:: TOPREAL5 semantic presentation
Lm1:
for a, b, c being real number holds
( c in [.a,b.] iff ( a <= c & c <= b ) )
by RCOMP_1:48;
Lm2:
for X, Y, Z being non empty TopSpace
for f being continuous Function of X,Y
for g being continuous Function of Y,Z holds g * f is continuous Function of X,Z
by TOPS_2:58;
theorem :: TOPREAL5:1
canceled;
theorem :: TOPREAL5:2
canceled;
theorem :: TOPREAL5:3
canceled;
theorem Th4: :: TOPREAL5:4
Lm3:
for X, Y being non empty TopSpace
for f being continuous Function of X,Y
for A being Subset of X st A is connected holds
f .: A is connected
by TOPS_2:75;
theorem :: TOPREAL5:5
canceled;
theorem Th6: :: TOPREAL5:6
Lm4:
for A being Subset of R^1
for a being real number st A = { r where r is Element of REAL : a < r } holds
A is open
by JORDAN2B:31;
Lm5:
for A being Subset of R^1
for a being real number st A = { r where r is Element of REAL : a > r } holds
A is open
by JORDAN2B:30;
theorem :: TOPREAL5:7
canceled;
theorem :: TOPREAL5:8
canceled;
theorem Th9: :: TOPREAL5:9
theorem :: TOPREAL5:10
theorem :: TOPREAL5:11
theorem Th12: :: TOPREAL5:12
theorem Th13: :: TOPREAL5:13
theorem :: TOPREAL5:14
theorem Th15: :: TOPREAL5:15
theorem Th16: :: TOPREAL5:16
theorem Th17: :: TOPREAL5:17
theorem Th18: :: TOPREAL5:18
theorem Th19: :: TOPREAL5:19
theorem Th20: :: TOPREAL5:20
theorem Th21: :: TOPREAL5:21
theorem Th22: :: TOPREAL5:22
theorem Th23: :: TOPREAL5:23
theorem :: TOPREAL5:24
theorem :: TOPREAL5:25