theorem
  for A,B being non empty bounded_below Subset of REAL holds lower_bound(A
  \/ B) = min(lower_bound A,lower_bound B)
proof
  let A,B be non empty bounded_below Subset of REAL;
  set r = min(lower_bound A,lower_bound B);
A1: r <= lower_bound B by XXREAL_0:17;
A2: for r1 being Real st for t being Real st t in A \/ B holds
  t >= r1 holds r >= r1
  proof
    let r1 be Real;
    assume
A3: for t being Real st t in A \/ B holds t >= r1;
    now
      let t be Real;
      assume t in B;
      then t in A \/ B by XBOOLE_0:def 3;
      hence t >= r1 by A3;
    end;
    then
A4: lower_bound B >= r1 by Th43;
    now
      let t be Real;
      assume t in A;
      then t in A \/ B by XBOOLE_0:def 3;
      hence t >= r1 by A3;
    end;
    then lower_bound A >= r1 by Th43;
    hence thesis by A4,XXREAL_0:20;
  end;
A5: r <= lower_bound A by XXREAL_0:17;
  for t being Real st t in A \/ B holds t >= r
  proof
    let t be Real;
    assume t in A \/ B;
    then t in A or t in B by XBOOLE_0:def 3;
    then lower_bound A <= t or lower_bound B <= t by Def2;
    hence thesis by A5,A1,XXREAL_0:2;
  end;
  hence thesis by A2,Th44;
end;
