reserve i,j,k,l,m,n for Nat,
  D for non empty set,
  f for FinSequence of D;
reserve X for compact Subset of TOP-REAL 2;
reserve r for Real;

theorem Th24:
  for f,g,h being FinSequence of TOP-REAL 2 st g is_in_the_area_of
  f & h is_in_the_area_of f holds g^h is_in_the_area_of f
proof
  let f,g,h be FinSequence of TOP-REAL 2 such that
A1: g is_in_the_area_of f and
A2: h is_in_the_area_of f;
  let n;
  assume
A3: n in dom(g^h);
  per cases by A3,FINSEQ_1:25;
  suppose
A4: n in dom g;
    then (g^h)/.n = g/.n by FINSEQ_4:68;
    hence thesis by A1,A4;
  end;
  suppose
    ex i be Nat st i in dom h & n = len g + i;
    then consider i be Nat such that
A5: i in dom h and
A6: n = len g + i;
    (g^h)/.n = h/.i by A5,A6,FINSEQ_4:69;
    hence thesis by A2,A5;
  end;
end;
