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

theorem Th32:
  for S, T being TopStruct, B being Basis of S st the TopStruct of
  S = the TopStruct of T holds B is Basis of T
proof
  let S, T be TopStruct, B be Basis of S;
A1: B c= the topology of S by TOPS_2:64;
  assume
A2: the TopStruct of S = the TopStruct of T;
  then the topology of T c= UniCl B by CANTOR_1:def 2;
  hence thesis by A2,A1,CANTOR_1:def 2,TOPS_2:64;
end;
