reserve A, B, X, Y for set;
reserve R, S, T for non empty TopSpace;

theorem Th33:
  for S, T being TopStruct, B being prebasis of S st the TopStruct
  of S = the TopStruct of T holds B is prebasis of T
proof
  let S, T be TopStruct, B be prebasis of S;
  consider F being Basis of S such that
A1: F c= FinMeetCl B by CANTOR_1:def 4;
  assume
A2: the TopStruct of S = the TopStruct of T;
  then B c= the topology of T & F is Basis of T by Th32,TOPS_2:64;
  hence thesis by A2,A1,CANTOR_1:def 4,TOPS_2:64;
end;
