reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem
  for X1 being non empty SubSpace of X, x being Point of X, x1 being
  Point of X1 st x = x1 holds Component_of x1 c= Component_of x
proof
  let X1 be non empty SubSpace of X, x be Point of X, x1 be Point of X1;
  consider F being Subset-Family of X such that
A1: for A being Subset of X holds A in F iff A is connected & x in A and
A2: union F = Component_of x by CONNSP_1:def 7;
  reconsider Z = Component_of x1 as Subset of X by PRE_TOPC:11;
A3: x1 in Z & Z is connected by CONNSP_1:23,38;
  assume x = x1;
  then Z in F by A1,A3;
  hence thesis by A2,ZFMISC_1:74;
end;
