reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty multMagma,
  P, Q, P1, Q1 for Subset of H,
  h for Element of H;
reserve G for Group,
  A, B for Subset of G,
  a for Element of G;

theorem Th32:
  id T = 1_HomeoGroup T
proof
  set G = HomeoGroup T;
  reconsider e = id T as Element of G by Def5;
  now
    let h be Element of G;
    reconsider h1 = h as Homeomorphism of T by Def5;
    thus h * e = id T * h1 by Def5
      .= h by FUNCT_2:17;
    thus e * h = h1 * id T by Def5
      .= h by FUNCT_2:17;
  end;
  hence thesis by GROUP_1:4;
end;
