reserve y,w for set;
reserve T for non empty TopSpace;

theorem Th7:
  for T,T1 being non empty TopSpace,f being continuous Function of
  T,T1 holds T1 is T_1 implies for w st w in the carrier of T_1-reflex T ex z
  being Element of T1 st z in rng f & w c= f"{z}
proof
  let T,T1 be non empty TopSpace;
  let f be continuous Function of T,T1;
  assume
A1: T1 is T_1;
  let w be set;
  assume w in the carrier of T_1-reflex T;
  then w in Intersection (Closed_Partitions T) by BORSUK_1:def 7;
  then consider x being Element of T such that
A2: w = EqClass(x,Intersection (Closed_Partitions T)) by EQREL_1:42;
  reconsider x as Element of T;
  reconsider fx = f.x as Element of T1;
  take fx;
  dom f = the carrier of T by FUNCT_2:def 1;
  hence thesis by A1,A2,Th6,FUNCT_1:def 3;
end;
