:: REAL semantic presentation

Lemma1: for b1, b2 being real number holds
( b1 <= b2 & b2 <= b1 implies b1 = b2 )
proof end;

Lemma2: for b1, b2, b3 being real number holds
( b1 <= b2 & b2 <= b3 implies b1 <= b3 )
proof end;

theorem Th1: :: REAL:1
for b1, b2 being real number holds
( b1 <= b2 & b1 is positive implies b2 is positive )
proof end;

theorem Th2: :: REAL:2
for b1, b2 being real number holds
( b1 <= b2 & b2 is negative implies b1 is negative )
proof end;

theorem Th3: :: REAL:3
for b1, b2 being real number holds
not ( b1 <= b2 & not b1 is negative & b2 is negative )
proof end;

theorem Th4: :: REAL:4
for b1, b2 being real number holds
not ( b1 <= b2 & not b2 is positive & b1 is positive )
proof end;

theorem Th5: :: REAL:5
for b1, b2 being real number holds
( b1 <= b2 & not b2 is empty & not b1 is negative implies b2 is positive )
proof end;

theorem Th6: :: REAL:6
for b1, b2 being real number holds
( b1 <= b2 & not b1 is empty & not b2 is positive implies b1 is negative )
proof end;

theorem Th7: :: REAL:7
for b1, b2 being real number holds
( not b1 <= b2 & not b1 is positive implies b2 is negative )
proof end;

theorem Th8: :: REAL:8
for b1, b2 being real number holds
( not b1 <= b2 & not b2 is negative implies b1 is positive )
proof end;