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 Th22:
  for f, g being FinSequence of TOP-REAL 2 st g is_in_the_area_of
  f for i,j st i in dom g & j in dom g holds mid(g,i,j) is_in_the_area_of f
proof
  let f, g be FinSequence of TOP-REAL 2 such that
A1: for n st n in dom g holds W-bound L~f <= (g/.n)`1 & (g/.n)`1 <=
  E-bound L~f & S-bound L~f <= (g/.n)`2 & (g/.n)`2 <= N-bound L~f;
  let i,j such that
A2: i in dom g & j in dom g;
  set h = mid(g,i,j);
  per cases;
  suppose
A3: i <= j;
    let n;
    assume n in dom h;
    then n+i-'1 in dom g & h/.n = g/.(n+i-'1) by A2,A3,Th1,Th3;
    hence thesis by A1;
  end;
  suppose
A4: i > j;
    let n;
    assume n in dom h;
    then i -' n + 1 in dom g & h/.n = g/.(i -' n + 1) by A2,A4,Th2,Th4;
    hence thesis by A1;
  end;
end;
