:: NUMBERS semantic presentation

Lemma11: 1 = succ 0
;

notation
synonym NAT for omega ;
end;

definition
func REAL -> set equals :: NUMBERS:def 1
(REAL+ \/ [:{0},REAL+ :]) \ {[0,0]};
coherence
(REAL+ \/ [:{0},REAL+ :]) \ {[0,0]} is set
;
end;

:: deftheorem Def1 defines REAL NUMBERS:def 1 :
REAL = (REAL+ \/ [:{0},REAL+ :]) \ {[0,0]};

Lemma12: REAL+ c= REAL
proof end;

registration
cluster REAL -> non empty ;
coherence
not REAL is empty
by , XBOOLE_1:3;
end;

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 :
COMPLEX = ((Funcs {0,1},REAL ) \ { x where x is Element of Funcs {0,1},REAL : x . 1 = 0 } ) \/ REAL ;

:: deftheorem Def3 defines RAT NUMBERS:def 3 :
RAT = (RAT+ \/ [:{0},RAT+ :]) \ {[0,0]};

:: deftheorem Def4 defines INT NUMBERS:def 4 :
INT = (NAT \/ [:{0},NAT :]) \ {[0,0]};

Lemma15: RAT+ c= RAT
proof end;

Lemma16: NAT c= INT
proof end;

registration
cluster COMPLEX -> non empty ;
coherence
not COMPLEX is empty
;
cluster RAT -> non empty ;
coherence
not RAT is empty
by , XBOOLE_1:3;
cluster INT -> non empty ;
coherence
not INT is empty
by , XBOOLE_1:3;
end;

Lemma22: for x, y, z being set st [x,y] = {z} holds
( z = {x} & x = y )
proof end;

Lemma26: for a, b being Element of REAL holds not 0,one --> a,b in REAL
proof end;

theorem Th1: :: NUMBERS:1
REAL c< COMPLEX
proof end;

Lemma62: RAT c= REAL
proof end;

Lemma64: for i, j being ordinal Element of RAT+ st i in j holds
i < j
proof end;

Lemma67: for i, j being ordinal Element of RAT+ st i c= j holds
i <=' j
proof end;

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
proof end;

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
proof end;

Lemma75: for i being Element of RAT+ holds i + i = two *' i
proof end;

theorem Th2: :: NUMBERS:2
RAT c< REAL
proof end;

theorem Th3: :: NUMBERS:3
RAT c< COMPLEX by , Lemma15, XBOOLE_1:56;

Lemma170: INT c= RAT
proof end;

theorem Th4: :: NUMBERS:4
INT c< RAT
proof end;

theorem Th5: :: NUMBERS:5
INT c< REAL by Lemma15, , XBOOLE_1:56;

theorem Th6: :: NUMBERS:6
INT c< COMPLEX by , , XBOOLE_1:56;

theorem Th7: :: NUMBERS:7
NAT c< INT
proof end;

theorem Th8: :: NUMBERS:8
NAT c< RAT by , , XBOOLE_1:56;

theorem Th9: :: NUMBERS:9
NAT c< REAL by Lemma15, Lemma22, XBOOLE_1:56;

theorem Th10: :: NUMBERS:10
NAT c< COMPLEX by , , XBOOLE_1:56;

theorem Th11: :: NUMBERS:11
REAL c= COMPLEX by , XBOOLE_0:def 8;

theorem Th12: :: NUMBERS:12
RAT c= REAL by Lemma15, XBOOLE_0:def 8;

theorem Th13: :: NUMBERS:13
RAT c= COMPLEX by Lemma16, XBOOLE_0:def 8;

theorem Th14: :: NUMBERS:14
INT c= RAT by , XBOOLE_0:def 8;

theorem Th15: :: NUMBERS:15
INT c= REAL by , XBOOLE_0:def 8;

theorem Th16: :: NUMBERS:16
INT c= COMPLEX by , XBOOLE_0:def 8;

theorem Th17: :: NUMBERS:17
NAT c= INT by , XBOOLE_0:def 8;

theorem Th18: :: NUMBERS:18
NAT c= RAT by Lemma22, XBOOLE_0:def 8;

theorem Th19: :: NUMBERS:19
NAT c= REAL ;

theorem Th20: :: NUMBERS:20
NAT c= COMPLEX by , XBOOLE_0:def 8;

theorem Th21: :: NUMBERS:21
REAL <> COMPLEX by ;

theorem Th22: :: NUMBERS:22
RAT <> REAL by Lemma15;

theorem Th23: :: NUMBERS:23
RAT <> COMPLEX by , Lemma15;

theorem Th24: :: NUMBERS:24
INT <> RAT by ;

theorem Th25: :: NUMBERS:25
INT <> REAL by Lemma15, ;

theorem Th26: :: NUMBERS:26
INT <> COMPLEX by , ;

theorem Th27: :: NUMBERS:27
NAT <> INT by ;

theorem Th28: :: NUMBERS:28
NAT <> RAT by , ;

theorem Th29: :: NUMBERS:29
NAT <> REAL by Lemma15, Lemma22;

theorem Th30: :: NUMBERS:30
NAT <> COMPLEX by , ;

definition
func ExtREAL -> set equals :: NUMBERS:def 5
REAL \/ {REAL ,[0,REAL ]};
coherence
REAL \/ {REAL ,[0,REAL ]} is set
;
end;

:: deftheorem Def5 defines ExtREAL NUMBERS:def 5 :
ExtREAL = REAL \/ {REAL ,[0,REAL ]};

registration
cluster ExtREAL -> non empty ;
coherence
not ExtREAL is empty
;
end;

theorem Th31: :: NUMBERS:31
REAL c= ExtREAL by XBOOLE_1:7;

theorem Th32: :: NUMBERS:32
REAL <> ExtREAL
proof end;

theorem Th33: :: NUMBERS:33
REAL c< ExtREAL by , Lemma26, XBOOLE_0:def 8;