:: NUMBERS semantic presentation begin Lm1: omega c= ( { [c,d] where c, d is Element of omega : ( c,d are_relative_prime & d <> {} ) } \ { [k,1] where k is Element of omega : verum } ) \/ omega by XBOOLE_1:7; notation synonym NAT for omega ; synonym 0 for {} ; end; Lm2: 1 = succ 0 ; definition func REAL -> set equals :: NUMBERS:def 1 (REAL+ \/ [:{0},REAL+:]) \ {[0,0]}; coherence (REAL+ \/ [:{0},REAL+:]) \ {[0,0]} is set ; end; :: deftheorem defines REAL NUMBERS:def_1_:_ REAL = (REAL+ \/ [:{0},REAL+:]) \ {[0,0]}; Lm3: REAL+ c= REAL proof REAL+ c= REAL+ \/ [:{0},REAL+:] by XBOOLE_1:7; hence REAL+ c= REAL by ARYTM_2:3, ZFMISC_1:34; ::_thesis: verum end; registration cluster REAL -> non empty ; coherence not REAL is empty by Lm3, 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 ; :: original: NAT redefine func NAT -> Subset of REAL; coherence NAT is Subset of REAL by Lm3, ARYTM_2:2, XBOOLE_1:1; end; :: deftheorem 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 defines RAT NUMBERS:def_3_:_ RAT = (RAT+ \/ [:{0},RAT+:]) \ {[0,0]}; :: deftheorem defines INT NUMBERS:def_4_:_ INT = (NAT \/ [:{0},NAT:]) \ {[0,0]}; Lm4: RAT+ c= RAT proof RAT+ c= RAT+ \/ [:{0},RAT+:] by XBOOLE_1:7; hence RAT+ c= RAT by ARYTM_3:61, ZFMISC_1:34; ::_thesis: verum end; Lm5: NAT c= INT proof NAT c= NAT \/ [:{0},NAT:] by XBOOLE_1:7; hence NAT c= INT by ARYTM_3:32, ZFMISC_1:34; ::_thesis: verum end; registration cluster COMPLEX -> non empty ; coherence not COMPLEX is empty ; cluster RAT -> non empty ; coherence not RAT is empty by Lm4, XBOOLE_1:3; cluster INT -> non empty ; coherence not INT is empty by Lm5, XBOOLE_1:3; end; Lm6: for x, y, z being set st [x,y] = {z} holds ( z = {x} & x = y ) proof let x, y, z be set ; ::_thesis: ( [x,y] = {z} implies ( z = {x} & x = y ) ) assume A1: [x,y] = {z} ; ::_thesis: ( z = {x} & x = y ) then {x} in {z} by TARSKI:def_2; hence A2: z = {x} by TARSKI:def_1; ::_thesis: x = y {x,y} in {z} by A1, TARSKI:def_2; then {x,y} = z by TARSKI:def_1; hence x = y by A2, ZFMISC_1:5; ::_thesis: verum end; Lm7: for a, b being Element of REAL holds not (0,one) --> (a,b) in REAL proof let a, b be Element of REAL ; ::_thesis: not (0,one) --> (a,b) in REAL set IR = { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s <=' r holds s in A ) & ex s being Element of RAT+ st ( s in A & r < s ) ) } ; set f = (0,one) --> (a,b); A1: now__::_thesis:_not_(0,one)_-->_(a,b)_in_[:{{}},REAL+:] (0,one) --> (a,b) = {[0,a],[one,b]} by FUNCT_4:67; then A2: [one,b] in (0,one) --> (a,b) by TARSKI:def_2; assume (0,one) --> (a,b) in [:{{}},REAL+:] ; ::_thesis: contradiction then consider x, y being set such that A3: x in {{}} and y in REAL+ and A4: (0,one) --> (a,b) = [x,y] by ZFMISC_1:84; A5: x = 0 by A3, TARSKI:def_1; percases ( {{one,b},{one}} = {0} or {{one,b},{one}} = {0,y} ) by A4, A5, A2, TARSKI:def_2; suppose {{one,b},{one}} = {0} ; ::_thesis: contradiction then 0 in {{one,b},{one}} by TARSKI:def_1; hence contradiction by TARSKI:def_2; ::_thesis: verum end; suppose {{one,b},{one}} = {0,y} ; ::_thesis: contradiction then 0 in {{one,b},{one}} by TARSKI:def_2; hence contradiction by TARSKI:def_2; ::_thesis: verum end; end; end; A6: (0,one) --> (a,b) = {[0,a],[one,b]} by FUNCT_4:67; now__::_thesis:_not_(0,one)_-->_(a,b)_in__{__[i,j]_where_i,_j_is_Element_of_NAT_:_(_i,j_are_relative_prime_&_j_<>_{}_)__}_ assume (0,one) --> (a,b) in { [i,j] where i, j is Element of NAT : ( i,j are_relative_prime & j <> {} ) } ; ::_thesis: contradiction then consider i, j being Element of NAT such that A7: (0,one) --> (a,b) = [i,j] and i,j are_relative_prime and j <> {} ; A8: ( {i} in (0,one) --> (a,b) & {i,j} in (0,one) --> (a,b) ) by A7, TARSKI:def_2; A9: now__::_thesis:_not_i_=_j assume i = j ; ::_thesis: contradiction then {i} = {i,j} by ENUMSET1:29; then A10: [i,j] = {{i}} by ENUMSET1:29; then [one,b] in {{i}} by A6, A7, TARSKI:def_2; then A11: [one,b] = {i} by TARSKI:def_1; [0,a] in {{i}} by A6, A7, A10, TARSKI:def_2; then [0,a] = {i} by TARSKI:def_1; hence contradiction by A11, XTUPLE_0:1; ::_thesis: verum end; percases ( ( {i,j} = [0,a] & {i} = [0,a] ) or ( {i,j} = [0,a] & {i} = [one,b] ) or ( {i,j} = [one,b] & {i} = [0,a] ) or ( {i,j} = [one,b] & {i} = [one,b] ) ) by A6, A8, TARSKI:def_2; suppose ( {i,j} = [0,a] & {i} = [0,a] ) ; ::_thesis: contradiction hence contradiction by A9, ZFMISC_1:5; ::_thesis: verum end; supposethat A12: {i,j} = [0,a] and A13: {i} = [one,b] ; ::_thesis: contradiction i in [0,a] by A12, TARSKI:def_2; then ( i = {0,a} or i = {0} ) by TARSKI:def_2; then A14: 0 in i by TARSKI:def_1, TARSKI:def_2; i = {one} by A13, Lm6; hence contradiction by A14, TARSKI:def_1; ::_thesis: verum end; supposethat A15: {i,j} = [one,b] and A16: {i} = [0,a] ; ::_thesis: contradiction i in [one,b] by A15, TARSKI:def_2; then ( i = {one,b} or i = {one} ) by TARSKI:def_2; then A17: one in i by TARSKI:def_1, TARSKI:def_2; i = {0} by A16, Lm6; hence contradiction by A17, TARSKI:def_1; ::_thesis: verum end; suppose ( {i,j} = [one,b] & {i} = [one,b] ) ; ::_thesis: contradiction hence contradiction by A9, ZFMISC_1:5; ::_thesis: verum end; end; end; then A18: not (0,one) --> (a,b) in { [i,j] where i, j is Element of NAT : ( i,j are_relative_prime & j <> {} ) } \ { [k,one] where k is Element of NAT : verum } ; for x, y being set holds not {[0,x],[one,y]} in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s <=' r holds s in A ) & ex s being Element of RAT+ st ( s in A & r < s ) ) } proof given x, y being set such that A19: {[0,x],[one,y]} in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s <=' r holds s in A ) & ex s being Element of RAT+ st ( s in A & r < s ) ) } ; ::_thesis: contradiction consider A being Subset of RAT+ such that A20: {[0,x],[one,y]} = A and A21: for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s <=' r holds s in A ) & ex s being Element of RAT+ st ( s in A & r < s ) ) by A19; ( [0,x] in A & ( for r, s being Element of RAT+ st r in A & s <=' r holds s in A ) ) by A20, A21, TARSKI:def_2; then consider r1, r2, r3 being Element of RAT+ such that A22: r1 in A and A23: r2 in A and A24: ( r3 in A & r1 <> r2 & r2 <> r3 & r3 <> r1 ) by ARYTM_3:75; A25: ( r2 = [0,x] or r2 = [one,y] ) by A20, A23, TARSKI:def_2; ( r1 = [0,x] or r1 = [one,y] ) by A20, A22, TARSKI:def_2; hence contradiction by A20, A24, A25, TARSKI:def_2; ::_thesis: verum end; then A26: not (0,one) --> (a,b) in DEDEKIND_CUTS by A6, ARYTM_2:def_1; now__::_thesis:_not_(0,one)_-->_(a,b)_in_omega assume (0,one) --> (a,b) in omega ; ::_thesis: contradiction then {} in (0,one) --> (a,b) by ORDINAL3:8; hence contradiction by A6, TARSKI:def_2; ::_thesis: verum end; then not (0,one) --> (a,b) in RAT+ by A18, XBOOLE_0:def_3; then not (0,one) --> (a,b) in REAL+ by A26, ARYTM_2:def_2, XBOOLE_0:def_3; hence not (0,one) --> (a,b) in REAL by A1, XBOOLE_0:def_3; ::_thesis: verum end; definition :: original: 0 redefine func 0 -> Element of NAT ; coherence 0 is Element of NAT by ORDINAL1:def_11; end; theorem Th1: :: NUMBERS:1 REAL c< COMPLEX proof set X = { x where x is Element of Funcs ({0,one},REAL) : x . one = 0 } ; thus REAL c= COMPLEX by XBOOLE_1:7; :: according to XBOOLE_0:def_8 ::_thesis: not REAL = COMPLEX A1: now__::_thesis:_not_(0,1)_-->_(0,1)_in__{__x_where_x_is_Element_of_Funcs_({0,one},REAL)_:_x_._one_=_0__}_ assume (0,1) --> (0,1) in { x where x is Element of Funcs ({0,one},REAL) : x . one = 0 } ; ::_thesis: contradiction then ex x being Element of Funcs ({0,one},REAL) st ( x = (0,1) --> (0,1) & x . one = 0 ) ; hence contradiction by FUNCT_4:63; ::_thesis: verum end; REAL+ c= REAL+ \/ [:{{}},REAL+:] by XBOOLE_1:7; then A2: REAL+ c= REAL by ARYTM_2:3, ZFMISC_1:34; then reconsider z = 0 , j = 1 as Element of REAL by ARYTM_2:20; A3: not (0,1) --> (z,j) in REAL by Lm7; ( rng ((0,1) --> (0,1)) c= {0,1} & {0,1} c= REAL ) by A2, ARYTM_2:20, FUNCT_4:62, ZFMISC_1:32; then ( dom ((0,1) --> (0,1)) = {0,1} & rng ((0,1) --> (0,1)) c= REAL ) by FUNCT_4:62, XBOOLE_1:1; then (0,1) --> (0,1) in Funcs ({0,one},REAL) by FUNCT_2:def_2; then (0,1) --> (0,1) in (Funcs ({0,one},REAL)) \ { x where x is Element of Funcs ({0,one},REAL) : x . one = 0 } by A1, XBOOLE_0:def_5; hence not REAL = COMPLEX by A3, XBOOLE_0:def_3; ::_thesis: verum end; Lm8: RAT c= REAL proof [:{0},RAT+:] c= [:{0},REAL+:] by ARYTM_2:1, ZFMISC_1:95; then RAT+ \/ [:{0},RAT+:] c= REAL+ \/ [:{0},REAL+:] by ARYTM_2:1, XBOOLE_1:13; hence RAT c= REAL by XBOOLE_1:33; ::_thesis: verum end; Lm9: for i, j being ordinal Element of RAT+ st i in j holds i < j proof let i, j be ordinal Element of RAT+ ; ::_thesis: ( i in j implies i < j ) A1: j in omega by ARYTM_3:31; i in omega by ARYTM_3:31; then reconsider x = i, y = j as Element of REAL+ by A1, ARYTM_2:2; assume A2: i in j ; ::_thesis: i < j then x <=' y by A1, ARYTM_2:18; then A3: ex x9, y9 being Element of RAT+ st ( x = x9 & y = y9 & x9 <=' y9 ) by ARYTM_2:def_5; i <> j by A2; hence i < j by A3, ARYTM_3:66; ::_thesis: verum end; Lm10: for i, j being ordinal Element of RAT+ st i c= j holds i <=' j proof let i, j be ordinal Element of RAT+ ; ::_thesis: ( i c= j implies i <=' j ) assume i c= j ; ::_thesis: i <=' j then consider C being ordinal number such that A1: j = i +^ C by ORDINAL3:27; C in omega by A1, ORDINAL3:74; then reconsider C = C as Element of RAT+ by Lm1; j = i + C by A1, ARYTM_3:58; hence i <=' j by ARYTM_3:def_13; ::_thesis: verum end; Lm11: 2 = succ 1 .= (succ 0) \/ {1} by ORDINAL1:def_1 .= (0 \/ {0}) \/ {1} by ORDINAL1:def_1 .= {0,1} by ENUMSET1:1 ; Lm12: for i, k being natural Ordinal st i *^ i = 2 *^ k holds ex k being natural Ordinal st i = 2 *^ k proof let i, k be natural Ordinal; ::_thesis: ( i *^ i = 2 *^ k implies ex k being natural Ordinal st i = 2 *^ k ) assume A1: i *^ i = 2 *^ k ; ::_thesis: ex k being natural Ordinal st i = 2 *^ k set id2 = i div^ 2; {} in 2 by ORDINAL1:14; then A2: i mod^ 2 in 2 by ARYTM_3:6; percases ( i mod^ 2 = 0 or i mod^ 2 = 1 ) by A2, Lm11, TARSKI:def_2; supposeA3: i mod^ 2 = 0 ; ::_thesis: ex k being natural Ordinal st i = 2 *^ k take k = i div^ 2; ::_thesis: i = 2 *^ k thus i = (k *^ 2) +^ 0 by A3, ORDINAL3:65 .= 2 *^ k by ORDINAL2:27 ; ::_thesis: verum end; suppose i mod^ 2 = 1 ; ::_thesis: ex k being natural Ordinal st i = 2 *^ k then i = ((i div^ 2) *^ 2) +^ 1 by ORDINAL3:65; then A4: i *^ i = (((i div^ 2) *^ 2) *^ (((i div^ 2) *^ 2) +^ 1)) +^ (one *^ (((i div^ 2) *^ 2) +^ 1)) by ORDINAL3:46 .= (((i div^ 2) *^ 2) *^ (((i div^ 2) *^ 2) +^ 1)) +^ ((one *^ ((i div^ 2) *^ 2)) +^ (one *^ 1)) by ORDINAL3:46 .= (((i div^ 2) *^ 2) *^ (((i div^ 2) *^ 2) +^ 1)) +^ ((one *^ ((i div^ 2) *^ 2)) +^ 1) by ORDINAL2:39 .= ((((i div^ 2) *^ 2) *^ (((i div^ 2) *^ 2) +^ 1)) +^ (one *^ ((i div^ 2) *^ 2))) +^ 1 by ORDINAL3:30 .= (((i div^ 2) *^ 2) *^ ((((i div^ 2) *^ 2) +^ 1) +^ one)) +^ 1 by ORDINAL3:46 .= (((i div^ 2) *^ ((((i div^ 2) *^ 2) +^ 1) +^ one)) *^ 2) +^ 1 by ORDINAL3:50 ; A5: 1 divides 2 by ARYTM_3:9; ( 2 divides ((i div^ 2) *^ ((((i div^ 2) *^ 2) +^ 1) +^ one)) *^ 2 & 2 divides i *^ i ) by A1, ARYTM_3:def_3; then 2 divides 1 by A4, ARYTM_3:11; hence ex k being natural Ordinal st i = 2 *^ k by A5, ARYTM_3:8; ::_thesis: verum end; end; end; 1 in omega ; then reconsider a9 = 1 as Element of RAT+ by Lm1; 2 in omega ; then reconsider two = 2 as ordinal Element of RAT+ by Lm1; Lm13: one + one = two proof 1 +^ 1 = succ (1 +^ {}) by Lm2, ORDINAL2:28 .= succ 1 by ORDINAL2:27 .= two ; hence one + one = two by ARYTM_3:58; ::_thesis: verum end; Lm14: for i being Element of RAT+ holds i + i = two *' i proof let i be Element of RAT+ ; ::_thesis: i + i = two *' i thus i + i = (one *' i) + i by ARYTM_3:53 .= (one *' i) + (one *' i) by ARYTM_3:53 .= two *' i by Lm13, ARYTM_3:57 ; ::_thesis: verum end; theorem Th2: :: NUMBERS:2 RAT c< REAL proof defpred S1[ Element of RAT+ ] means $1 *' $1 < two; set X = { s where s is Element of RAT+ : S1[s] } ; reconsider X = { s where s is Element of RAT+ : S1[s] } as Subset of RAT+ from DOMAIN_1:sch_7(); A1: ( 2 *^ 2 = two *' two & 1 *^ 2 = 2 ) by ARYTM_3:59, ORDINAL2:39; 2 = succ 1 .= 1 \/ {1} by ORDINAL1:def_1 ; then A2: a9 <=' two by Lm10, XBOOLE_1:7; then A3: a9 < two by ARYTM_3:68; A4: a9 *' a9 = a9 by ARYTM_3:53; then A5: 1 in X by A3; A6: for r, t being Element of RAT+ st r in X & t <=' r holds t in X proof let r, t be Element of RAT+ ; ::_thesis: ( r in X & t <=' r implies t in X ) assume r in X ; ::_thesis: ( not t <=' r or t in X ) then A7: ex s being Element of RAT+ st ( r = s & s *' s < two ) ; assume t <=' r ; ::_thesis: t in X then ( t *' t <=' t *' r & t *' r <=' r *' r ) by ARYTM_3:82; then t *' t <=' r *' r by ARYTM_3:67; then t *' t < two by A7, ARYTM_3:69; hence t in X ; ::_thesis: verum end; then A8: 0 in X by A5, ARYTM_3:64; now__::_thesis:_not_X_=_[0,0] assume X = [0,0] ; ::_thesis: contradiction then X = {{0},{0}} by ENUMSET1:29 .= {{0}} by ENUMSET1:29 ; hence contradiction by A8, TARSKI:def_1; ::_thesis: verum end; then A9: not X in {[0,0]} by TARSKI:def_1; reconsider 09 = 0 as Element of RAT+ by Lm1; set DD = { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s <=' r holds s in A ) & ex s being Element of RAT+ st ( s in A & r < s ) ) } ; consider half being Element of RAT+ such that A10: a9 = two *' half by ARYTM_3:55; A11: one <=' two by Lm13, ARYTM_3:def_13; then A12: one < two by ARYTM_3:68; A13: now__::_thesis:_not_X_in__{___{__s_where_s_is_Element_of_RAT+_:_s_<_t__}__where_t_is_Element_of_RAT+_:_t_<>_0__}_ assume X in { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> 0 } ; ::_thesis: contradiction then consider t0 being Element of RAT+ such that A14: X = { s where s is Element of RAT+ : s < t0 } and A15: t0 <> 0 ; set n = numerator t0; set d = denominator t0; now__::_thesis:_not_t0_*'_t0_<>_two assume A16: t0 *' t0 <> two ; ::_thesis: contradiction percases ( t0 *' t0 < two or two < t0 *' t0 ) by A16, ARYTM_3:66; suppose t0 *' t0 < two ; ::_thesis: contradiction then t0 in X ; then ex s being Element of RAT+ st ( s = t0 & s < t0 ) by A14; hence contradiction ; ::_thesis: verum end; supposeA17: two < t0 *' t0 ; ::_thesis: contradiction consider es being Element of RAT+ such that A18: ( two + es = t0 *' t0 or (t0 *' t0) + es = two ) by ARYTM_3:92; A19: now__::_thesis:_not_09_=_es assume 09 = es ; ::_thesis: contradiction then two + es = two by ARYTM_3:50; hence contradiction by A17, A18, ARYTM_3:def_13; ::_thesis: verum end; 09 <=' es by ARYTM_3:64; then 09 < es by A19, ARYTM_3:68; then consider s being Element of RAT+ such that A20: 09 < s and A21: s < es by ARYTM_3:93; now__::_thesis:_contradiction percases ( s < one or one <=' s ) ; supposeA22: s < one ; ::_thesis: contradiction A23: s <> 0 by A20; then s *' s < s *' one by A22, ARYTM_3:80; then A24: s *' s < s by ARYTM_3:53; A25: now__::_thesis:_not_t0_<='_one assume A26: t0 <=' one ; ::_thesis: contradiction then t0 *' t0 <=' t0 *' one by ARYTM_3:82; then t0 *' t0 <=' t0 by ARYTM_3:53; then t0 *' t0 <=' one by A26, ARYTM_3:67; hence contradiction by A11, A17, ARYTM_3:69; ::_thesis: verum end; then A27: one *' one < one *' t0 by ARYTM_3:80; one *' t0 < two *' t0 by A12, A15, ARYTM_3:80; then A28: one *' one < two *' t0 by A27, ARYTM_3:70; consider t02s2 being Element of RAT+ such that A29: ( (s *' s) + t02s2 = t0 *' t0 or (t0 *' t0) + t02s2 = s *' s ) by ARYTM_3:92; s < t0 by A22, A25, ARYTM_3:70; then A30: s *' s < t0 by A24, ARYTM_3:70; consider 2t9 being Element of RAT+ such that A31: (two *' t0) *' 2t9 = one by A15, ARYTM_3:55, ARYTM_3:78; set x = (s *' s) *' 2t9; consider t0x being Element of RAT+ such that A32: ( ((s *' s) *' 2t9) + t0x = t0 or t0 + t0x = (s *' s) *' 2t9 ) by ARYTM_3:92; ((s *' s) *' 2t9) *' (two *' t0) = (s *' s) *' one by A31, ARYTM_3:52; then ( (s *' s) *' 2t9 <=' s *' s or two *' t0 <=' one ) by ARYTM_3:83; then A33: (s *' s) *' 2t9 < t0 by A28, A30, ARYTM_3:53, ARYTM_3:69; then A34: t0x <=' t0 by A32, ARYTM_3:def_13; A35: ((((s *' s) *' 2t9) *' t0x) + (((s *' s) *' 2t9) *' t0)) + (((s *' s) *' 2t9) *' ((s *' s) *' 2t9)) = ((((s *' s) *' 2t9) *' t0x) + (((s *' s) *' 2t9) *' ((s *' s) *' 2t9))) + (((s *' s) *' 2t9) *' t0) by ARYTM_3:51 .= (((s *' s) *' 2t9) *' t0) + (((s *' s) *' 2t9) *' t0) by A32, A33, ARYTM_3:57, ARYTM_3:def_13 .= ((((s *' s) *' 2t9) *' t0) *' one) + (((s *' s) *' 2t9) *' t0) by ARYTM_3:53 .= ((((s *' s) *' 2t9) *' t0) *' one) + ((((s *' s) *' 2t9) *' t0) *' one) by ARYTM_3:53 .= (t0 *' ((s *' s) *' 2t9)) *' two by Lm13, ARYTM_3:57 .= ((s *' s) *' 2t9) *' (t0 *' two) by ARYTM_3:52 .= (s *' s) *' one by A31, ARYTM_3:52 .= s *' s by ARYTM_3:53 ; es <=' t0 *' t0 by A17, A18, ARYTM_3:def_13; then s < t0 *' t0 by A21, ARYTM_3:69; then A36: s *' s < t0 *' t0 by A24, ARYTM_3:70; then (t02s2 + (((s *' s) *' 2t9) *' ((s *' s) *' 2t9))) + (s *' s) = ((t0x + ((s *' s) *' 2t9)) *' t0) + (((s *' s) *' 2t9) *' ((s *' s) *' 2t9)) by A29, A32, A33, ARYTM_3:51, ARYTM_3:def_13 .= ((t0x *' (t0x + ((s *' s) *' 2t9))) + (((s *' s) *' 2t9) *' t0)) + (((s *' s) *' 2t9) *' ((s *' s) *' 2t9)) by A32, A33, ARYTM_3:57, ARYTM_3:def_13 .= (((t0x *' t0x) + (((s *' s) *' 2t9) *' t0x)) + (((s *' s) *' 2t9) *' t0)) + (((s *' s) *' 2t9) *' ((s *' s) *' 2t9)) by ARYTM_3:57 .= ((t0x *' t0x) + (((s *' s) *' 2t9) *' t0x)) + ((((s *' s) *' 2t9) *' t0) + (((s *' s) *' 2t9) *' ((s *' s) *' 2t9))) by ARYTM_3:51 .= (t0x *' t0x) + ((((s *' s) *' 2t9) *' t0x) + ((((s *' s) *' 2t9) *' t0) + (((s *' s) *' 2t9) *' ((s *' s) *' 2t9)))) by ARYTM_3:51 .= (t0x *' t0x) + (s *' s) by A35, ARYTM_3:51 ; then t0x *' t0x = t02s2 + (((s *' s) *' 2t9) *' ((s *' s) *' 2t9)) by ARYTM_3:62; then A37: t02s2 <=' t0x *' t0x by ARYTM_3:def_13; now__::_thesis:_not_(s_*'_s)_*'_2t9_=_0 assume A38: (s *' s) *' 2t9 = 0 ; ::_thesis: contradiction percases ( s *' s = 0 or 2t9 = 0 ) by A38, ARYTM_3:78; suppose s *' s = 0 ; ::_thesis: contradiction hence contradiction by A23, ARYTM_3:78; ::_thesis: verum end; suppose 2t9 = 0 ; ::_thesis: contradiction hence contradiction by A31, ARYTM_3:48; ::_thesis: verum end; end; end; then t0x <> t0 by A32, A33, ARYTM_3:84, ARYTM_3:def_13; then t0x < t0 by A34, ARYTM_3:68; then t0x in X by A14; then A39: ex s being Element of RAT+ st ( s = t0x & s *' s < two ) ; s *' s < es by A21, A24, ARYTM_3:70; then two + (s *' s) < two + es by ARYTM_3:76; then two < t02s2 by A17, A18, A29, A36, ARYTM_3:76, ARYTM_3:def_13; hence contradiction by A37, A39, ARYTM_3:69; ::_thesis: verum end; supposeA40: one <=' s ; ::_thesis: contradiction half *' two = one *' one by A10, ARYTM_3:53; then A41: half <=' one by A12, ARYTM_3:83; half <> one by A10, ARYTM_3:53; then A42: half < one by A41, ARYTM_3:68; then half < s by A40, ARYTM_3:69; then A43: half < es by A21, ARYTM_3:70; one <=' two by Lm13, ARYTM_3:def_13; then one < two by ARYTM_3:68; then A44: one *' t0 < two *' t0 by A15, ARYTM_3:80; A45: now__::_thesis:_not_t0_<='_one assume A46: t0 <=' one ; ::_thesis: contradiction then t0 *' t0 <=' t0 *' one by ARYTM_3:82; then t0 *' t0 <=' t0 by ARYTM_3:53; then t0 *' t0 <=' one by A46, ARYTM_3:67; hence contradiction by A11, A17, ARYTM_3:69; ::_thesis: verum end; then one *' one < one *' t0 by ARYTM_3:80; then A47: one *' one < two *' t0 by A44, ARYTM_3:70; set s = half; consider t02s2 being Element of RAT+ such that A48: ( (half *' half) + t02s2 = t0 *' t0 or (t0 *' t0) + t02s2 = half *' half ) by ARYTM_3:92; A49: half <> 0 by A10, ARYTM_3:48; then half *' half < half *' one by A42, ARYTM_3:80; then A50: half *' half < half by ARYTM_3:53; half < t0 by A42, A45, ARYTM_3:70; then A51: half *' half < t0 by A50, ARYTM_3:70; consider 2t9 being Element of RAT+ such that A52: (two *' t0) *' 2t9 = one by A15, ARYTM_3:55, ARYTM_3:78; set x = (half *' half) *' 2t9; consider t0x being Element of RAT+ such that A53: ( ((half *' half) *' 2t9) + t0x = t0 or t0 + t0x = (half *' half) *' 2t9 ) by ARYTM_3:92; ((half *' half) *' 2t9) *' (two *' t0) = (half *' half) *' one by A52, ARYTM_3:52; then ( (half *' half) *' 2t9 <=' half *' half or two *' t0 <=' one ) by ARYTM_3:83; then A54: (half *' half) *' 2t9 < t0 by A47, A51, ARYTM_3:53, ARYTM_3:69; then A55: t0x <=' t0 by A53, ARYTM_3:def_13; A56: ((((half *' half) *' 2t9) *' t0x) + (((half *' half) *' 2t9) *' t0)) + (((half *' half) *' 2t9) *' ((half *' half) *' 2t9)) = ((((half *' half) *' 2t9) *' t0x) + (((half *' half) *' 2t9) *' ((half *' half) *' 2t9))) + (((half *' half) *' 2t9) *' t0) by ARYTM_3:51 .= (((half *' half) *' 2t9) *' t0) + (((half *' half) *' 2t9) *' t0) by A53, A54, ARYTM_3:57, ARYTM_3:def_13 .= ((((half *' half) *' 2t9) *' t0) *' one) + (((half *' half) *' 2t9) *' t0) by ARYTM_3:53 .= ((((half *' half) *' 2t9) *' t0) *' one) + ((((half *' half) *' 2t9) *' t0) *' one) by ARYTM_3:53 .= (t0 *' ((half *' half) *' 2t9)) *' two by Lm13, ARYTM_3:57 .= ((half *' half) *' 2t9) *' (t0 *' two) by ARYTM_3:52 .= (half *' half) *' one by A52, ARYTM_3:52 .= half *' half by ARYTM_3:53 ; es <=' t0 *' t0 by A17, A18, ARYTM_3:def_13; then half < t0 *' t0 by A43, ARYTM_3:69; then A57: half *' half < t0 *' t0 by A50, ARYTM_3:70; then (t02s2 + (((half *' half) *' 2t9) *' ((half *' half) *' 2t9))) + (half *' half) = (t0 *' t0) + (((half *' half) *' 2t9) *' ((half *' half) *' 2t9)) by A48, ARYTM_3:51, ARYTM_3:def_13 .= ((t0x *' (t0x + ((half *' half) *' 2t9))) + (((half *' half) *' 2t9) *' t0)) + (((half *' half) *' 2t9) *' ((half *' half) *' 2t9)) by A53, A54, ARYTM_3:57, ARYTM_3:def_13 .= (((t0x *' t0x) + (((half *' half) *' 2t9) *' t0x)) + (((half *' half) *' 2t9) *' t0)) + (((half *' half) *' 2t9) *' ((half *' half) *' 2t9)) by ARYTM_3:57 .= ((t0x *' t0x) + (((half *' half) *' 2t9) *' t0x)) + ((((half *' half) *' 2t9) *' t0) + (((half *' half) *' 2t9) *' ((half *' half) *' 2t9))) by ARYTM_3:51 .= (t0x *' t0x) + ((((half *' half) *' 2t9) *' t0x) + ((((half *' half) *' 2t9) *' t0) + (((half *' half) *' 2t9) *' ((half *' half) *' 2t9)))) by ARYTM_3:51 .= (t0x *' t0x) + (half *' half) by A56, ARYTM_3:51 ; then t0x *' t0x = t02s2 + (((half *' half) *' 2t9) *' ((half *' half) *' 2t9)) by ARYTM_3:62; then A58: t02s2 <=' t0x *' t0x by ARYTM_3:def_13; now__::_thesis:_not_(half_*'_half)_*'_2t9_=_0 assume A59: (half *' half) *' 2t9 = 0 ; ::_thesis: contradiction percases ( half *' half = 0 or 2t9 = 0 ) by A59, ARYTM_3:78; suppose half *' half = 0 ; ::_thesis: contradiction hence contradiction by A49, ARYTM_3:78; ::_thesis: verum end; suppose 2t9 = 0 ; ::_thesis: contradiction hence contradiction by A52, ARYTM_3:48; ::_thesis: verum end; end; end; then t0x <> t0 by A53, A54, ARYTM_3:84, ARYTM_3:def_13; then t0x < t0 by A55, ARYTM_3:68; then t0x in X by A14; then A60: ex s being Element of RAT+ st ( s = t0x & s *' s < two ) ; half *' half < es by A50, A43, ARYTM_3:70; then two + (half *' half) < two + es by ARYTM_3:76; then two < t02s2 by A17, A18, A48, A57, ARYTM_3:76, ARYTM_3:def_13; hence contradiction by A58, A60, ARYTM_3:69; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; then A61: two / 1 = ((numerator t0) *^ (numerator t0)) / ((denominator t0) *^ (denominator t0)) by ARYTM_3:40; denominator t0 <> 0 by ARYTM_3:35; then (denominator t0) *^ (denominator t0) <> {} by ORDINAL3:31; then A62: two *^ ((denominator t0) *^ (denominator t0)) = 1 *^ ((numerator t0) *^ (numerator t0)) by A61, ARYTM_3:45 .= (numerator t0) *^ (numerator t0) by ORDINAL2:39 ; then consider k being natural Ordinal such that A63: numerator t0 = 2 *^ k by Lm12; two *^ ((denominator t0) *^ (denominator t0)) = 2 *^ (k *^ (2 *^ k)) by A62, A63, ORDINAL3:50; then (denominator t0) *^ (denominator t0) = k *^ (2 *^ k) by ORDINAL3:33 .= 2 *^ (k *^ k) by ORDINAL3:50 ; then A64: ex p being natural Ordinal st denominator t0 = 2 *^ p by Lm12; numerator t0, denominator t0 are_relative_prime by ARYTM_3:34; hence contradiction by A63, A64, ARYTM_3:def_2; ::_thesis: verum end; 2 = succ 1 ; then 1 in 2 by ORDINAL1:6; then A65: 1 *^ 2 in 2 *^ 2 by ORDINAL3:19; A66: 09 <=' a9 by ARYTM_3:64; now__::_thesis:_for_r_being_Element_of_RAT+_st_r_in_X_holds_ (_(_for_t_being_Element_of_RAT+_st_t_<='_r_holds_ t_in_X_)_&_ex_a9_being_Element_of_RAT+_st_ (_a9_in_X_&_r_<_a9_)_) let r be Element of RAT+ ; ::_thesis: ( r in X implies ( ( for t being Element of RAT+ st t <=' r holds t in X ) & ex a9 being Element of RAT+ st ( b2 in X & a9 < b2 ) ) ) assume A67: r in X ; ::_thesis: ( ( for t being Element of RAT+ st t <=' r holds t in X ) & ex a9 being Element of RAT+ st ( b2 in X & a9 < b2 ) ) then A68: ex s being Element of RAT+ st ( r = s & s *' s < two ) ; thus for t being Element of RAT+ st t <=' r holds t in X by A6, A67; ::_thesis: ex a9 being Element of RAT+ st ( b2 in X & a9 < b2 ) percases ( r = 0 or r <> 0 ) ; supposeA69: r = 0 ; ::_thesis: ex a9 being Element of RAT+ st ( b2 in X & a9 < b2 ) take a9 = a9; ::_thesis: ( a9 in X & r < a9 ) thus a9 in X by A4, A3; ::_thesis: r < a9 thus r < a9 by A66, A69, ARYTM_3:68; ::_thesis: verum end; supposeA70: r <> 0 ; ::_thesis: ex t being Element of RAT+ st ( b2 in X & t < b2 ) then consider 3r9 being Element of RAT+ such that A71: ((r + r) + r) *' 3r9 = one by ARYTM_3:54, ARYTM_3:63; consider rr being Element of RAT+ such that A72: ( (r *' r) + rr = two or two + rr = r *' r ) by ARYTM_3:92; set eps = rr *' 3r9; A73: now__::_thesis:_not_rr_*'_3r9_=_0 assume A74: rr *' 3r9 = 0 ; ::_thesis: contradiction percases ( rr = 09 or 3r9 = 09 ) by A74, ARYTM_3:78; suppose rr = 09 ; ::_thesis: contradiction then r *' r = two by A72, ARYTM_3:50; hence contradiction by A68; ::_thesis: verum end; suppose 3r9 = 09 ; ::_thesis: contradiction hence contradiction by A71, ARYTM_3:48; ::_thesis: verum end; end; end; now__::_thesis:_ex_t_being_Element_of_RAT+_st_ (_t_in_X_&_r_<_t_) percases ( rr *' 3r9 < r or r <=' rr *' 3r9 ) ; suppose rr *' 3r9 < r ; ::_thesis: ex t being Element of RAT+ st ( t in X & r < t ) then (rr *' 3r9) *' (rr *' 3r9) < r *' (rr *' 3r9) by A73, ARYTM_3:80; then A75: ((r *' (rr *' 3r9)) + ((rr *' 3r9) *' r)) + ((rr *' 3r9) *' (rr *' 3r9)) < ((r *' (rr *' 3r9)) + ((rr *' 3r9) *' r)) + (r *' (rr *' 3r9)) by ARYTM_3:76; take t = r + (rr *' 3r9); ::_thesis: ( t in X & r < t ) A76: t *' t = (r *' t) + ((rr *' 3r9) *' t) by ARYTM_3:57 .= ((r *' r) + (r *' (rr *' 3r9))) + ((rr *' 3r9) *' t) by ARYTM_3:57 .= ((r *' r) + (r *' (rr *' 3r9))) + (((rr *' 3r9) *' r) + ((rr *' 3r9) *' (rr *' 3r9))) by ARYTM_3:57 .= (r *' r) + ((r *' (rr *' 3r9)) + (((rr *' 3r9) *' r) + ((rr *' 3r9) *' (rr *' 3r9)))) by ARYTM_3:51 .= (r *' r) + (((r *' (rr *' 3r9)) + ((rr *' 3r9) *' r)) + ((rr *' 3r9) *' (rr *' 3r9))) by ARYTM_3:51 ; ((r *' (rr *' 3r9)) + ((rr *' 3r9) *' r)) + (r *' (rr *' 3r9)) = ((rr *' 3r9) *' (r + r)) + (r *' (rr *' 3r9)) by ARYTM_3:57 .= (rr *' 3r9) *' ((r + r) + r) by ARYTM_3:57 .= rr *' one by A71, ARYTM_3:52 .= rr by ARYTM_3:53 ; then t *' t < two by A68, A72, A75, A76, ARYTM_3:76, ARYTM_3:def_13; hence t in X ; ::_thesis: r < t 09 <=' rr *' 3r9 by ARYTM_3:64; then 09 < rr *' 3r9 by A73, ARYTM_3:68; then r + 09 < r + (rr *' 3r9) by ARYTM_3:76; hence r < t by ARYTM_3:50; ::_thesis: verum end; supposeA77: r <=' rr *' 3r9 ; ::_thesis: ex t being Element of RAT+ st ( t in X & r < t ) (rr *' 3r9) *' ((r + r) + r) = rr *' one by A71, ARYTM_3:52 .= rr by ARYTM_3:53 ; then A78: r *' ((r + r) + r) <=' rr by A77, ARYTM_3:82; take t = (a9 + half) *' r; ::_thesis: ( t in X & r < t ) a9 < two *' one by A3, ARYTM_3:53; then half < one by A10, ARYTM_3:82; then one + half < two by Lm13, ARYTM_3:76; then A79: t < two *' r by A70, ARYTM_3:80; then A80: (two *' r) *' t < (two *' r) *' (two *' r) by A70, ARYTM_3:78, ARYTM_3:80; a9 + half <> 0 by ARYTM_3:63; then t *' t < (two *' r) *' t by A70, A79, ARYTM_3:78, ARYTM_3:80; then A81: t *' t < (two *' r) *' (two *' r) by A80, ARYTM_3:70; (r *' ((r + r) + r)) + (r *' r) = ((r *' (r + r)) + (r *' r)) + (r *' r) by ARYTM_3:57 .= (r *' (r + r)) + ((r *' r) + (r *' r)) by ARYTM_3:51 .= (r *' (two *' r)) + ((r *' r) + (r *' r)) by Lm14 .= (r *' (two *' r)) + (two *' (r *' r)) by Lm14 .= (two *' (r *' r)) + (two *' (r *' r)) by ARYTM_3:52 .= two *' (two *' (r *' r)) by Lm14 .= two *' ((two *' r) *' r) by ARYTM_3:52 .= (two *' r) *' (two *' r) by ARYTM_3:52 ; then (two *' r) *' (two *' r) <=' two by A68, A72, A78, ARYTM_3:76, ARYTM_3:def_13; then t *' t < two by A81, ARYTM_3:69; hence t in X ; ::_thesis: r < t ( 09 <> half & 09 <=' half ) by A10, ARYTM_3:48, ARYTM_3:64; then 09 < half by ARYTM_3:68; then one + 09 < one + half by ARYTM_3:76; then one < one + half by ARYTM_3:50; then one *' r < t by A70, ARYTM_3:80; hence r < t by ARYTM_3:53; ::_thesis: verum end; end; end; hence ex t being Element of RAT+ st ( t in X & r < t ) ; ::_thesis: verum end; end; end; then A82: X in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s <=' r holds s in A ) & ex s being Element of RAT+ st ( s in A & r < s ) ) } ; a9 *' half = half by ARYTM_3:53; then A83: half in X by A10, A6, A2, A5, ARYTM_3:82; A84: now__::_thesis:_not_X_in_RAT assume A85: X in RAT ; ::_thesis: contradiction percases ( X in RAT+ or X in [:{0},RAT+:] ) by A85, XBOOLE_0:def_3; supposeA86: X in RAT+ ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( X in { [i,j] where i, j is Element of omega : ( i,j are_relative_prime & j <> {} ) } \ { [k,one] where k is Element of omega : verum } or X in omega ) by A86, XBOOLE_0:def_3; suppose X in { [i,j] where i, j is Element of omega : ( i,j are_relative_prime & j <> {} ) } \ { [k,one] where k is Element of omega : verum } ; ::_thesis: contradiction then X in { [i,j] where i, j is Element of omega : ( i,j are_relative_prime & j <> {} ) } ; then ex i, j being Element of omega st ( X = [i,j] & i,j are_relative_prime & j <> {} ) ; hence contradiction by A8, TARSKI:def_2; ::_thesis: verum end; supposeA87: X in omega ; ::_thesis: contradiction 2 c= X by A5, A8, Lm11, ZFMISC_1:32; then A88: not X in 2 by ORDINAL1:5; now__::_thesis:_contradiction percases ( X = two or two in X ) by A87, A88, ORDINAL1:14; suppose X = two ; ::_thesis: contradiction then ( half = 0 or half = 1 ) by A83, Lm11, TARSKI:def_2; hence contradiction by A10, ARYTM_3:48, ARYTM_3:53; ::_thesis: verum end; suppose two in X ; ::_thesis: contradiction then ex s being Element of RAT+ st ( s = two & s *' s < two ) ; hence contradiction by A1, A65, Lm9; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; suppose X in [:{0},RAT+:] ; ::_thesis: contradiction then ex x, y being set st X = [x,y] by RELAT_1:def_1; hence contradiction by A8, TARSKI:def_2; ::_thesis: verum end; end; end; now__::_thesis:_not_two_in_X assume two in X ; ::_thesis: contradiction then ex s being Element of RAT+ st ( two = s & s *' s < two ) ; hence contradiction by A1, A65, Lm9; ::_thesis: verum end; then X <> RAT+ ; then not X in {RAT+} by TARSKI:def_1; then X in DEDEKIND_CUTS by A82, ARYTM_2:def_1, XBOOLE_0:def_5; then X in RAT+ \/ DEDEKIND_CUTS by XBOOLE_0:def_3; then X in REAL+ by A13, ARYTM_2:def_2, XBOOLE_0:def_5; then X in REAL+ \/ [:{0},REAL+:] by XBOOLE_0:def_3; then X in REAL by A9, XBOOLE_0:def_5; hence RAT c< REAL by A84, Lm8, XBOOLE_0:def_8; ::_thesis: verum end; theorem Th3: :: NUMBERS:3 RAT c< COMPLEX by Th1, Th2, XBOOLE_1:56; Lm15: INT c= RAT proof [:{0},NAT:] c= [:{0},RAT+:] by Lm1, ZFMISC_1:95; then NAT \/ [:{0},NAT:] c= RAT+ \/ [:{0},RAT+:] by Lm1, XBOOLE_1:13; hence INT c= RAT by XBOOLE_1:33; ::_thesis: verum end; theorem Th4: :: NUMBERS:4 INT c< RAT proof 1,2 are_relative_prime proof let c, d1, d2 be Ordinal; :: according to ARYTM_3:def_2 ::_thesis: ( not 1 = c *^ d1 or not 2 = c *^ d2 or c = 1 ) assume that A1: 1 = c *^ d1 and 2 = c *^ d2 ; ::_thesis: c = 1 thus c = 1 by A1, ORDINAL3:37; ::_thesis: verum end; then A2: [1,2] in RAT+ by ARYTM_3:33; not 1 in {0} by TARSKI:def_1; then ( not [1,2] in NAT & not [1,2] in [:{0},NAT:] ) by ARYTM_3:32, ZFMISC_1:87; then not [1,2] in NAT \/ [:{0},NAT:] by XBOOLE_0:def_3; then INT <> RAT by A2, Lm4, XBOOLE_0:def_5; hence INT c< RAT by Lm15, XBOOLE_0:def_8; ::_thesis: verum end; theorem Th5: :: NUMBERS:5 INT c< REAL by Th2, Th4, XBOOLE_1:56; theorem Th6: :: NUMBERS:6 INT c< COMPLEX by Th1, Th5, XBOOLE_1:56; theorem Th7: :: NUMBERS:7 NAT c< INT proof 0 in {0} by TARSKI:def_1; then [0,1] in [:{0},NAT:] by ZFMISC_1:87; then A1: [0,1] in NAT \/ [:{0},NAT:] by XBOOLE_0:def_3; A2: not [0,1] in NAT by ARYTM_3:32; [0,1] <> [0,0] by XTUPLE_0:1; then not [0,1] in {[0,0]} by TARSKI:def_1; then [0,1] in INT by A1, XBOOLE_0:def_5; hence NAT c< INT by A2, Lm5, XBOOLE_0:def_8; ::_thesis: verum end; theorem Th8: :: NUMBERS:8 NAT c< RAT by Th4, Th7, XBOOLE_1:56; theorem Th9: :: NUMBERS:9 NAT c< REAL by Th2, Th8, XBOOLE_1:56; theorem Th10: :: NUMBERS:10 NAT c< COMPLEX by Th1, Th9, XBOOLE_1:56; begin theorem :: NUMBERS:11 REAL c= COMPLEX by Th1, XBOOLE_0:def_8; theorem :: NUMBERS:12 RAT c= REAL by Th2, XBOOLE_0:def_8; theorem :: NUMBERS:13 RAT c= COMPLEX by Th3, XBOOLE_0:def_8; theorem :: NUMBERS:14 INT c= RAT by Th4, XBOOLE_0:def_8; theorem :: NUMBERS:15 INT c= REAL by Th5, XBOOLE_0:def_8; theorem :: NUMBERS:16 INT c= COMPLEX by Th6, XBOOLE_0:def_8; theorem :: NUMBERS:17 NAT c= INT by Lm5; theorem Th18: :: NUMBERS:18 NAT c= RAT by Th8, XBOOLE_0:def_8; theorem Th19: :: NUMBERS:19 NAT c= REAL ; theorem Th20: :: NUMBERS:20 NAT c= COMPLEX by Th10, XBOOLE_0:def_8; theorem :: NUMBERS:21 REAL <> COMPLEX by Th1; theorem :: NUMBERS:22 RAT <> REAL by Th2; theorem :: NUMBERS:23 RAT <> COMPLEX by Th1, Th2; theorem :: NUMBERS:24 INT <> RAT by Th4; theorem :: NUMBERS:25 INT <> REAL by Th2, Th4; theorem :: NUMBERS:26 INT <> COMPLEX by Th1, Th2, Th4, XBOOLE_1:56; theorem :: NUMBERS:27 NAT <> INT by Th7; theorem :: NUMBERS:28 NAT <> RAT by Th4, Th7; theorem :: NUMBERS:29 NAT <> REAL by Th2, Th4, Th7, XBOOLE_1:56; theorem :: NUMBERS:30 NAT <> COMPLEX by Th1, Th2, Th8, XBOOLE_1:56; definition func ExtREAL -> set equals :: NUMBERS:def 5 REAL \/ {REAL,[0,REAL]}; coherence REAL \/ {REAL,[0,REAL]} is set ; end; :: deftheorem 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 REAL in {REAL,[0,REAL]} by TARSKI:def_2; then REAL in ExtREAL by XBOOLE_0:def_3; hence REAL <> ExtREAL ; ::_thesis: verum end; theorem :: NUMBERS:33 REAL c< ExtREAL by Th31, Th32, XBOOLE_0:def_8; registration cluster INT -> infinite ; coherence not INT is finite by Lm5, FINSET_1:1; cluster RAT -> infinite ; coherence not RAT is finite by Th18, FINSET_1:1; cluster REAL -> infinite ; coherence not REAL is finite by Th19, FINSET_1:1; cluster COMPLEX -> infinite ; coherence not COMPLEX is finite by Th20, FINSET_1:1; end;