without_nonatomicND_wrt V &
   (for T being TypeSCNominativeData of V,A holds loc/.1 is_a_value_on T &
   loc/.2 is_a_value_on T & loc/.3 is_a_value_on T &
   loc/.4 is_a_value_on T & loc/.6 is_a_value_on T &
   loc/.7 is_a_value_on T & loc/.8 is_a_value_on T &
   loc/.9 is_a_value_on T & loc/.10 is_a_value_on T) &
   loc is one-to-one & loc,val are_different_wrt 10
  implies
  <* valid_Lucas_input(V,A,val,x0,y0,p0,q0,n0),
     Lucas_program(A,loc,val,z),
     valid_Lucas_output(A,z,x0,y0,p0,q0,n0) *>
  is SFHT of ND(V,A)
  proof
    let V be non empty set;
    let loc be V-valued 10-element FinSequence;
    let val be 10-element FinSequence;
    let z be Element of V;
    set i = loc/.1, j = loc/.2, n = loc/.3, s = loc/.4, b = loc/.5, c = loc/.6;
    set p = loc/.7, q = loc/.8, ps = loc/.9, qc = loc/.10;
    set i1 = val.1, j1 = val.2, n1 = val.3, s1 = val.4;
    set D = ND(V,A);
    set P = valid_Lucas_input(V,A,val,x0,y0,p0,q0,n0);
    set F = Lucas_main_part(A,loc,val);
    set G = SC_assignment(denaming(V,A,s),z);
    set Q = valid_Lucas_output(A,z,x0,y0,p0,q0,n0);
    set inv = Lucas_inv(A,loc,x0,y0,p0,q0,n0);
    set E = Equality(A,i,n);
    assume that
A1: A is complex-containing & A is_without_nonatomicND_wrt V
    and
A2: for T being TypeSCNominativeData of V,A
    holds i is_a_value_on T & j is_a_value_on T & n is_a_value_on T &
    s is_a_value_on T & c is_a_value_on T & p is_a_value_on T &
    q is_a_value_on T & ps is_a_value_on T & qc is_a_value_on T;
    assume loc is one-to-one & loc,val are_different_wrt 10;
    then
A3: <*P,F,PP_and(E,inv)*> is SFHT of D by A1,A2,Th18;
A4: <*PP_and(E,inv),G,Q*> is SFHT of D by A1,A2,Th20;
    <*PP_inversion(PP_and(E,inv)),G,Q*> is SFHT of D by A2,Th21;
    hence thesis by A3,A4,NOMIN_3:25;
  end;
