reserve a,b,c for set;

theorem Th1: :: (B1)
  for X being set, B being Subset-Family of X holds B is covering
  iff for x being set st x in X ex U being Subset of X st U in B & x in U
proof
  let X be set;
  let B be Subset-Family of X;
  hereby
    assume B is covering;
    then
A1: union B = X by ABIAN:4;
    let x be set;
    assume x in X;
    then consider U being set such that
A2: x in U and
A3: U in B by A1,TARSKI:def 4;
    reconsider U as Subset of X by A3;
    take U;
    thus U in B & x in U by A2,A3;
  end;
  assume
A4: for x being set st x in X ex U being Subset of X st U in B & x in U;
  union B = X
  proof
    thus union B c= X;
    let a be object;
    assume a in X;
    then ex U being Subset of X st U in B & a in U by A4;
    hence thesis by TARSKI:def 4;
  end;
  hence thesis by ABIAN:4;
end;
