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 Th11:
  (for T holds loc/.1 is_a_value_on T) & (for T holds loc/.4 is_a_value_on T)
  implies
  <* PP_inversion(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 i = loc/.1, j = loc/.2, b = loc/.3, n = loc/.4, s = loc/.5;
    set D = ND(V,A);
    set inv = power_inv(A,loc,b0,n0);
    set O = valid_power_output(A,z,b0,n0);
    set Di = denaming(V,A,i);
    set Dn = denaming(V,A,n);
    set Ds = denaming(V,A,s);
    set g = SC_assignment(Ds,z);
    set E = Equality(A,i,n);
    set q = PP_and(E,inv);
    set P = PP_inversion(q);
    assume
A1: (for T holds i is_a_value_on T) & (for T holds n is_a_value_on T);
    now
      let d be TypeSCNominativeData of V,A such that
A2:   d in dom P and
      P.d = TRUE and
      d in dom g and
      g.d in dom O;
A3:   dom q = {d where d is Element of D:
      d in dom E & E.d = FALSE or d in dom inv & inv.d = FALSE
      or d in dom E & E.d = TRUE & d in dom inv & inv.d = TRUE} by PARTPR_1:16;
A4:   dom Di = {d where d is NonatomicND of V,A: i in dom d} by NOMIN_1:def 18;
A5:   dom Dn = {d where d is NonatomicND of V,A: n in dom d} by NOMIN_1:def 18;
A6:   dom E = dom Di /\ dom Dn by A1,NOMIN_4:11;
A7:   not d in dom q by A2,PARTPR_2:9;
      dom E c= dom q by PARTPR_2:3;
      then not d in dom E by A2,PARTPR_2:9;
      then
A8:   not d in dom Di or not d in dom Dn by A6,XBOOLE_0:def 4;
      dom inv = D by Def12;
      then
A9:   d in dom inv;
      then inv.d <> FALSE by A3,A7;
      then power_inv_pred A,loc,b0,n0,d by A9,Def12;
      then consider d1 being NonatomicND of V,A such that
A10:  d = d1 & {i,j,b,n,s} c= dom d1 and
      d1.(loc/.2) = 1 & d1.(loc/.3) = b0 & d1.(loc/.4) = n0 &
      ex S being Complex, I being Nat st I = d1.(loc/.1)
      & S = d1.(loc/.5) & S = b0|^I;
      i in {i,j,b,n,s} & n in {i,j,b,n,s} by ENUMSET1:def 3;
      hence O.(g.d) = TRUE by A4,A5,A8,A10;
    end;
    hence thesis by NOMIN_3:28;
  end;
