reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;
reserve D for Subset of X;
reserve Y0 for SubSpace of X;
reserve X0 for SubSpace of X;
reserve X0 for non empty SubSpace of X;

theorem
  for X1, X2 being 1-sorted holds the carrier of X1 = the carrier of X2
implies for C1 being Subset of X1, C2 being Subset of X2 holds C1 = C2 iff C1`
  = C2`
proof
  let X1, X2 be 1-sorted;
  assume
A1: the carrier of X1 = the carrier of X2;
  let C1 be Subset of X1, C2 be Subset of X2;
  thus C1 = C2 implies C1` = C2` by A1;
  thus C1` = C2` implies C1 = C2
  proof
    assume C1` = C2`;
    hence C1 = [#]X2 \ C2` by A1,PRE_TOPC:3
      .= C2 by PRE_TOPC:3;
  end;
end;
