theorem Th77:
  H is_subformula_of x 'in' y iff H = x 'in' y
proof
  thus H is_subformula_of x 'in' y implies H = x 'in' y
  proof
    assume
A1: H is_subformula_of x 'in' y;
    assume H <> x 'in' y;
    then H is_proper_subformula_of x 'in' y by A1;
    then ex F st F is_immediate_constituent_of x 'in' y by Th63;
    hence contradiction by Th51;
  end;
  assume H = x 'in' y;
  hence thesis by Th59;
end;
