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

theorem Th32:
  for XX being non empty TopSpace, X being non empty SubSpace of
  XX, D being non empty a_partition of the carrier of X, x being Point of XX st
  not x in the carrier of X holds {x} in TrivExt D
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, x be Point of XX;
  union TrivExt D = the carrier of XX by EQREL_1:def 4;
  then consider A being set such that
A1: x in A and
A2: A in TrivExt D by TARSKI:def 4;
  assume not x in the carrier of X;
  then not A in D by A1;
  then A in {{p} where p is Point of XX : not p in the carrier of X} by A2,
XBOOLE_0:def 3;
  then ex p being Point of XX st A = {p} & not p in the carrier of X;
  hence thesis by A1,A2,TARSKI:def 1;
end;
