theorem Th22:
  x in Int K iff ex Q st Q is open & Q c= K & x in Q
proof
  thus x in Int K implies ex Q st Q is open & Q c= K & x in Q by Th16;
  given Q such that
A1: Q is open and
A2: Q c= K and
A3: x in Q;
  K` c= Q` by A2,SUBSET_1:12;
  then Cl K` c= Cl Q` by PRE_TOPC:19;
  then Cl K` c= Q` by A1,PRE_TOPC:22;
  then Q`` c= (Cl K`)` by SUBSET_1:12;
  hence thesis by A3;
end;
