reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th71:
  for T being non empty TopSpace, S being SubSpace of T for B
being Basis of T holds {A/\[#]S where A is Subset of T: A in B & A meets [#]S}
  is Basis of S
proof
  let T be non empty TopSpace;
  let S be SubSpace of T;
  let B be Basis of T;
  set X = {A/\[#]S where A is Subset of T: A in B & A meets [#]S};
  X c= bool the carrier of S
  proof
    let u be object;
    assume u in X;
    then ex A being Subset of T st u = A/\[#]S & A in B & A meets [#]S;
    hence thesis;
  end;
  then reconsider X as Subset-Family of S;
A1: now
    let U be Subset of S;
    assume U is open;
    then consider U0 being Subset of T such that
A2: U0 is open and
A3: U = U0 /\ the carrier of S by TSP_1:def 1;
    let x be Point of S;
    assume
A4: x in U;
    then x in U0 by A3,XBOOLE_0:def 4;
    then consider V0 being Subset of T such that
A5: V0 in B and
A6: x in V0 and
A7: V0 c= U0 by A2,YELLOW_9:31;
    reconsider V = V0 /\ [#]S as Subset of S;
    take V;
    V0 meets [#]S by A4,A6,XBOOLE_0:3;
    hence V in X by A5;
    thus x in V by A4,A6,XBOOLE_0:def 4;
    thus V c= U by A3,A7,XBOOLE_1:26;
  end;
  X c= the topology of S
  proof
    let u be object;
    assume u in X;
    then
A8: ex A being Subset of T st u = A/\[#]S & A in B & A meets [#]S;
    B c= the topology of T by TOPS_2:64;
    hence thesis by A8,PRE_TOPC:def 4;
  end;
  hence thesis by A1,YELLOW_9:32;
end;
