reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th43:
  for f,g be FinSequence holds
     {f}^{g} = {f^g}
proof
  let f,g be FinSequence;
  thus {f}^{g} c= {f^g}
  proof
    let x;
    assume x in {f}^{g};
    then consider p, q be FinSequence such that
A1:   x = p^q & p in {f} & q in {g} by POLNOT_1:def 2;
    p=f & q=g by A1,TARSKI:def 1;
    hence thesis by A1,TARSKI:def 1;
  end;
  let x;
A2: f in {f} & g in {g} by TARSKI:def 1;
  assume x in {f^g};
  then x = f^g by TARSKI:def 1;
  hence thesis by A2, POLNOT_1:def 2;
end;
