
theorem Th2:
  for x,y be Real st y > 0 holds min(x,y)/max(x,y) <= 1
proof
  let x,y be Real;
  assume
A1: y > 0;
  per cases;
  suppose
A2: x > 0;
    now
      per cases;
      suppose
A3:     x >= y;
        then max(x,y) = x & min(x,y) = y by XXREAL_0:def 9,def 10;
        hence thesis by A1,A3,XREAL_1:183;
      end;
      suppose
A4:     x < y;
        then max(x,y) = y & min(x,y) = x by XXREAL_0:def 9,def 10;
        hence thesis by A2,A4,XREAL_1:183;
      end;
    end;
    hence thesis;
  end;
  suppose
A5: x <= 0;
    then min (x,y) = x & max (x,y) = y by A1,XXREAL_0:def 9,def 10;
    hence thesis by A1,A5;
  end;
end;
