:: HEINE semantic presentation
theorem Th1: :: HEINE:1
theorem Th2: :: HEINE:2
Lemma43:
for x, a, b being real number st x >= 0 & a + x <= b holds
a <= b
by XREAL_1:42;
Lemma44:
for x, a, b being real number st x >= 0 & a - x >= b holds
a >= b
by XREAL_1:53;
theorem Th3: :: HEINE:3
canceled;
theorem Th4: :: HEINE:4
canceled;
theorem Th5: :: HEINE:5
canceled;
theorem Th6: :: HEINE:6
:: deftheorem Def1 defines to_power HEINE:def 1 :
defpred S1[ Element of NAT ] means 2 |^ a1 >= a1 + 1;
Lemma54:
S1[0]
by NEWTON:9;
Lemma55:
for n being Element of NAT st S1[n] holds
S1[n + 1]
theorem Th7: :: HEINE:7
theorem Th8: :: HEINE:8
theorem Th9: :: HEINE:9
theorem Th10: :: HEINE:10
theorem Th11: :: HEINE:11