
theorem
  for A,B being non empty Interval, a,b being Real st
    a in A & b in B & sup A <= inf B holds a <= b
proof
  let A,B be non empty Interval, a,b be Real;
  assume that
A1: a in A and
A2: b in B;
A3: inf B <= b by A2,XXREAL_2:3;
  assume
A4: sup A <= inf B;
  a <= sup A by A1,XXREAL_2:4;
  then a <= inf B by A4,XXREAL_0:2;
  hence thesis by A3,XXREAL_0:2;
end;
