reserve x for set;

theorem Th20:
  for T1, T2 being non empty TopSpace, B being prebasis of T1 st B
  c= the topology of T2 & the carrier of T1 in the topology of T2 holds the
  topology of T1 c= the topology of T2
proof
  let T1, T2 be non empty TopSpace;
  let B be prebasis of T1 such that
A1: B c= the topology of T2 and
A2: the carrier of T1 in the topology of T2;
  let x be object;
  FinMeetCl B is Basis of T1 by YELLOW_9:23;
  then
A3: the topology of T1 = UniCl FinMeetCl B by YELLOW_9:22;
  assume x in the topology of T1;
  then consider Y being Subset-Family of T1 such that
A4: Y c= FinMeetCl B and
A5: x = union Y by A3,CANTOR_1:def 1;
A6: Y c= the topology of T2
  proof
    let y be object;
    assume y in Y;
    then consider Z being Subset-Family of T1 such that
A7: Z c= B and
A8: Z is finite and
A9: y = Intersect Z by A4,CANTOR_1:def 3;
    Z c= the topology of T2 by A1,A7;
    then reconsider Z9 = Z as Subset-Family of T2 by XBOOLE_1:1;
    y = the carrier of T1 or Z9 c= the topology of T2 & y = meet Z9 &
    meet Z9 = Intersect Z9 by A1,A7,A9,SETFAM_1:def 9;
    then y = the carrier of T1 or y in FinMeetCl the topology of T2 by A8,
CANTOR_1:def 3;
    hence thesis by A2,CANTOR_1:5;
  end;
  then reconsider Y as Subset-Family of T2 by XBOOLE_1:1;
  union Y in UniCl the topology of T2 by A6,CANTOR_1:def 1;
  hence thesis by A5,CANTOR_1:6;
end;
