Article REAL, MML version 4.99.1005

:: REAL:th 1
theorem
for b1, b2 being real set
      st b1 <= b2 & b1 is positive
   holds b2 is positive;

:: REAL:th 2
theorem
for b1, b2 being real set
      st b1 <= b2 & b2 is negative
   holds b1 is negative;

:: REAL:th 3
theorem
for b1, b2 being real set
      st b1 <= b2 & b1 is not negative
   holds b2 is not negative;

:: REAL:th 4
theorem
for b1, b2 being real set
      st b1 <= b2 & b2 is not positive
   holds b1 is not positive;

:: REAL:th 5
theorem
for b1, b2 being real set
      st b1 <= b2 & b2 is not empty & b1 is not negative
   holds b2 is positive;

:: REAL:th 6
theorem
for b1, b2 being real set
      st b1 <= b2 & b1 is not empty & b2 is not positive
   holds b1 is negative;

:: REAL:th 7
theorem
for b1, b2 being real set
      st b2 < b1 & b1 is not positive
   holds b2 is negative;

:: REAL:th 8
theorem
for b1, b2 being real set
      st b2 < b1 & b2 is not negative
   holds b1 is positive;