reserve i,j,k for Nat,
  r,s,r1,r2,s1,s2,sb,tb for Real,
  x for set,
  GX for non empty TopSpace;

theorem Th1:
  for A being Subset of GX, p being Point of GX st p in A & A is
  connected holds A c= Component_of p
proof
  let A be Subset of GX, p be Point of GX;
  consider F being Subset-Family of GX such that
A1: for B being Subset of GX holds B in F iff B is connected & p in B and
A2: union F = Component_of p by CONNSP_1:def 7;
  assume p in A & A is connected;
  then
A3: A in F by A1;
  A c= union F
  by A3,TARSKI:def 4;
  hence thesis by A2;
end;
