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

theorem Th73:
  for T being TopSpace for U,V being Subset of T for B being set
  st U in B & V in B & B \/ {U \/ V} is Basis of T holds B is Basis of T
proof
  let T be TopSpace;
  let U,V be Subset of T;
  let B be set;
  assume that
A1: U in B and
A2: V in B and
A3: B \/ {U \/ V} is Basis of T;
A4: B c= B \/ {U \/ V} by XBOOLE_1:7;
  then reconsider B9 = B as Subset-Family of T by A3,XBOOLE_1:1;
A5: now
A6: V c= U\/V by XBOOLE_1:7;
A7: U c= U\/V by XBOOLE_1:7;
    let A be Subset of T such that
A8: A is open;
    let p be Point of T;
    assume p in A;
    then consider A9 being Subset of T such that
A9: A9 in B \/ {U\/V} and
A10: p in A9 and
A11: A9 c= A by A3,A8,YELLOW_9:31;
    assume
A12: not ex a being Subset of T st a in B9 & p in a & a c= A;
    A9 in B or A9 = U \/ V by A9,ZFMISC_1:136;
    then p in U & U c= A or p in V & V c= A by A10,A11,A12,A7,A6,XBOOLE_0:def 3
;
    hence contradiction by A1,A2,A12;
  end;
  B \/ {U \/ V} c= the topology of T by A3,TOPS_2:64;
  hence thesis by A5,A4,XBOOLE_1:1,YELLOW_9:32;
end;
