reserve D for non empty set;
reserve f1,f2,f3,f4,f5 for BinominativeFunction of D;
reserve p,q,r,t,w,u for PartialPredicate of D;
reserve d,v,v1 for object;
reserve V,A for set;
reserve i,j,b,n,s,z for Element of V;
reserve i1,j1,b1,n1,s1 for object;
reserve d1,Li,Lj,Lb,Ln,Ls for NonatomicND of V,A;
reserve Di,Dj,Db,Dn,Ds for SCBinominativeFunction 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;
reserve b0 for Complex;

theorem
  V is non empty & A is complex-containing & A is_without_nonatomicND_wrt V &
  loc/.1, loc/.2, loc/.3, loc/.4, loc/.5 are_mutually_distinct &
  loc,val are_compatible_wrt_5_locs &
  (for T holds loc/.1 is_a_value_on T) & (for T holds loc/.4 is_a_value_on T)
  implies
  <* valid_power_input(V,A,val,b0,n0),
     power_program(A,loc,val,z),
     valid_power_output(A,z,b0,n0) *>
  is SFHT of ND(V,A)
  proof
    set i = loc/.1, j = loc/.2, b = loc/.3, n = loc/.4, s = loc/.5;
    set D = ND(V,A);
    set p = valid_power_input(V,A,val,b0,n0);
    set f = power_main_part(A,loc,val);
    set g = SC_assignment(denaming(V,A,s),z);
    set q = valid_power_output(A,z,b0,n0);
    set inv = power_inv(A,loc,b0,n0);
    set E = Equality(A,i,n);
    assume that
A1: V is non empty & A is complex-containing & A is_without_nonatomicND_wrt V
    & i,j,b,n,s are_mutually_distinct and
A2: loc,val are_compatible_wrt_5_locs and
A3: (for T holds i is_a_value_on T) & (for T holds n is_a_value_on T);
A4: <*p,f,PP_and(E,inv)*> is SFHT of D by A1,A2,Th8;
A5: <*PP_and(E,inv),g,q*> is SFHT of D by A1,A3,Th10;
    <*PP_inversion(PP_and(E,inv)),g,q*> is SFHT of D by A3,Th11;
    hence thesis by A4,A5,NOMIN_3:25;
  end;
