theorem
  f is_a_proof_wrt X & X c= Y implies f is_a_proof_wrt Y
proof
  assume that
A1: f is_a_proof_wrt X and
A2: X c= Y;
  thus f <> {} by A1;
  let n;
  assume
A3: 1 <= n & n <= len f;
then A4: f,n is_a_correct_step_wrt X by A1;
  (f.n)`2 = 0 or ... or (f.n)`2 = 9 by A3,Th19;
  then per cases;
  case
 (f.n)`2 = 0;
then  (f.n)`1 in X by A4,Def4;
    hence thesis by A2;
  end;
  case
 (f.n)`2 = 1;
    hence thesis by A4,Def4;
  end;
  case
 (f.n)`2 = 2;
    hence thesis by A4,Def4;
  end;
  case
 (f.n)`2 = 3;
    hence thesis by A4,Def4;
  end;
  case
 (f.n)`2 = 4;
    hence thesis by A4,Def4;
  end;
  case
 (f.n)`2 = 5;
    hence thesis by A4,Def4;
  end;
  case
 (f.n)`2 = 6;
    hence thesis by A4,Def4;
  end;
  case
 (f.n)`2 = 7;
    hence thesis by A4,Def4;
  end;
  case
 (f.n)`2 = 8;
    hence thesis by A4,Def4;
  end;
  case
 (f.n)`2 = 9;
    hence thesis by A4,Def4;
  end;
end;
