reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th28:
  for f be natural-valued Function st n>1 holds
    Coim(n|^f,n|^k) = Coim(f,k)
proof
  let f be natural-valued Function such that A1:n>1;
  thus Coim(n|^f,n|^k) c= Coim(f,k)
  proof
    let x be object;
    assume x in Coim(n|^f,n|^k);
    then x in (n|^f)"{n|^k} by RELAT_1:def 17;
    then x in dom (n|^f) & (n|^f).x in {n|^k} by FUNCT_1:def 7;
    then A2:x in dom f & (n|^f).x =n|^k by TARSKI:def 1,Def4;
    then (n|^f).x = n to_power (f.x) by Def4
                 .=n |^ (f.x);
    then k=f.x by A2,A1,PEPIN:30;
    then f.x in {k} by TARSKI:def 1;
    then x in f"{k} by A2,FUNCT_1:def 7;
    hence thesis by RELAT_1:def 17;
  end;
  let x be object;
  assume x in Coim(f,k);
  then x in f"{k} by RELAT_1:def 17;
  then A3:x in dom f & f.x in {k} by FUNCT_1:def 7;
  then A4:f.x=k & x in dom (n|^f) by TARSKI:def 1,Def4;
  then (n|^f).x = n to_power k by Def4,A3
               .= n|^k;
  then (n|^f).x in {n|^k} by TARSKI:def 1;
  then x in (n|^f)"{n|^k} by FUNCT_1:def 7,A4;
  hence thesis by RELAT_1:def 17;
end;
