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 Th10:
  V is non empty & A is_without_nonatomicND_wrt V &
  (for T holds loc/.1 is_a_value_on T) & (for T holds loc/.4 is_a_value_on T)
  implies
  <* PP_and(Equality(A,loc/.1,loc/.4),power_inv(A,loc,b0,n0)),
     SC_assignment(denaming(V,A,loc/.5),z),
     valid_power_output(A,z,b0,n0) *> is SFHT of ND(V,A)
  proof
    set s = loc/.5;
    <*SC_Psuperpos(valid_power_output(A,z,b0,n0),denaming(V,A,s),z),
      SC_assignment(denaming(V,A,s),z),
      valid_power_output(A,z,b0,n0)*> is SFHT of ND(V,A) by NOMIN_3:29;
    hence thesis by Th9,NOMIN_3:15;
  end;
