:: NUMBERS semantic presentation
Lemma11:
1 = succ 0
;
:: deftheorem Def1 defines REAL NUMBERS:def 1 :
Lemma12:
REAL+ c= REAL
definition
func COMPLEX -> set equals :: NUMBERS:def 2
((Funcs {0,1},REAL ) \ { x where x is Element of Funcs {0,1},REAL : x . 1 = 0 } ) \/ REAL ;
coherence
((Funcs {0,1},REAL ) \ { x where x is Element of Funcs {0,1},REAL : x . 1 = 0 } ) \/ REAL is set
;
func RAT -> set equals :: NUMBERS:def 3
(RAT+ \/ [:{0},RAT+ :]) \ {[0,0]};
coherence
(RAT+ \/ [:{0},RAT+ :]) \ {[0,0]} is set
;
func INT -> set equals :: NUMBERS:def 4
(NAT \/ [:{0},NAT :]) \ {[0,0]};
coherence
(NAT \/ [:{0},NAT :]) \ {[0,0]} is set
;
redefine func NAT as
NAT -> Subset of
REAL ;
coherence
NAT is Subset of REAL
by , ARYTM_2:2, XBOOLE_1:1;
end;
:: deftheorem Def2 defines COMPLEX NUMBERS:def 2 :
:: deftheorem Def3 defines RAT NUMBERS:def 3 :
:: deftheorem Def4 defines INT NUMBERS:def 4 :
Lemma15:
RAT+ c= RAT
Lemma16:
NAT c= INT
Lemma22:
for x, y, z being set st [x,y] = {z} holds
( z = {x} & x = y )
Lemma26:
for a, b being Element of REAL holds not 0,one --> a,b in REAL
theorem Th1: :: NUMBERS:1
Lemma62:
RAT c= REAL
Lemma64:
for i, j being ordinal Element of RAT+ st i in j holds
i < j
Lemma67:
for i, j being ordinal Element of RAT+ st i c= j holds
i <=' j
E69: 2 =
succ 1
.=
(succ 0) \/ {1}
by ORDINAL1:def 1
.=
(0 \/ {0}) \/ {1}
by ORDINAL1:def 1
.=
{0,1}
by ENUMSET1:41
;
Lemma70:
for i, k being natural Ordinal st i *^ i = 2 *^ k holds
ex k being natural Ordinal st i = 2 *^ k
1 in omega
;
then reconsider 1' = 1 as Element of RAT+ by ARYTM_3:34;
2 in omega
;
then reconsider two = 2 as ordinal Element of RAT+ by ARYTM_3:34;
Lemma74:
one + one = two
Lemma75:
for i being Element of RAT+ holds i + i = two *' i
theorem Th2: :: NUMBERS:2
theorem Th3: :: NUMBERS:3
Lemma170:
INT c= RAT
theorem Th4: :: NUMBERS:4
theorem Th5: :: NUMBERS:5
theorem Th6: :: NUMBERS:6
theorem Th7: :: NUMBERS:7
theorem Th8: :: NUMBERS:8
theorem Th9: :: NUMBERS:9
theorem Th10: :: NUMBERS:10
theorem Th11: :: NUMBERS:11
theorem Th12: :: NUMBERS:12
theorem Th13: :: NUMBERS:13
theorem Th14: :: NUMBERS:14
theorem Th15: :: NUMBERS:15
theorem Th16: :: NUMBERS:16
theorem Th17: :: NUMBERS:17
theorem Th18: :: NUMBERS:18
theorem Th19: :: NUMBERS:19
theorem Th20: :: NUMBERS:20
theorem Th21: :: NUMBERS:21
theorem Th22: :: NUMBERS:22
theorem Th23: :: NUMBERS:23
theorem Th24: :: NUMBERS:24
theorem Th25: :: NUMBERS:25
theorem Th26: :: NUMBERS:26
theorem Th27: :: NUMBERS:27
theorem Th28: :: NUMBERS:28
theorem Th29: :: NUMBERS:29
theorem Th30: :: NUMBERS:30
:: deftheorem Def5 defines ExtREAL NUMBERS:def 5 :
theorem Th31: :: NUMBERS:31
theorem Th32: :: NUMBERS:32
theorem Th33: :: NUMBERS:33