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

theorem Th37:
  for XX being non empty TopSpace , X being non empty SubSpace of
  XX for D being non empty a_partition of the carrier of X for e st e in the
  carrier of X holds Proj TrivExt D.e in the carrier of space D
proof
  let XX be non empty TopSpace , X be non empty SubSpace of XX;
  let D be non empty a_partition of the carrier of X;
  let e;
  assume
A1: e in the carrier of X;
  then reconsider x = e as Point of X;
  the carrier of X c= the carrier of XX by Th1;
  then Proj D.x = Proj TrivExt D.x by A1,Th33;
  hence thesis;
end;
