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 Th17:
  for V being non empty set
  for loc being V-valued 10-element FinSequence 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/.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
  implies
  <* Lucas_inv(A,loc,x0,y0,p0,q0,n0),
     Lucas_main_loop(A,loc),
     PP_and(Equality(A,loc/.1,loc/.3),Lucas_inv(A,loc,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;
    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 D = ND(V,A);
    set inv = Lucas_inv(A,loc,x0,y0,p0,q0,n0);
    set B = Lucas_loop_body(A,loc);
    set E = Equality(A,i,n);
    set N = PP_inversion(inv);
    assume A is complex-containing &
    A is_without_nonatomicND_wrt V &
    (for T being TypeSCNominativeData of V,A
    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 &  p is_a_value_on T &
    q is_a_value_on T & ps is_a_value_on T & qc is_a_value_on T) &
    loc is one-to-one;
    then <*inv,B,inv*> is SFHT of D by Th16;
    then
A1: <*PP_and(PP_not(E),inv),B,inv*> is SFHT of D by NOMIN_3:3,15;
    <*N,B,inv*> is SFHT of D by NOMIN_3:19;
    then <*PP_and(PP_not(E),N),B,inv*> is SFHT of D by NOMIN_3:3,15;
    hence thesis by A1,NOMIN_3:26;
  end;
