reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th31:
  for XX being non empty TopSpace, X being non empty SubSpace of
XX, D being non empty a_partition of the carrier of X, A being Subset of XX st
A in TrivExt D holds A in D or ex x being Point of XX st not x in [#]X & A = {x
  }
proof
  let XX be non empty TopSpace, X be non empty SubSpace of XX, D be non empty
  a_partition of the carrier of X, A be Subset of XX;
  assume
A1: A in TrivExt D;
  now
    per cases by A1,XBOOLE_0:def 3;
    case
      A in D;
      hence A in D;
    end;
    case
      A in {{p} where p is Point of XX : not p in the carrier of X};
      then consider x being Point of XX such that
A2:   A = {x} and
A3:   not x in the carrier of X;
      take x;
      thus not x in [#]X by A3;
      thus A = {x} by A2;
    end;
  end;
  hence thesis;
end;
