reserve a, b, r, s for Real;

theorem Th46:
  for S being 1-element 1-sorted, s being Point of S, F
  being Subset-Family of S st F is Cover of S holds {s} in F
proof
  let S be 1-element 1-sorted, s be Point of S, F be Subset-Family of
  S such that
A1: the carrier of S c= union F;
  consider d being Element of S such that
A2: the carrier of S = {d} by TEX_1:def 1;
  s in the carrier of S;
  then consider Z being set such that
A3: s in Z and
A4: Z in F by A1,TARSKI:def 4;
A5: s = d by ZFMISC_1:def 10;
  Z = {s}
  proof
    thus for x being object st x in Z holds x in {s} by A4,A2,A5;
    let x be object;
    thus thesis by A3,TARSKI:def 1;
  end;
  hence thesis by A4;
end;
