:: CALCUL_1 semantic presentation begin definition let D be non empty set ; let f be FinSequence of D; func Ant f -> FinSequence of D means :Def1: :: CALCUL_1:def 1 for i being Element of NAT st len f = i + 1 holds it = f | (Seg i) if len f > 0 otherwise it = {} ; existence ( ( len f > 0 implies ex b1 being FinSequence of D st for i being Element of NAT st len f = i + 1 holds b1 = f | (Seg i) ) & ( not len f > 0 implies ex b1 being FinSequence of D st b1 = {} ) ) proof A1: ( len f > 0 implies ex g being FinSequence of D st for i being Element of NAT st len f = i + 1 holds g = f | (Seg i) ) proof assume len f > 0 ; ::_thesis: ex g being FinSequence of D st for i being Element of NAT st len f = i + 1 holds g = f | (Seg i) then consider j being Nat such that A2: len f = j + 1 by NAT_1:6; reconsider j = j as Element of NAT by ORDINAL1:def_12; take g = f | (Seg j); ::_thesis: ( g is FinSequence of D & ( for i being Element of NAT st len f = i + 1 holds g = f | (Seg i) ) ) reconsider g = g as FinSequence by FINSEQ_1:15; now__::_thesis:_for_a_being_set_st_a_in_rng_g_holds_ a_in_D A3: rng g c= rng f by RELAT_1:70; let a be set ; ::_thesis: ( a in rng g implies a in D ) assume a in rng g ; ::_thesis: a in D then a in rng f by A3; hence a in D ; ::_thesis: verum end; then rng g c= D by TARSKI:def_3; then reconsider g = g as FinSequence of D by FINSEQ_1:def_4; for i being Element of NAT st len f = i + 1 holds g = f | (Seg i) by A2; hence ( g is FinSequence of D & ( for i being Element of NAT st len f = i + 1 holds g = f | (Seg i) ) ) ; ::_thesis: verum end; <*> D = {} ; hence ( ( len f > 0 implies ex b1 being FinSequence of D st for i being Element of NAT st len f = i + 1 holds b1 = f | (Seg i) ) & ( not len f > 0 implies ex b1 being FinSequence of D st b1 = {} ) ) by A1; ::_thesis: verum end; uniqueness for b1, b2 being FinSequence of D holds ( ( len f > 0 & ( for i being Element of NAT st len f = i + 1 holds b1 = f | (Seg i) ) & ( for i being Element of NAT st len f = i + 1 holds b2 = f | (Seg i) ) implies b1 = b2 ) & ( not len f > 0 & b1 = {} & b2 = {} implies b1 = b2 ) ) proof let g, h be FinSequence of D; ::_thesis: ( ( len f > 0 & ( for i being Element of NAT st len f = i + 1 holds g = f | (Seg i) ) & ( for i being Element of NAT st len f = i + 1 holds h = f | (Seg i) ) implies g = h ) & ( not len f > 0 & g = {} & h = {} implies g = h ) ) ( len f > 0 & ( for i being Element of NAT st len f = i + 1 holds g = f | (Seg i) ) & ( for i being Element of NAT st len f = i + 1 holds h = f | (Seg i) ) implies g = h ) proof assume that A4: len f > 0 and A5: for i being Element of NAT st len f = i + 1 holds g = f | (Seg i) and A6: for i being Element of NAT st len f = i + 1 holds h = f | (Seg i) ; ::_thesis: g = h consider j being Nat such that A7: len f = j + 1 by A4, NAT_1:6; reconsider j = j as Element of NAT by ORDINAL1:def_12; g = f | (Seg j) by A5, A7; hence g = h by A6, A7; ::_thesis: verum end; hence ( ( len f > 0 & ( for i being Element of NAT st len f = i + 1 holds g = f | (Seg i) ) & ( for i being Element of NAT st len f = i + 1 holds h = f | (Seg i) ) implies g = h ) & ( not len f > 0 & g = {} & h = {} implies g = h ) ) ; ::_thesis: verum end; consistency for b1 being FinSequence of D holds verum ; end; :: deftheorem Def1 defines Ant CALCUL_1:def_1_:_ for D being non empty set for f, b3 being FinSequence of D holds ( ( len f > 0 implies ( b3 = Ant f iff for i being Element of NAT st len f = i + 1 holds b3 = f | (Seg i) ) ) & ( not len f > 0 implies ( b3 = Ant f iff b3 = {} ) ) ); definition let Al be QC-alphabet ; let f be FinSequence of CQC-WFF Al; func Suc f -> Element of CQC-WFF Al equals :Def2: :: CALCUL_1:def 2 f . (len f) if len f > 0 otherwise VERUM Al; coherence ( ( len f > 0 implies f . (len f) is Element of CQC-WFF Al ) & ( not len f > 0 implies VERUM Al is Element of CQC-WFF Al ) ) proof percases ( len f > 0 or not len f > 0 ) ; suppose len f > 0 ; ::_thesis: ( ( len f > 0 implies f . (len f) is Element of CQC-WFF Al ) & ( not len f > 0 implies VERUM Al is Element of CQC-WFF Al ) ) then 0 + 1 <= len f by NAT_1:13; then len f in dom f by FINSEQ_3:25; then f . (len f) in rng f by FUNCT_1:3; hence ( ( len f > 0 implies f . (len f) is Element of CQC-WFF Al ) & ( not len f > 0 implies VERUM Al is Element of CQC-WFF Al ) ) ; ::_thesis: verum end; supposeA1: not len f > 0 ; ::_thesis: ( ( len f > 0 implies f . (len f) is Element of CQC-WFF Al ) & ( not len f > 0 implies VERUM Al is Element of CQC-WFF Al ) ) thus ( ( len f > 0 implies f . (len f) is Element of CQC-WFF Al ) & ( not len f > 0 implies VERUM Al is Element of CQC-WFF Al ) ) by A1; ::_thesis: verum end; end; end; consistency for b1 being Element of CQC-WFF Al holds verum ; end; :: deftheorem Def2 defines Suc CALCUL_1:def_2_:_ for Al being QC-alphabet for f being FinSequence of CQC-WFF Al holds ( ( len f > 0 implies Suc f = f . (len f) ) & ( not len f > 0 implies Suc f = VERUM Al ) ); definition let f be Relation; let p be set ; predp is_tail_of f means :Def3: :: CALCUL_1:def 3 p in rng f; end; :: deftheorem Def3 defines is_tail_of CALCUL_1:def_3_:_ for f being Relation for p being set holds ( p is_tail_of f iff p in rng f ); Lm1: now__::_thesis:_for_f_being_FinSequence for_p_being_set_st_p_is_tail_of_f_holds_ ex_i_being_Element_of_NAT_st_ (_i_in_dom_f_&_f_._i_=_p_) let f be FinSequence; ::_thesis: for p being set st p is_tail_of f holds ex i being Element of NAT st ( i in dom f & f . i = p ) let p be set ; ::_thesis: ( p is_tail_of f implies ex i being Element of NAT st ( i in dom f & f . i = p ) ) assume p is_tail_of f ; ::_thesis: ex i being Element of NAT st ( i in dom f & f . i = p ) then p in rng f by Def3; then consider i being set such that A1: i in dom f and A2: f . i = p by FUNCT_1:def_3; reconsider i = i as Element of NAT by A1; take i = i; ::_thesis: ( i in dom f & f . i = p ) thus ( i in dom f & f . i = p ) by A1, A2; ::_thesis: verum end; Lm2: now__::_thesis:_for_f_being_FinSequence for_p_being_set_st_ex_i_being_Element_of_NAT_st_ (_i_in_dom_f_&_f_._i_=_p_)_holds_ p_is_tail_of_f let f be FinSequence; ::_thesis: for p being set st ex i being Element of NAT st ( i in dom f & f . i = p ) holds p is_tail_of f let p be set ; ::_thesis: ( ex i being Element of NAT st ( i in dom f & f . i = p ) implies p is_tail_of f ) assume ex i being Element of NAT st ( i in dom f & f . i = p ) ; ::_thesis: p is_tail_of f then p in rng f by FUNCT_1:def_3; hence p is_tail_of f by Def3; ::_thesis: verum end; definition let Al be QC-alphabet ; let f, g be FinSequence of CQC-WFF Al; predf is_Subsequence_of g means :Def4: :: CALCUL_1:def 4 ex N being Subset of NAT st f c= Seq (g | N); end; :: deftheorem Def4 defines is_Subsequence_of CALCUL_1:def_4_:_ for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al holds ( f is_Subsequence_of g iff ex N being Subset of NAT st f c= Seq (g | N) ); theorem Th1: :: CALCUL_1:1 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al st f is_Subsequence_of g holds ( rng f c= rng g & ex N being Subset of NAT st rng f c= rng (g | N) ) proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al st f is_Subsequence_of g holds ( rng f c= rng g & ex N being Subset of NAT st rng f c= rng (g | N) ) let f, g be FinSequence of CQC-WFF Al; ::_thesis: ( f is_Subsequence_of g implies ( rng f c= rng g & ex N being Subset of NAT st rng f c= rng (g | N) ) ) assume f is_Subsequence_of g ; ::_thesis: ( rng f c= rng g & ex N being Subset of NAT st rng f c= rng (g | N) ) then consider N being Subset of NAT such that A1: f c= Seq (g | N) by Def4; A2: rng (g | N) c= rng g by RELAT_1:70; A3: now__::_thesis:_for_a_being_set_st_a_in_rng_f_holds_ a_in_rng_(g_|_N) rng (Seq (g | N)) = rng ((g | N) * (Sgm (dom (g | N)))) by FINSEQ_1:def_14; then A4: rng (Seq (g | N)) c= rng (g | N) by RELAT_1:26; let a be set ; ::_thesis: ( a in rng f implies a in rng (g | N) ) assume a in rng f ; ::_thesis: a in rng (g | N) then consider n being Nat such that A5: n in dom f and A6: f . n = a by FINSEQ_2:10; [n,(f . n)] in f by A5, FUNCT_1:1; then A7: (Seq (g | N)) . n = a by A1, A6, FUNCT_1:1; dom f c= dom (Seq (g | N)) by A1, RELAT_1:11; then a in rng (Seq (g | N)) by A5, A7, FUNCT_1:3; hence a in rng (g | N) by A4; ::_thesis: verum end; then rng f c= rng (g | N) by TARSKI:def_3; hence rng f c= rng g by A2, XBOOLE_1:1; ::_thesis: ex N being Subset of NAT st rng f c= rng (g | N) take N ; ::_thesis: rng f c= rng (g | N) thus rng f c= rng (g | N) by A3, TARSKI:def_3; ::_thesis: verum end; theorem Th2: :: CALCUL_1:2 for Al being QC-alphabet for f being FinSequence of CQC-WFF Al st len f > 0 holds ( (len (Ant f)) + 1 = len f & len (Ant f) < len f ) proof let Al be QC-alphabet ; ::_thesis: for f being FinSequence of CQC-WFF Al st len f > 0 holds ( (len (Ant f)) + 1 = len f & len (Ant f) < len f ) let f be FinSequence of CQC-WFF Al; ::_thesis: ( len f > 0 implies ( (len (Ant f)) + 1 = len f & len (Ant f) < len f ) ) assume len f > 0 ; ::_thesis: ( (len (Ant f)) + 1 = len f & len (Ant f) < len f ) then consider i being Nat such that A1: len f = i + 1 by NAT_1:6; reconsider i = i as Element of NAT by ORDINAL1:def_12; Ant f = f | (Seg i) by A1, Def1; then dom (Ant f) = (dom f) /\ (Seg i) by RELAT_1:61; then Seg (len (Ant f)) = (dom f) /\ (Seg i) by FINSEQ_1:def_3; then A2: Seg (len (Ant f)) = (Seg (len f)) /\ (Seg i) by FINSEQ_1:def_3; i <= len f by A1, NAT_1:11; then A3: Seg i c= Seg (len f) by FINSEQ_1:5; hence (len (Ant f)) + 1 = len f by A1, A2, FINSEQ_1:6, XBOOLE_1:28; ::_thesis: len (Ant f) < len f len (Ant f) = i by A2, A3, FINSEQ_1:6, XBOOLE_1:28; hence len (Ant f) < len f by A1, NAT_1:13; ::_thesis: verum end; theorem Th3: :: CALCUL_1:3 for Al being QC-alphabet for f being FinSequence of CQC-WFF Al st len f > 0 holds ( f = (Ant f) ^ <*(Suc f)*> & rng f = (rng (Ant f)) \/ {(Suc f)} ) proof let Al be QC-alphabet ; ::_thesis: for f being FinSequence of CQC-WFF Al st len f > 0 holds ( f = (Ant f) ^ <*(Suc f)*> & rng f = (rng (Ant f)) \/ {(Suc f)} ) let f be FinSequence of CQC-WFF Al; ::_thesis: ( len f > 0 implies ( f = (Ant f) ^ <*(Suc f)*> & rng f = (rng (Ant f)) \/ {(Suc f)} ) ) assume A1: len f > 0 ; ::_thesis: ( f = (Ant f) ^ <*(Suc f)*> & rng f = (rng (Ant f)) \/ {(Suc f)} ) then A2: len f = (len (Ant f)) + 1 by Th2; A3: dom f = Seg (len f) by FINSEQ_1:def_3; A4: now__::_thesis:_for_j_being_Nat_st_j_in_dom_f_holds_ f_._j_=_((Ant_f)_^_<*(Suc_f)*>)_._j let j be Nat; ::_thesis: ( j in dom f implies f . j = ((Ant f) ^ <*(Suc f)*>) . j ) assume A5: j in dom f ; ::_thesis: f . j = ((Ant f) ^ <*(Suc f)*>) . j A6: 1 <= j by A3, A5, FINSEQ_1:1; A7: now__::_thesis:_(_j_<=_len_(Ant_f)_implies_f_._j_=_((Ant_f)_^_<*(Suc_f)*>)_._j_) assume j <= len (Ant f) ; ::_thesis: f . j = ((Ant f) ^ <*(Suc f)*>) . j then A8: j in dom (Ant f) by A6, FINSEQ_3:25; Ant f = f | (Seg (len (Ant f))) by A2, Def1; then Ant f = f | (dom (Ant f)) by FINSEQ_1:def_3; then f . j = (Ant f) . j by A8, FUNCT_1:49; hence f . j = ((Ant f) ^ <*(Suc f)*>) . j by A8, FINSEQ_1:def_7; ::_thesis: verum end; A9: now__::_thesis:_(_j_=_(len_(Ant_f))_+_1_implies_f_._j_=_((Ant_f)_^_<*(Suc_f)*>)_._j_) 1 in Seg 1 by FINSEQ_1:1; then A10: 1 in dom <*(Suc f)*> by FINSEQ_1:38; assume A11: j = (len (Ant f)) + 1 ; ::_thesis: f . j = ((Ant f) ^ <*(Suc f)*>) . j then j = len f by A1, Th2; then f . j = Suc f by A1, Def2; then f . j = <*(Suc f)*> . 1 by FINSEQ_1:40; hence f . j = ((Ant f) ^ <*(Suc f)*>) . j by A11, A10, FINSEQ_1:def_7; ::_thesis: verum end; j <= (len (Ant f)) + 1 by A2, A3, A5, FINSEQ_1:1; hence f . j = ((Ant f) ^ <*(Suc f)*>) . j by A7, A9, NAT_1:8; ::_thesis: verum end; len f = (len (Ant f)) + (len <*(Suc f)*>) by A2, FINSEQ_1:39; then A12: len f = len ((Ant f) ^ <*(Suc f)*>) by FINSEQ_1:22; then f = (Ant f) ^ <*(Suc f)*> by A4, FINSEQ_2:9; then rng f = (rng (Ant f)) \/ (rng <*(Suc f)*>) by FINSEQ_1:31; hence ( f = (Ant f) ^ <*(Suc f)*> & rng f = (rng (Ant f)) \/ {(Suc f)} ) by A12, A4, FINSEQ_1:38, FINSEQ_2:9; ::_thesis: verum end; theorem Th4: :: CALCUL_1:4 for Al being QC-alphabet for f being FinSequence of CQC-WFF Al st len f > 1 holds len (Ant f) > 0 proof let Al be QC-alphabet ; ::_thesis: for f being FinSequence of CQC-WFF Al st len f > 1 holds len (Ant f) > 0 let f be FinSequence of CQC-WFF Al; ::_thesis: ( len f > 1 implies len (Ant f) > 0 ) assume len f > 1 ; ::_thesis: len (Ant f) > 0 then (len (Ant f)) + 1 > 1 by Th2; hence len (Ant f) > 0 by NAT_1:13; ::_thesis: verum end; theorem Th5: :: CALCUL_1:5 for Al being QC-alphabet for p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al holds ( Suc (f ^ <*p*>) = p & Ant (f ^ <*p*>) = f ) proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al holds ( Suc (f ^ <*p*>) = p & Ant (f ^ <*p*>) = f ) let p be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al holds ( Suc (f ^ <*p*>) = p & Ant (f ^ <*p*>) = f ) let f be FinSequence of CQC-WFF Al; ::_thesis: ( Suc (f ^ <*p*>) = p & Ant (f ^ <*p*>) = f ) set fin = f ^ <*p*>; A1: len (f ^ <*p*>) = (len f) + 1 by FINSEQ_2:16; then (f ^ <*p*>) . (len (f ^ <*p*>)) = p by FINSEQ_1:42; hence Suc (f ^ <*p*>) = p by Def2; ::_thesis: Ant (f ^ <*p*>) = f thus Ant (f ^ <*p*>) = f ::_thesis: verum proof set fin = f ^ <*p*>; now__::_thesis:_for_a_being_set_st_a_in_f_holds_ a_in_f_^_<*p*> let a be set ; ::_thesis: ( a in f implies a in f ^ <*p*> ) assume a in f ; ::_thesis: a in f ^ <*p*> then consider k being Nat such that A2: k in dom f and A3: a = [k,(f . k)] by FINSEQ_1:12; ( k in dom (f ^ <*p*>) & f . k = (f ^ <*p*>) . k ) by A2, FINSEQ_1:def_7, FINSEQ_2:15; hence a in f ^ <*p*> by A3, FUNCT_1:1; ::_thesis: verum end; then f c= f ^ <*p*> by TARSKI:def_3; then f = (f ^ <*p*>) | (dom f) by GRFUNC_1:23; then f = (f ^ <*p*>) | (Seg (len f)) by FINSEQ_1:def_3; hence Ant (f ^ <*p*>) = f by A1, Def1; ::_thesis: verum end; end; theorem Th6: :: CALCUL_1:6 for fin, fin1 being FinSequence holds ( len fin <= len (fin ^ fin1) & len fin1 <= len (fin ^ fin1) & ( fin <> {} implies ( 1 <= len fin & len fin1 < len (fin1 ^ fin) ) ) ) proof let fin, fin1 be FinSequence; ::_thesis: ( len fin <= len (fin ^ fin1) & len fin1 <= len (fin ^ fin1) & ( fin <> {} implies ( 1 <= len fin & len fin1 < len (fin1 ^ fin) ) ) ) len (fin ^ fin1) = (len fin) + (len fin1) by FINSEQ_1:22; hence ( len fin <= len (fin ^ fin1) & len fin1 <= len (fin ^ fin1) ) by NAT_1:12; ::_thesis: ( fin <> {} implies ( 1 <= len fin & len fin1 < len (fin1 ^ fin) ) ) assume fin <> {} ; ::_thesis: ( 1 <= len fin & len fin1 < len (fin1 ^ fin) ) then A1: 0 + 1 <= len fin by NAT_1:13; then (len fin1) + 1 <= (len fin) + (len fin1) by XREAL_1:6; then (len fin1) + 1 <= len (fin1 ^ fin) by FINSEQ_1:22; hence ( 1 <= len fin & len fin1 < len (fin1 ^ fin) ) by A1, NAT_1:13; ::_thesis: verum end; theorem Th7: :: CALCUL_1:7 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al holds Seq ((f ^ g) | (dom f)) = (f ^ g) | (dom f) proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al holds Seq ((f ^ g) | (dom f)) = (f ^ g) | (dom f) let f, g be FinSequence of CQC-WFF Al; ::_thesis: Seq ((f ^ g) | (dom f)) = (f ^ g) | (dom f) (f ^ g) | (dom f) = f by FINSEQ_1:21; hence Seq ((f ^ g) | (dom f)) = (f ^ g) | (dom f) by FINSEQ_3:116; ::_thesis: verum end; theorem Th8: :: CALCUL_1:8 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al holds f is_Subsequence_of f ^ g proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al holds f is_Subsequence_of f ^ g let f, g be FinSequence of CQC-WFF Al; ::_thesis: f is_Subsequence_of f ^ g set a = len f; take N = Seg (len f); :: according to CALCUL_1:def_4 ::_thesis: f c= Seq ((f ^ g) | N) reconsider f1 = (f ^ g) | N as FinSequence by FINSEQ_1:15; A1: N = dom f by FINSEQ_1:def_3; then f c= f1 by FINSEQ_1:21; hence f c= Seq ((f ^ g) | N) by A1, Th7; ::_thesis: verum end; theorem Th9: :: CALCUL_1:9 for b, c being set for fin being FinSequence holds 1 < len ((fin ^ <*b*>) ^ <*c*>) proof let b, c be set ; ::_thesis: for fin being FinSequence holds 1 < len ((fin ^ <*b*>) ^ <*c*>) let fin be FinSequence; ::_thesis: 1 < len ((fin ^ <*b*>) ^ <*c*>) len ((fin ^ <*b*>) ^ <*c*>) = (len (fin ^ <*b*>)) + (len <*c*>) by FINSEQ_1:22; then len ((fin ^ <*b*>) ^ <*c*>) = (len (fin ^ <*b*>)) + 1 by FINSEQ_1:39; then len ((fin ^ <*b*>) ^ <*c*>) = ((len fin) + (len <*b*>)) + 1 by FINSEQ_1:22; then len ((fin ^ <*b*>) ^ <*c*>) = ((len fin) + 1) + 1 by FINSEQ_1:39; then len ((fin ^ <*b*>) ^ <*c*>) = (len fin) + (1 + 1) ; then 1 + 1 <= len ((fin ^ <*b*>) ^ <*c*>) by NAT_1:11; hence 1 < len ((fin ^ <*b*>) ^ <*c*>) by NAT_1:13; ::_thesis: verum end; theorem Th10: :: CALCUL_1:10 for b being set for fin being FinSequence holds ( 1 <= len (fin ^ <*b*>) & len (fin ^ <*b*>) in dom (fin ^ <*b*>) ) proof let b be set ; ::_thesis: for fin being FinSequence holds ( 1 <= len (fin ^ <*b*>) & len (fin ^ <*b*>) in dom (fin ^ <*b*>) ) let fin be FinSequence; ::_thesis: ( 1 <= len (fin ^ <*b*>) & len (fin ^ <*b*>) in dom (fin ^ <*b*>) ) len (fin ^ <*b*>) = (len fin) + 1 by FINSEQ_2:16; then 1 <= len (fin ^ <*b*>) by NAT_1:11; hence ( 1 <= len (fin ^ <*b*>) & len (fin ^ <*b*>) in dom (fin ^ <*b*>) ) by FINSEQ_3:25; ::_thesis: verum end; theorem Th11: :: CALCUL_1:11 for m, n being Element of NAT st 0 < m holds len (Sgm ((Seg n) \/ {(n + m)})) = n + 1 proof let m, n be Element of NAT ; ::_thesis: ( 0 < m implies len (Sgm ((Seg n) \/ {(n + m)})) = n + 1 ) A1: m <= n + m by NAT_1:11; assume A2: 0 < m ; ::_thesis: len (Sgm ((Seg n) \/ {(n + m)})) = n + 1 then card ((Seg n) \/ {(n + m)}) = (card (Seg n)) + 1 by CARD_2:41, FINSEQ_3:10; then A3: card ((Seg n) \/ {(n + m)}) = n + 1 by FINSEQ_1:57; 0 + 1 <= m by A2, NAT_1:13; then 1 <= m + n by A1, XXREAL_0:2; then n + m in Seg (n + m) by FINSEQ_1:1; then A4: {(n + m)} c= Seg (n + m) by ZFMISC_1:31; Seg n c= Seg (n + m) by FINSEQ_3:18; hence len (Sgm ((Seg n) \/ {(n + m)})) = n + 1 by A3, A4, FINSEQ_3:39, XBOOLE_1:8; ::_thesis: verum end; theorem Th12: :: CALCUL_1:12 for m, n being Element of NAT st 0 < m holds dom (Sgm ((Seg n) \/ {(n + m)})) = Seg (n + 1) proof let m, n be Element of NAT ; ::_thesis: ( 0 < m implies dom (Sgm ((Seg n) \/ {(n + m)})) = Seg (n + 1) ) assume 0 < m ; ::_thesis: dom (Sgm ((Seg n) \/ {(n + m)})) = Seg (n + 1) then len (Sgm ((Seg n) \/ {(n + m)})) = n + 1 by Th11; hence dom (Sgm ((Seg n) \/ {(n + m)})) = Seg (n + 1) by FINSEQ_1:def_3; ::_thesis: verum end; theorem Th13: :: CALCUL_1:13 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al st 0 < len f holds f is_Subsequence_of ((Ant f) ^ g) ^ <*(Suc f)*> proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al st 0 < len f holds f is_Subsequence_of ((Ant f) ^ g) ^ <*(Suc f)*> let f, g be FinSequence of CQC-WFF Al; ::_thesis: ( 0 < len f implies f is_Subsequence_of ((Ant f) ^ g) ^ <*(Suc f)*> ) set n = len (Ant f); set m = len g; set N = (Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))}; set f1 = ((Ant f) ^ g) ^ <*(Suc f)*>; reconsider f2 = (((Ant f) ^ g) ^ <*(Suc f)*>) | ((Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))}) as FinSubsequence ; assume A1: 0 < len f ; ::_thesis: f is_Subsequence_of ((Ant f) ^ g) ^ <*(Suc f)*> take (Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))} ; :: according to CALCUL_1:def_4 ::_thesis: f c= Seq ((((Ant f) ^ g) ^ <*(Suc f)*>) | ((Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))})) now__::_thesis:_for_b_being_set_st_b_in_f_holds_ b_in_Seq_f2 now__::_thesis:_for_b_being_set_st_b_in_(Seg_(len_(Ant_f)))_\/_{((len_(Ant_f))_+_((len_g)_+_1))}_holds_ b_in_dom_(((Ant_f)_^_g)_^_<*(Suc_f)*>) let b be set ; ::_thesis: ( b in (Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))} implies b in dom (((Ant f) ^ g) ^ <*(Suc f)*>) ) assume A2: b in (Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))} ; ::_thesis: b in dom (((Ant f) ^ g) ^ <*(Suc f)*>) reconsider i = b as Element of NAT by A2; A3: now__::_thesis:_(_i_in_{((len_(Ant_f))_+_((len_g)_+_1))}_implies_i_in_dom_(((Ant_f)_^_g)_^_<*(Suc_f)*>)_) assume i in {((len (Ant f)) + ((len g) + 1))} ; ::_thesis: i in dom (((Ant f) ^ g) ^ <*(Suc f)*>) then A4: i = ((len (Ant f)) + (len g)) + 1 by TARSKI:def_1; then A5: 1 <= i by NAT_1:11; len (((Ant f) ^ g) ^ <*(Suc f)*>) = (len ((Ant f) ^ g)) + (len <*(Suc f)*>) by FINSEQ_1:22; then len (((Ant f) ^ g) ^ <*(Suc f)*>) = ((len (Ant f)) + (len g)) + (len <*(Suc f)*>) by FINSEQ_1:22; then i <= len (((Ant f) ^ g) ^ <*(Suc f)*>) by A4, FINSEQ_1:39; hence i in dom (((Ant f) ^ g) ^ <*(Suc f)*>) by A5, FINSEQ_3:25; ::_thesis: verum end; now__::_thesis:_(_i_in_Seg_(len_(Ant_f))_implies_i_in_dom_(((Ant_f)_^_g)_^_<*(Suc_f)*>)_) ((Ant f) ^ g) ^ <*(Suc f)*> = (Ant f) ^ (g ^ <*(Suc f)*>) by FINSEQ_1:32; then A6: len (Ant f) <= len (((Ant f) ^ g) ^ <*(Suc f)*>) by Th6; assume A7: i in Seg (len (Ant f)) ; ::_thesis: i in dom (((Ant f) ^ g) ^ <*(Suc f)*>) then A8: 1 <= i by FINSEQ_1:1; i <= len (Ant f) by A7, FINSEQ_1:1; then i <= len (((Ant f) ^ g) ^ <*(Suc f)*>) by A6, XXREAL_0:2; hence i in dom (((Ant f) ^ g) ^ <*(Suc f)*>) by A8, FINSEQ_3:25; ::_thesis: verum end; hence b in dom (((Ant f) ^ g) ^ <*(Suc f)*>) by A2, A3, XBOOLE_0:def_3; ::_thesis: verum end; then A9: (Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))} c= dom (((Ant f) ^ g) ^ <*(Suc f)*>) by TARSKI:def_3; dom f2 = (dom (((Ant f) ^ g) ^ <*(Suc f)*>)) /\ ((Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))}) by RELAT_1:61; then A10: dom f2 = (Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))} by A9, XBOOLE_1:28; then A11: dom (Sgm (dom f2)) = Seg ((len (Ant f)) + 1) by Th12; now__::_thesis:_for_i_being_Element_of_NAT_holds_ (_i_in_dom_f_iff_i_in_Seg_((len_(Ant_f))_+_1)_) let i be Element of NAT ; ::_thesis: ( i in dom f iff i in Seg ((len (Ant f)) + 1) ) ( i in dom f iff ( 1 <= i & i <= len f ) ) by FINSEQ_3:25; then ( i in dom f iff ( 1 <= i & i <= (len (Ant f)) + 1 ) ) by A1, Th2; hence ( i in dom f iff i in Seg ((len (Ant f)) + 1) ) by FINSEQ_1:1; ::_thesis: verum end; then for b being set holds ( b in dom f iff b in Seg ((len (Ant f)) + 1) ) ; then A12: dom (Sgm (dom f2)) = dom f by A11, TARSKI:1; A13: now__::_thesis:_for_i,_j_being_Element_of_NAT_st_i_in_Seg_(len_(Ant_f))_&_j_in_{((len_(Ant_f))_+_((len_g)_+_1))}_holds_ i_<_j let i, j be Element of NAT ; ::_thesis: ( i in Seg (len (Ant f)) & j in {((len (Ant f)) + ((len g) + 1))} implies i < j ) assume that A14: i in Seg (len (Ant f)) and A15: j in {((len (Ant f)) + ((len g) + 1))} ; ::_thesis: i < j A16: i <= len (Ant f) by A14, FINSEQ_1:1; (len (Ant f)) + 1 <= ((len (Ant f)) + 1) + (len g) by NAT_1:11; then len (Ant f) < (len (Ant f)) + ((len g) + 1) by NAT_1:13; then len (Ant f) < j by A15, TARSKI:def_1; hence i < j by A16, XXREAL_0:2; ::_thesis: verum end; let b be set ; ::_thesis: ( b in f implies b in Seq f2 ) assume A17: b in f ; ::_thesis: b in Seq f2 consider c, d being set such that A18: b = [c,d] by A17, RELAT_1:def_1; A19: c in dom f by A17, A18, FUNCT_1:1; then reconsider i = c as Element of NAT ; ( 0 + 1 <= (len g) + 1 & (len g) + 1 <= (len (Ant f)) + ((len g) + 1) ) by NAT_1:11; then 1 <= (len (Ant f)) + ((len g) + 1) by XXREAL_0:2; then (len (Ant f)) + ((len g) + 1) in Seg ((len (Ant f)) + ((len g) + 1)) by FINSEQ_1:1; then {((len (Ant f)) + ((len g) + 1))} c= Seg ((len (Ant f)) + ((len g) + 1)) by ZFMISC_1:31; then Sgm (dom f2) = (Sgm (Seg (len (Ant f)))) ^ (Sgm {((len (Ant f)) + ((len g) + 1))}) by A10, A13, FINSEQ_3:42; then Sgm (dom f2) = (Sgm (Seg (len (Ant f)))) ^ <*((len (Ant f)) + ((len g) + 1))*> by FINSEQ_3:44; then A20: Sgm (dom f2) = (idseq (len (Ant f))) ^ <*((len (Ant f)) + ((len g) + 1))*> by FINSEQ_3:48; A21: now__::_thesis:_(_i_in_Seg_(len_(Ant_f))_implies_(Seq_f2)_._i_=_f_._i_) assume A22: i in Seg (len (Ant f)) ; ::_thesis: (Seq f2) . i = f . i then A23: i in dom (Ant f) by FINSEQ_1:def_3; i in dom (idseq (len (Ant f))) by A22, RELAT_1:45; then (Sgm (dom f2)) . i = (idseq (len (Ant f))) . i by A20, FINSEQ_1:def_7; then A24: (Sgm (dom f2)) . i = i by A22, FUNCT_1:18; ( i in dom (Sgm (dom f2)) & Seq f2 = f2 * (Sgm (dom f2)) ) by A17, A18, A12, FINSEQ_1:def_14, FUNCT_1:1; then (Seq f2) . i = f2 . i by A24, FUNCT_1:13; then (Seq f2) . i = (f2 | (Seg (len (Ant f)))) . i by A22, FUNCT_1:49; then A25: (Seq f2) . i = ((((Ant f) ^ g) ^ <*(Suc f)*>) | (Seg (len (Ant f)))) . i by RELAT_1:74, XBOOLE_1:7; ( ((Ant f) ^ g) ^ <*(Suc f)*> = (Ant f) ^ (g ^ <*(Suc f)*>) & Seg (len (Ant f)) = dom (Ant f) ) by FINSEQ_1:32, FINSEQ_1:def_3; then A26: (Seq f2) . i = (Ant f) . i by A25, FINSEQ_1:21; f = (Ant f) ^ <*(Suc f)*> by A1, Th3; hence (Seq f2) . i = f . i by A26, A23, FINSEQ_1:def_7; ::_thesis: verum end; rng (Sgm (dom ((((Ant f) ^ g) ^ <*(Suc f)*>) | ((Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))})))) = dom ((((Ant f) ^ g) ^ <*(Suc f)*>) | ((Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))})) by FINSEQ_1:50; then dom f = dom (((((Ant f) ^ g) ^ <*(Suc f)*>) | ((Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))})) * (Sgm (dom ((((Ant f) ^ g) ^ <*(Suc f)*>) | ((Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))}))))) by A12, RELAT_1:27; then A27: dom f = dom (Seq f2) by FINSEQ_1:def_14; A28: now__::_thesis:_(_i_=_(len_(Ant_f))_+_1_implies_(Seq_f2)_._i_=_f_._i_) 1 in Seg 1 by FINSEQ_1:1; then A29: ( len ((Ant f) ^ g) = (len (Ant f)) + (len g) & 1 in dom <*(Suc f)*> ) by FINSEQ_1:22, FINSEQ_1:38; A30: ( i in dom (Sgm (dom f2)) & Seq f2 = f2 * (Sgm (dom f2)) ) by A17, A18, A12, FINSEQ_1:def_14, FUNCT_1:1; assume A31: i = (len (Ant f)) + 1 ; ::_thesis: (Seq f2) . i = f . i len (idseq (len (Ant f))) = len (Ant f) by CARD_1:def_7; then (Sgm (dom f2)) . i = (len (Ant f)) + ((len g) + 1) by A20, A31, FINSEQ_1:42; then A32: (Seq f2) . i = f2 . ((len (Ant f)) + ((len g) + 1)) by A30, FUNCT_1:13; ( (len (Ant f)) + ((len g) + 1) in {((len (Ant f)) + ((len g) + 1))} & {((len (Ant f)) + ((len g) + 1))} c= (Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))} ) by TARSKI:def_1, XBOOLE_1:7; then (Seq f2) . i = (((Ant f) ^ g) ^ <*(Suc f)*>) . (((len (Ant f)) + (len g)) + 1) by A32, FUNCT_1:49; then A33: (Seq f2) . i = <*(Suc f)*> . 1 by A29, FINSEQ_1:def_7; f . i = f . (len f) by A1, A31, Th2; then f . i = Suc f by A1, Def2; hence (Seq f2) . i = f . i by A33, FINSEQ_1:40; ::_thesis: verum end; d = f . c by A17, A18, FUNCT_1:1; hence b in Seq f2 by A18, A19, A11, A12, A21, A28, A27, FINSEQ_2:7, FUNCT_1:1; ::_thesis: verum end; hence f c= Seq ((((Ant f) ^ g) ^ <*(Suc f)*>) | ((Seg (len (Ant f))) \/ {((len (Ant f)) + ((len g) + 1))})) by TARSKI:def_3; ::_thesis: verum end; theorem Th14: :: CALCUL_1:14 for Al being QC-alphabet for c, d being set for f being FinSequence of CQC-WFF Al holds ( 1 in dom <*c,d*> & 2 in dom <*c,d*> & (f ^ <*c,d*>) . ((len f) + 1) = c & (f ^ <*c,d*>) . ((len f) + 2) = d ) proof let Al be QC-alphabet ; ::_thesis: for c, d being set for f being FinSequence of CQC-WFF Al holds ( 1 in dom <*c,d*> & 2 in dom <*c,d*> & (f ^ <*c,d*>) . ((len f) + 1) = c & (f ^ <*c,d*>) . ((len f) + 2) = d ) let c, d be set ; ::_thesis: for f being FinSequence of CQC-WFF Al holds ( 1 in dom <*c,d*> & 2 in dom <*c,d*> & (f ^ <*c,d*>) . ((len f) + 1) = c & (f ^ <*c,d*>) . ((len f) + 2) = d ) let f be FinSequence of CQC-WFF Al; ::_thesis: ( 1 in dom <*c,d*> & 2 in dom <*c,d*> & (f ^ <*c,d*>) . ((len f) + 1) = c & (f ^ <*c,d*>) . ((len f) + 2) = d ) A1: 2 <= len <*c,d*> by FINSEQ_1:44; then 2 in dom <*c,d*> by FINSEQ_3:25; then A2: (f ^ <*c,d*>) . ((len f) + 2) = <*c,d*> . 2 by FINSEQ_1:def_7; 1 <= 2 ; then A3: 1 <= len <*c,d*> by FINSEQ_1:44; then 1 in dom <*c,d*> by FINSEQ_3:25; then (f ^ <*c,d*>) . ((len f) + 1) = <*c,d*> . 1 by FINSEQ_1:def_7; hence ( 1 in dom <*c,d*> & 2 in dom <*c,d*> & (f ^ <*c,d*>) . ((len f) + 1) = c & (f ^ <*c,d*>) . ((len f) + 2) = d ) by A3, A1, A2, FINSEQ_1:44, FINSEQ_3:25; ::_thesis: verum end; begin definition let Al be QC-alphabet ; let f be FinSequence of CQC-WFF Al; func still_not-bound_in f -> Subset of (bound_QC-variables Al) means :Def5: :: CALCUL_1:def 5 for a being set holds ( a in it iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ); existence ex b1 being Subset of (bound_QC-variables Al) st for a being set holds ( a in b1 iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) proof defpred S1[ set ] means ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & \$1 in still_not-bound_in p ); consider X being set such that A1: for a being set holds ( a in X iff ( a in bound_QC-variables Al & S1[a] ) ) from XBOOLE_0:sch_1(); take X ; ::_thesis: ( X is Subset of (bound_QC-variables Al) & ( for a being set holds ( a in X iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) ) ) for a being set st a in X holds a in bound_QC-variables Al by A1; hence X is Subset of (bound_QC-variables Al) by TARSKI:def_3; ::_thesis: for a being set holds ( a in X iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) thus for a being set holds ( a in X iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Subset of (bound_QC-variables Al) st ( for a being set holds ( a in b1 iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) ) & ( for a being set holds ( a in b2 iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) ) holds b1 = b2 proof let X, Y be Subset of (bound_QC-variables Al); ::_thesis: ( ( for a being set holds ( a in X iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) ) & ( for a being set holds ( a in Y iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) ) implies X = Y ) assume that A2: for a being set holds ( a in X iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) and A3: for a being set holds ( a in Y iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) ; ::_thesis: X = Y now__::_thesis:_for_a_being_set_holds_ (_a_in_X_iff_a_in_Y_) let a be set ; ::_thesis: ( a in X iff a in Y ) ( a in X iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) by A2; hence ( a in X iff a in Y ) by A3; ::_thesis: verum end; hence X = Y by TARSKI:1; ::_thesis: verum end; end; :: deftheorem Def5 defines still_not-bound_in CALCUL_1:def_5_:_ for Al being QC-alphabet for f being FinSequence of CQC-WFF Al for b3 being Subset of (bound_QC-variables Al) holds ( b3 = still_not-bound_in f iff for a being set holds ( a in b3 iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) ); definition let Al be QC-alphabet ; func set_of_CQC-WFF-seq Al -> set means :Def6: :: CALCUL_1:def 6 for a being set holds ( a in it iff a is FinSequence of CQC-WFF Al ); existence ex b1 being set st for a being set holds ( a in b1 iff a is FinSequence of CQC-WFF Al ) proof defpred S1[ set ] means \$1 is FinSequence of CQC-WFF Al; consider X being set such that A1: for a being set holds ( a in X iff ( a in bool [:NAT,(CQC-WFF Al):] & S1[a] ) ) from XBOOLE_0:sch_1(); take X ; ::_thesis: for a being set holds ( a in X iff a is FinSequence of CQC-WFF Al ) thus for a being set holds ( a in X iff a is FinSequence of CQC-WFF Al ) by A1; ::_thesis: verum end; uniqueness for b1, b2 being set st ( for a being set holds ( a in b1 iff a is FinSequence of CQC-WFF Al ) ) & ( for a being set holds ( a in b2 iff a is FinSequence of CQC-WFF Al ) ) holds b1 = b2 proof let X, Y be set ; ::_thesis: ( ( for a being set holds ( a in X iff a is FinSequence of CQC-WFF Al ) ) & ( for a being set holds ( a in Y iff a is FinSequence of CQC-WFF Al ) ) implies X = Y ) assume that A2: for a being set holds ( a in X iff a is FinSequence of CQC-WFF Al ) and A3: for a being set holds ( a in Y iff a is FinSequence of CQC-WFF Al ) ; ::_thesis: X = Y now__::_thesis:_for_a_being_set_holds_ (_a_in_X_iff_a_in_Y_) let a be set ; ::_thesis: ( a in X iff a in Y ) ( a in X iff a is FinSequence of CQC-WFF Al ) by A2; hence ( a in X iff a in Y ) by A3; ::_thesis: verum end; hence X = Y by TARSKI:1; ::_thesis: verum end; end; :: deftheorem Def6 defines set_of_CQC-WFF-seq CALCUL_1:def_6_:_ for Al being QC-alphabet for b2 being set holds ( b2 = set_of_CQC-WFF-seq Al iff for a being set holds ( a in b2 iff a is FinSequence of CQC-WFF Al ) ); definition let Al be QC-alphabet ; let PR be FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:]; let n be Nat; predPR,n is_a_correct_step means :Def7: :: CALCUL_1:def 7 ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) if (PR . n) `2 = 0 ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> if (PR . n) `2 = 1 ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) if (PR . n) `2 = 2 ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) if (PR . n) `2 = 3 ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) if (PR . n) `2 = 4 ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) if (PR . n) `2 = 5 ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) if (PR . n) `2 = 6 ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) if (PR . n) `2 = 7 ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) if (PR . n) `2 = 8 ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) if (PR . n) `2 = 9 ; consistency ( ( (PR . n) `2 = 0 & (PR . n) `2 = 1 implies ( ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 2 implies ( ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 3 implies ( ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 4 implies ( ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 5 implies ( ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 6 implies ( ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 7 implies ( ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 8 implies ( ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 9 implies ( ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 2 implies ( ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> iff ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 3 implies ( ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 4 implies ( ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 5 implies ( ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 6 implies ( ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 7 implies ( ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 8 implies ( ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 9 implies ( ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 3 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 4 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 5 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 6 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 7 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 8 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 9 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 4 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 5 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 6 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 7 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 8 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 9 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 & (PR . n) `2 = 5 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 & (PR . n) `2 = 6 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 & (PR . n) `2 = 7 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 & (PR . n) `2 = 8 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 & (PR . n) `2 = 9 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 & (PR . n) `2 = 6 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 & (PR . n) `2 = 7 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 & (PR . n) `2 = 8 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 & (PR . n) `2 = 9 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 6 & (PR . n) `2 = 7 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 6 & (PR . n) `2 = 8 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 6 & (PR . n) `2 = 9 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 7 & (PR . n) `2 = 8 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 7 & (PR . n) `2 = 9 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 8 & (PR . n) `2 = 9 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) ) ; end; :: deftheorem Def7 defines is_a_correct_step CALCUL_1:def_7_:_ for Al being QC-alphabet for PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] for n being Nat holds ( ( (PR . n) `2 = 0 implies ( PR,n is_a_correct_step iff ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) ) ) & ( (PR . n) `2 = 1 implies ( PR,n is_a_correct_step iff ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> ) ) & ( (PR . n) `2 = 2 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) ) ) & ( (PR . n) `2 = 3 implies ( PR,n is_a_correct_step iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 implies ( PR,n is_a_correct_step iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 implies ( PR,n is_a_correct_step iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 6 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 7 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 8 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 9 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) ); definition let Al be QC-alphabet ; let PR be FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:]; attrPR is a_proof means :Def8: :: CALCUL_1:def 8 ( PR <> {} & ( for n being Nat st 1 <= n & n <= len PR holds PR,n is_a_correct_step ) ); end; :: deftheorem Def8 defines a_proof CALCUL_1:def_8_:_ for Al being QC-alphabet for PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] holds ( PR is a_proof iff ( PR <> {} & ( for n being Nat st 1 <= n & n <= len PR holds PR,n is_a_correct_step ) ) ); definition let Al be QC-alphabet ; let f be FinSequence of CQC-WFF Al; pred |- f means :Def9: :: CALCUL_1:def 9 ex PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] st ( PR is a_proof & f = (PR . (len PR)) `1 ); end; :: deftheorem Def9 defines |- CALCUL_1:def_9_:_ for Al being QC-alphabet for f being FinSequence of CQC-WFF Al holds ( |- f iff ex PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] st ( PR is a_proof & f = (PR . (len PR)) `1 ) ); definition let Al be QC-alphabet ; let p be Element of CQC-WFF Al; let X be Subset of (CQC-WFF Al); predp is_formal_provable_from X means :Def10: :: CALCUL_1:def 10 ex f being FinSequence of CQC-WFF Al st ( rng (Ant f) c= X & Suc f = p & |- f ); end; :: deftheorem Def10 defines is_formal_provable_from CALCUL_1:def_10_:_ for Al being QC-alphabet for p being Element of CQC-WFF Al for X being Subset of (CQC-WFF Al) holds ( p is_formal_provable_from X iff ex f being FinSequence of CQC-WFF Al st ( rng (Ant f) c= X & Suc f = p & |- f ) ); definition let Al be QC-alphabet ; let X be Subset of (CQC-WFF Al); let A be non empty set ; let J be interpretation of Al,A; let v be Element of Valuations_in (Al,A); predJ,v |= X means :Def11: :: CALCUL_1:def 11 for p being Element of CQC-WFF Al st p in X holds J,v |= p; end; :: deftheorem Def11 defines |= CALCUL_1:def_11_:_ for Al being QC-alphabet for X being Subset of (CQC-WFF Al) for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) holds ( J,v |= X iff for p being Element of CQC-WFF Al st p in X holds J,v |= p ); definition let Al be QC-alphabet ; let X be Subset of (CQC-WFF Al); let p be Element of CQC-WFF Al; predX |= p means :: CALCUL_1:def 12 for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= X holds J,v |= p; end; :: deftheorem defines |= CALCUL_1:def_12_:_ for Al being QC-alphabet for X being Subset of (CQC-WFF Al) for p being Element of CQC-WFF Al holds ( X |= p iff for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= X holds J,v |= p ); definition let Al be QC-alphabet ; let p be Element of CQC-WFF Al; pred |= p means :: CALCUL_1:def 13 {} (CQC-WFF Al) |= p; end; :: deftheorem defines |= CALCUL_1:def_13_:_ for Al being QC-alphabet for p being Element of CQC-WFF Al holds ( |= p iff {} (CQC-WFF Al) |= p ); definition let Al be QC-alphabet ; let f be FinSequence of CQC-WFF Al; let A be non empty set ; let J be interpretation of Al,A; let v be Element of Valuations_in (Al,A); predJ,v |= f means :Def14: :: CALCUL_1:def 14 J,v |= rng f; end; :: deftheorem Def14 defines |= CALCUL_1:def_14_:_ for Al being QC-alphabet for f being FinSequence of CQC-WFF Al for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) holds ( J,v |= f iff J,v |= rng f ); definition let Al be QC-alphabet ; let f be FinSequence of CQC-WFF Al; let p be Element of CQC-WFF Al; predf |= p means :Def15: :: CALCUL_1:def 15 for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= f holds J,v |= p; end; :: deftheorem Def15 defines |= CALCUL_1:def_15_:_ for Al being QC-alphabet for f being FinSequence of CQC-WFF Al for p being Element of CQC-WFF Al holds ( f |= p iff for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= f holds J,v |= p ); theorem Th15: :: CALCUL_1:15 for Al being QC-alphabet for f being FinSequence of CQC-WFF Al st Suc f is_tail_of Ant f holds Ant f |= Suc f proof let Al be QC-alphabet ; ::_thesis: for f being FinSequence of CQC-WFF Al st Suc f is_tail_of Ant f holds Ant f |= Suc f let f be FinSequence of CQC-WFF Al; ::_thesis: ( Suc f is_tail_of Ant f implies Ant f |= Suc f ) assume Suc f is_tail_of Ant f ; ::_thesis: Ant f |= Suc f then ex i being Element of NAT st ( i in dom (Ant f) & (Ant f) . i = Suc f ) by Lm1; then A1: Suc f in rng (Ant f) by FUNCT_1:3; let A be non empty set ; :: according to CALCUL_1:def_15 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= Suc f let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= Suc f let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= Ant f implies J,v |= Suc f ) assume J,v |= rng (Ant f) ; :: according to CALCUL_1:def_14 ::_thesis: J,v |= Suc f hence J,v |= Suc f by A1, Def11; ::_thesis: verum end; theorem Th16: :: CALCUL_1:16 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al st Ant f is_Subsequence_of Ant g & Suc f = Suc g & Ant f |= Suc f holds Ant g |= Suc g proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al st Ant f is_Subsequence_of Ant g & Suc f = Suc g & Ant f |= Suc f holds Ant g |= Suc g let f, g be FinSequence of CQC-WFF Al; ::_thesis: ( Ant f is_Subsequence_of Ant g & Suc f = Suc g & Ant f |= Suc f implies Ant g |= Suc g ) assume that A1: Ant f is_Subsequence_of Ant g and A2: ( Suc f = Suc g & Ant f |= Suc f ) ; ::_thesis: Ant g |= Suc g let A be non empty set ; :: according to CALCUL_1:def_15 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= Ant g holds J,v |= Suc g let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant g holds J,v |= Suc g let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= Ant g implies J,v |= Suc g ) assume A3: J,v |= rng (Ant g) ; :: according to CALCUL_1:def_14 ::_thesis: J,v |= Suc g now__::_thesis:_for_p_being_Element_of_CQC-WFF_Al_st_p_in_rng_(Ant_f)_holds_ J,v_|=_p let p be Element of CQC-WFF Al; ::_thesis: ( p in rng (Ant f) implies J,v |= p ) assume A4: p in rng (Ant f) ; ::_thesis: J,v |= p rng (Ant f) c= rng (Ant g) by A1, Th1; hence J,v |= p by A3, A4, Def11; ::_thesis: verum end; then J,v |= rng (Ant f) by Def11; then J,v |= Ant f by Def14; hence J,v |= Suc g by A2, Def15; ::_thesis: verum end; theorem Th17: :: CALCUL_1:17 for Al being QC-alphabet for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) for f being FinSequence of CQC-WFF Al st len f > 0 holds ( ( J,v |= Ant f & J,v |= Suc f ) iff J,v |= f ) proof let Al be QC-alphabet ; ::_thesis: for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) for f being FinSequence of CQC-WFF Al st len f > 0 holds ( ( J,v |= Ant f & J,v |= Suc f ) iff J,v |= f ) let A be non empty set ; ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) for f being FinSequence of CQC-WFF Al st len f > 0 holds ( ( J,v |= Ant f & J,v |= Suc f ) iff J,v |= f ) let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) for f being FinSequence of CQC-WFF Al st len f > 0 holds ( ( J,v |= Ant f & J,v |= Suc f ) iff J,v |= f ) let v be Element of Valuations_in (Al,A); ::_thesis: for f being FinSequence of CQC-WFF Al st len f > 0 holds ( ( J,v |= Ant f & J,v |= Suc f ) iff J,v |= f ) let f be FinSequence of CQC-WFF Al; ::_thesis: ( len f > 0 implies ( ( J,v |= Ant f & J,v |= Suc f ) iff J,v |= f ) ) assume A1: len f > 0 ; ::_thesis: ( ( J,v |= Ant f & J,v |= Suc f ) iff J,v |= f ) thus ( J,v |= Ant f & J,v |= Suc f implies J,v |= f ) ::_thesis: ( J,v |= f implies ( J,v |= Ant f & J,v |= Suc f ) ) proof assume that A2: J,v |= Ant f and A3: J,v |= Suc f ; ::_thesis: J,v |= f let p be Element of CQC-WFF Al; :: according to CALCUL_1:def_11,CALCUL_1:def_14 ::_thesis: ( p in rng f implies J,v |= p ) assume p in rng f ; ::_thesis: J,v |= p then p in (rng (Ant f)) \/ {(Suc f)} by A1, Th3; then A4: ( p in rng (Ant f) or p in {(Suc f)} ) by XBOOLE_0:def_3; J,v |= rng (Ant f) by A2, Def14; hence J,v |= p by A3, A4, Def11, TARSKI:def_1; ::_thesis: verum end; thus ( J,v |= f implies ( J,v |= Ant f & J,v |= Suc f ) ) ::_thesis: verum proof assume A5: J,v |= rng f ; :: according to CALCUL_1:def_14 ::_thesis: ( J,v |= Ant f & J,v |= Suc f ) thus J,v |= rng (Ant f) :: according to CALCUL_1:def_14 ::_thesis: J,v |= Suc f proof A6: rng (Ant f) c= (rng (Ant f)) \/ {(Suc f)} by XBOOLE_1:7; let p be Element of CQC-WFF Al; :: according to CALCUL_1:def_11 ::_thesis: ( p in rng (Ant f) implies J,v |= p ) assume p in rng (Ant f) ; ::_thesis: J,v |= p then p in (rng (Ant f)) \/ {(Suc f)} by A6; then p in rng f by A1, Th3; hence J,v |= p by A5, Def11; ::_thesis: verum end; 0 + 1 <= len f by A1, NAT_1:13; then A7: len f in dom f by FINSEQ_3:25; Suc f = f . (len f) by A1, Def2; then Suc f in rng f by A7, FUNCT_1:3; hence J,v |= Suc f by A5, Def11; ::_thesis: verum end; end; theorem Th18: :: CALCUL_1:18 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al st len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & Ant f |= Suc f & Ant g |= Suc g holds Ant (Ant f) |= Suc f proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al st len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & Ant f |= Suc f & Ant g |= Suc g holds Ant (Ant f) |= Suc f let f, g be FinSequence of CQC-WFF Al; ::_thesis: ( len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & Ant f |= Suc f & Ant g |= Suc g implies Ant (Ant f) |= Suc f ) assume that A1: len f > 1 and A2: len g > 1 and A3: Ant (Ant f) = Ant (Ant g) and A4: 'not' (Suc (Ant f)) = Suc (Ant g) and A5: Suc f = Suc g and A6: Ant f |= Suc f and A7: Ant g |= Suc g ; ::_thesis: Ant (Ant f) |= Suc f let A be non empty set ; :: according to CALCUL_1:def_15 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= Ant (Ant f) holds J,v |= Suc f let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant (Ant f) holds J,v |= Suc f let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= Ant (Ant f) implies J,v |= Suc f ) assume A8: J,v |= Ant (Ant f) ; ::_thesis: J,v |= Suc f A9: len (Ant g) > 0 by A2, Th4; A10: now__::_thesis:_(_not_J,v_|=_Suc_(Ant_f)_implies_J,v_|=_Suc_f_) assume not J,v |= Suc (Ant f) ; ::_thesis: J,v |= Suc f then J,v |= Suc (Ant g) by A4, VALUAT_1:17; then J,v |= Ant g by A3, A9, A8, Th17; hence J,v |= Suc f by A5, A7, Def15; ::_thesis: verum end; A11: len (Ant f) > 0 by A1, Th4; now__::_thesis:_(_J,v_|=_Suc_(Ant_f)_implies_J,v_|=_Suc_f_) assume J,v |= Suc (Ant f) ; ::_thesis: J,v |= Suc f then J,v |= Ant f by A11, A8, Th17; hence J,v |= Suc f by A6, Def15; ::_thesis: verum end; hence J,v |= Suc f by A10; ::_thesis: verum end; theorem Th19: :: CALCUL_1:19 for Al being QC-alphabet for p being Element of CQC-WFF Al for f, g being FinSequence of CQC-WFF Al st len f > 1 & Ant f = Ant g & 'not' p = Suc (Ant f) & 'not' (Suc f) = Suc g & Ant f |= Suc f & Ant g |= Suc g holds Ant (Ant f) |= p proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for f, g being FinSequence of CQC-WFF Al st len f > 1 & Ant f = Ant g & 'not' p = Suc (Ant f) & 'not' (Suc f) = Suc g & Ant f |= Suc f & Ant g |= Suc g holds Ant (Ant f) |= p let p be Element of CQC-WFF Al; ::_thesis: for f, g being FinSequence of CQC-WFF Al st len f > 1 & Ant f = Ant g & 'not' p = Suc (Ant f) & 'not' (Suc f) = Suc g & Ant f |= Suc f & Ant g |= Suc g holds Ant (Ant f) |= p let f, g be FinSequence of CQC-WFF Al; ::_thesis: ( len f > 1 & Ant f = Ant g & 'not' p = Suc (Ant f) & 'not' (Suc f) = Suc g & Ant f |= Suc f & Ant g |= Suc g implies Ant (Ant f) |= p ) assume that A1: len f > 1 and A2: Ant f = Ant g and A3: 'not' p = Suc (Ant f) and A4: ( 'not' (Suc f) = Suc g & Ant f |= Suc f & Ant g |= Suc g ) ; ::_thesis: Ant (Ant f) |= p A5: len (Ant f) > 0 by A1, Th4; A6: now__::_thesis:_for_A_being_non_empty_set_ for_J_being_interpretation_of_Al,A for_v_being_Element_of_Valuations_in_(Al,A)_holds_not_J,v_|=_Ant_f given A being non empty set , J being interpretation of Al,A, v being Element of Valuations_in (Al,A) such that A7: J,v |= Ant f ; ::_thesis: contradiction ( J,v |= Suc f & J,v |= 'not' (Suc f) ) by A2, A4, A7, Def15; hence contradiction by VALUAT_1:17; ::_thesis: verum end; let A be non empty set ; :: according to CALCUL_1:def_15 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= Ant (Ant f) holds J,v |= p let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant (Ant f) holds J,v |= p let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= Ant (Ant f) implies J,v |= p ) assume A8: J,v |= Ant (Ant f) ; ::_thesis: J,v |= p now__::_thesis:_not_J,v_|=_Suc_(Ant_f) assume J,v |= Suc (Ant f) ; ::_thesis: contradiction then J,v |= Ant f by A5, A8, Th17; hence contradiction by A6; ::_thesis: verum end; hence J,v |= p by A3, VALUAT_1:17; ::_thesis: verum end; theorem Th20: :: CALCUL_1:20 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al st Ant f = Ant g & Ant f |= Suc f & Ant g |= Suc g holds Ant f |= (Suc f) '&' (Suc g) proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al st Ant f = Ant g & Ant f |= Suc f & Ant g |= Suc g holds Ant f |= (Suc f) '&' (Suc g) let f, g be FinSequence of CQC-WFF Al; ::_thesis: ( Ant f = Ant g & Ant f |= Suc f & Ant g |= Suc g implies Ant f |= (Suc f) '&' (Suc g) ) assume A1: ( Ant f = Ant g & Ant f |= Suc f & Ant g |= Suc g ) ; ::_thesis: Ant f |= (Suc f) '&' (Suc g) let A be non empty set ; :: according to CALCUL_1:def_15 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= (Suc f) '&' (Suc g) let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= (Suc f) '&' (Suc g) let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= Ant f implies J,v |= (Suc f) '&' (Suc g) ) assume J,v |= Ant f ; ::_thesis: J,v |= (Suc f) '&' (Suc g) then ( J,v |= Suc f & J,v |= Suc g ) by A1, Def15; hence J,v |= (Suc f) '&' (Suc g) by VALUAT_1:18; ::_thesis: verum end; theorem Th21: :: CALCUL_1:21 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds Ant f |= p proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds Ant f |= p let p, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds Ant f |= p let f be FinSequence of CQC-WFF Al; ::_thesis: ( Ant f |= p '&' q implies Ant f |= p ) assume A1: Ant f |= p '&' q ; ::_thesis: Ant f |= p let A be non empty set ; :: according to CALCUL_1:def_15 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= p let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= p let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= Ant f implies J,v |= p ) assume J,v |= Ant f ; ::_thesis: J,v |= p then J,v |= p '&' q by A1, Def15; hence J,v |= p by VALUAT_1:18; ::_thesis: verum end; theorem Th22: :: CALCUL_1:22 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds Ant f |= q proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds Ant f |= q let p, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds Ant f |= q let f be FinSequence of CQC-WFF Al; ::_thesis: ( Ant f |= p '&' q implies Ant f |= q ) assume A1: Ant f |= p '&' q ; ::_thesis: Ant f |= q let A be non empty set ; :: according to CALCUL_1:def_15 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= q let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= q let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= Ant f implies J,v |= q ) assume J,v |= Ant f ; ::_thesis: J,v |= q then J,v |= p '&' q by A1, Def15; hence J,v |= q by VALUAT_1:18; ::_thesis: verum end; theorem Th23: :: CALCUL_1:23 for Al being QC-alphabet for p being Element of CQC-WFF Al for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) for Sub being CQC_Substitution of Al holds ( J,v |= [p,Sub] iff J,v |= p ) proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) for Sub being CQC_Substitution of Al holds ( J,v |= [p,Sub] iff J,v |= p ) let p be Element of CQC-WFF Al; ::_thesis: for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) for Sub being CQC_Substitution of Al holds ( J,v |= [p,Sub] iff J,v |= p ) let A be non empty set ; ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) for Sub being CQC_Substitution of Al holds ( J,v |= [p,Sub] iff J,v |= p ) let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) for Sub being CQC_Substitution of Al holds ( J,v |= [p,Sub] iff J,v |= p ) let v be Element of Valuations_in (Al,A); ::_thesis: for Sub being CQC_Substitution of Al holds ( J,v |= [p,Sub] iff J,v |= p ) let Sub be CQC_Substitution of Al; ::_thesis: ( J,v |= [p,Sub] iff J,v |= p ) ( J,v |= [p,Sub] iff J,v |= [p,Sub] `1 ) by SUBLEMMA:def_2; hence ( J,v |= [p,Sub] iff J,v |= p ) ; ::_thesis: verum end; theorem Th24: :: CALCUL_1:24 for Al being QC-alphabet for p being Element of CQC-WFF Al for x, y being bound_QC-variable of Al for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) holds ( J,v |= p . (x,y) iff ex a being Element of A st ( v . y = a & J,v . (x | a) |= p ) ) proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for x, y being bound_QC-variable of Al for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) holds ( J,v |= p . (x,y) iff ex a being Element of A st ( v . y = a & J,v . (x | a) |= p ) ) let p be Element of CQC-WFF Al; ::_thesis: for x, y being bound_QC-variable of Al for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) holds ( J,v |= p . (x,y) iff ex a being Element of A st ( v . y = a & J,v . (x | a) |= p ) ) let x, y be bound_QC-variable of Al; ::_thesis: for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) holds ( J,v |= p . (x,y) iff ex a being Element of A st ( v . y = a & J,v . (x | a) |= p ) ) let A be non empty set ; ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) holds ( J,v |= p . (x,y) iff ex a being Element of A st ( v . y = a & J,v . (x | a) |= p ) ) let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) holds ( J,v |= p . (x,y) iff ex a being Element of A st ( v . y = a & J,v . (x | a) |= p ) ) let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= p . (x,y) iff ex a being Element of A st ( v . y = a & J,v . (x | a) |= p ) ) A1: ( J,v |= CQC_Sub [p,(Sbst (x,y))] iff J,v . (Val_S (v,[p,(Sbst (x,y))])) |= [p,(Sbst (x,y))] ) by SUBLEMMA:89; A2: ( J,v . (Val_S (v,[p,(Sbst (x,y))])) |= [p,(Sbst (x,y))] iff J,v . (Val_S (v,[p,(Sbst (x,y))])) |= p ) by Th23; Val_S (v,[p,(Sbst (x,y))]) = v * (@ ([p,(Sbst (x,y))] `2)) by SUBLEMMA:def_1; then Val_S (v,[p,(Sbst (x,y))]) = v * (@ (Sbst (x,y))) ; then A3: Val_S (v,[p,(Sbst (x,y))]) = v * (x .--> y) by SUBSTUT1:def_2; y in bound_QC-variables Al ; then y in dom v by SUBLEMMA:58; then Val_S (v,[p,(Sbst (x,y))]) = x .--> (v . y) by A3, FUNCOP_1:17; hence ( J,v |= p . (x,y) iff ex a being Element of A st ( v . y = a & J,v . (x | a) |= p ) ) by A1, A2, SUBSTUT2:def_1; ::_thesis: verum end; theorem Th25: :: CALCUL_1:25 for Al being QC-alphabet for p being Element of CQC-WFF Al for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds for y being bound_QC-variable of Al holds Ant f |= p . (x,y) proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds for y being bound_QC-variable of Al holds Ant f |= p . (x,y) let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds for y being bound_QC-variable of Al holds Ant f |= p . (x,y) let x be bound_QC-variable of Al; ::_thesis: for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & Ant f |= Suc f holds for y being bound_QC-variable of Al holds Ant f |= p . (x,y) let f be FinSequence of CQC-WFF Al; ::_thesis: ( Suc f = All (x,p) & Ant f |= Suc f implies for y being bound_QC-variable of Al holds Ant f |= p . (x,y) ) assume A1: ( Suc f = All (x,p) & Ant f |= Suc f ) ; ::_thesis: for y being bound_QC-variable of Al holds Ant f |= p . (x,y) let y be bound_QC-variable of Al; ::_thesis: Ant f |= p . (x,y) let A be non empty set ; :: according to CALCUL_1:def_15 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= p . (x,y) let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= p . (x,y) let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= Ant f implies J,v |= p . (x,y) ) assume J,v |= Ant f ; ::_thesis: J,v |= p . (x,y) then A2: J,v |= All (x,p) by A1, Def15; ex a being Element of A st ( v . y = a & J,v . (x | a) |= p ) proof take v . y ; ::_thesis: ( v . y = v . y & J,v . (x | (v . y)) |= p ) thus ( v . y = v . y & J,v . (x | (v . y)) |= p ) by A2, SUBLEMMA:50; ::_thesis: verum end; hence J,v |= p . (x,y) by Th24; ::_thesis: verum end; theorem Th26: :: CALCUL_1:26 for Al being QC-alphabet for x being bound_QC-variable of Al for A being non empty set for v being Element of Valuations_in (Al,A) for a being Element of A for X being set st X c= bound_QC-variables Al & not x in X holds (v . (x | a)) | X = v | X proof let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al for A being non empty set for v being Element of Valuations_in (Al,A) for a being Element of A for X being set st X c= bound_QC-variables Al & not x in X holds (v . (x | a)) | X = v | X let x be bound_QC-variable of Al; ::_thesis: for A being non empty set for v being Element of Valuations_in (Al,A) for a being Element of A for X being set st X c= bound_QC-variables Al & not x in X holds (v . (x | a)) | X = v | X let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A) for a being Element of A for X being set st X c= bound_QC-variables Al & not x in X holds (v . (x | a)) | X = v | X let v be Element of Valuations_in (Al,A); ::_thesis: for a being Element of A for X being set st X c= bound_QC-variables Al & not x in X holds (v . (x | a)) | X = v | X let a be Element of A; ::_thesis: for X being set st X c= bound_QC-variables Al & not x in X holds (v . (x | a)) | X = v | X let X be set ; ::_thesis: ( X c= bound_QC-variables Al & not x in X implies (v . (x | a)) | X = v | X ) assume that A1: X c= bound_QC-variables Al and A2: not x in X ; ::_thesis: (v . (x | a)) | X = v | X set f2 = v | X; set f1 = (v . (x | a)) | X; A3: dom ((v . (x | a)) | X) = dom (v | X) by A1, SUBLEMMA:63; now__::_thesis:_for_b_being_set_st_b_in_dom_((v_._(x_|_a))_|_X)_holds_ ((v_._(x_|_a))_|_X)_._b_=_(v_|_X)_._b let b be set ; ::_thesis: ( b in dom ((v . (x | a)) | X) implies ((v . (x | a)) | X) . b = (v | X) . b ) assume A4: b in dom ((v . (x | a)) | X) ; ::_thesis: ((v . (x | a)) | X) . b = (v | X) . b x <> b by A2, A4; then A5: (v . (x | a)) . b = v . b by SUBLEMMA:48; (v . (x | a)) . b = ((v . (x | a)) | X) . b by A4, FUNCT_1:47; hence ((v . (x | a)) | X) . b = (v | X) . b by A3, A4, A5, FUNCT_1:47; ::_thesis: verum end; hence (v . (x | a)) | X = v | X by A3, FUNCT_1:2; ::_thesis: verum end; theorem Th27: :: CALCUL_1:27 for Al being QC-alphabet for A being non empty set for J being interpretation of Al,A for f being FinSequence of CQC-WFF Al for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in f) = w | (still_not-bound_in f) & J,v |= f holds J,w |= f proof let Al be QC-alphabet ; ::_thesis: for A being non empty set for J being interpretation of Al,A for f being FinSequence of CQC-WFF Al for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in f) = w | (still_not-bound_in f) & J,v |= f holds J,w |= f let A be non empty set ; ::_thesis: for J being interpretation of Al,A for f being FinSequence of CQC-WFF Al for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in f) = w | (still_not-bound_in f) & J,v |= f holds J,w |= f let J be interpretation of Al,A; ::_thesis: for f being FinSequence of CQC-WFF Al for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in f) = w | (still_not-bound_in f) & J,v |= f holds J,w |= f let f be FinSequence of CQC-WFF Al; ::_thesis: for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in f) = w | (still_not-bound_in f) & J,v |= f holds J,w |= f let v, w be Element of Valuations_in (Al,A); ::_thesis: ( v | (still_not-bound_in f) = w | (still_not-bound_in f) & J,v |= f implies J,w |= f ) assume A1: v | (still_not-bound_in f) = w | (still_not-bound_in f) ; ::_thesis: ( not J,v |= f or J,w |= f ) assume J,v |= f ; ::_thesis: J,w |= f then A2: J,v |= rng f by Def14; let p be Element of CQC-WFF Al; :: according to CALCUL_1:def_11,CALCUL_1:def_14 ::_thesis: ( p in rng f implies J,w |= p ) assume A3: p in rng f ; ::_thesis: J,w |= p ex i being Nat st ( i in dom f & p = f . i ) by A3, FINSEQ_2:10; then for b being set st b in still_not-bound_in p holds b in still_not-bound_in f by Def5; then still_not-bound_in p c= still_not-bound_in f by TARSKI:def_3; then A4: v | (still_not-bound_in p) = w | (still_not-bound_in p) by A1, RELAT_1:153; J,v |= p by A2, A3, Def11; hence J,w |= p by A4, SUBLEMMA:68; ::_thesis: verum end; theorem Th28: :: CALCUL_1:28 for Al being QC-alphabet for p being Element of CQC-WFF Al for y, x being bound_QC-variable of Al for A being non empty set for v being Element of Valuations_in (Al,A) for a being Element of A st not y in still_not-bound_in (All (x,p)) holds ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for y, x being bound_QC-variable of Al for A being non empty set for v being Element of Valuations_in (Al,A) for a being Element of A st not y in still_not-bound_in (All (x,p)) holds ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) let p be Element of CQC-WFF Al; ::_thesis: for y, x being bound_QC-variable of Al for A being non empty set for v being Element of Valuations_in (Al,A) for a being Element of A st not y in still_not-bound_in (All (x,p)) holds ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) let y, x be bound_QC-variable of Al; ::_thesis: for A being non empty set for v being Element of Valuations_in (Al,A) for a being Element of A st not y in still_not-bound_in (All (x,p)) holds ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A) for a being Element of A st not y in still_not-bound_in (All (x,p)) holds ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) let v be Element of Valuations_in (Al,A); ::_thesis: for a being Element of A st not y in still_not-bound_in (All (x,p)) holds ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) let a be Element of A; ::_thesis: ( not y in still_not-bound_in (All (x,p)) implies ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) ) A1: (v . (y | a)) . (x | a) = v +* ((y | a) +* (x | a)) by FUNCT_4:14; assume A2: not y in still_not-bound_in (All (x,p)) ; ::_thesis: ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) now__::_thesis:_(_x_<>_y_implies_((v_._(y_|_a))_._(x_|_a))_|_(still_not-bound_in_p)_=_(v_._(x_|_a))_|_(still_not-bound_in_p)_) assume A3: x <> y ; ::_thesis: ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) ( dom (x | a) = {x} & dom (y | a) = {y} ) by SUBLEMMA:58; then (v . (y | a)) . (x | a) = v +* ((x | a) +* (y | a)) by A1, A3, FUNCT_4:35, ZFMISC_1:11; then A4: (v . (y | a)) . (x | a) = (v +* (x | a)) +* (y | a) by FUNCT_4:14; not y in (still_not-bound_in p) \ {x} by A2, QC_LANG3:12; then not y in still_not-bound_in p by A3, ZFMISC_1:56; hence ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) by A4, Th26; ::_thesis: verum end; hence ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) by A1; ::_thesis: verum end; theorem Th29: :: CALCUL_1:29 for Al being QC-alphabet for p being Element of CQC-WFF Al for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = p . (x,y) & Ant f |= Suc f & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) holds Ant f |= All (x,p) proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = p . (x,y) & Ant f |= Suc f & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) holds Ant f |= All (x,p) let p be Element of CQC-WFF Al; ::_thesis: for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = p . (x,y) & Ant f |= Suc f & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) holds Ant f |= All (x,p) let x, y be bound_QC-variable of Al; ::_thesis: for f being FinSequence of CQC-WFF Al st Suc f = p . (x,y) & Ant f |= Suc f & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) holds Ant f |= All (x,p) let f be FinSequence of CQC-WFF Al; ::_thesis: ( Suc f = p . (x,y) & Ant f |= Suc f & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) implies Ant f |= All (x,p) ) assume that A1: ( Suc f = p . (x,y) & Ant f |= Suc f ) and A2: not y in still_not-bound_in (Ant f) and A3: not y in still_not-bound_in (All (x,p)) ; ::_thesis: Ant f |= All (x,p) let A be non empty set ; :: according to CALCUL_1:def_15 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= All (x,p) let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds J,v |= All (x,p) let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= Ant f implies J,v |= All (x,p) ) assume A4: J,v |= Ant f ; ::_thesis: J,v |= All (x,p) for a being Element of A holds J,v . (x | a) |= p proof let a be Element of A; ::_thesis: J,v . (x | a) |= p (v . (y | a)) | (still_not-bound_in (Ant f)) = v | (still_not-bound_in (Ant f)) by A2, Th26; then J,v . (y | a) |= Ant f by A4, Th27; then J,v . (y | a) |= p . (x,y) by A1, Def15; then ex a1 being Element of A st ( (v . (y | a)) . y = a1 & J,(v . (y | a)) . (x | a1) |= p ) by Th24; then A5: J,(v . (y | a)) . (x | a) |= p by SUBLEMMA:49; ((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p) by A3, Th28; hence J,v . (x | a) |= p by A5, SUBLEMMA:68; ::_thesis: verum end; hence J,v |= All (x,p) by SUBLEMMA:50; ::_thesis: verum end; theorem Th30: :: CALCUL_1:30 for Al being QC-alphabet for f being FinSequence of CQC-WFF Al holds Ant (f ^ <*(VERUM Al)*>) |= Suc (f ^ <*(VERUM Al)*>) proof let Al be QC-alphabet ; ::_thesis: for f being FinSequence of CQC-WFF Al holds Ant (f ^ <*(VERUM Al)*>) |= Suc (f ^ <*(VERUM Al)*>) let f be FinSequence of CQC-WFF Al; ::_thesis: Ant (f ^ <*(VERUM Al)*>) |= Suc (f ^ <*(VERUM Al)*>) let A be non empty set ; :: according to CALCUL_1:def_15 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= Ant (f ^ <*(VERUM Al)*>) holds J,v |= Suc (f ^ <*(VERUM Al)*>) let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant (f ^ <*(VERUM Al)*>) holds J,v |= Suc (f ^ <*(VERUM Al)*>) let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= Ant (f ^ <*(VERUM Al)*>) implies J,v |= Suc (f ^ <*(VERUM Al)*>) ) assume J,v |= Ant (f ^ <*(VERUM Al)*>) ; ::_thesis: J,v |= Suc (f ^ <*(VERUM Al)*>) Suc (f ^ <*(VERUM Al)*>) = VERUM Al by Th5; hence J,v |= Suc (f ^ <*(VERUM Al)*>) by VALUAT_1:32; ::_thesis: verum end; theorem Th31: :: CALCUL_1:31 for Al being QC-alphabet for PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] for n being Nat st 1 <= n & n <= len PR & not (PR . n) `2 = 0 & not (PR . n) `2 = 1 & not (PR . n) `2 = 2 & not (PR . n) `2 = 3 & not (PR . n) `2 = 4 & not (PR . n) `2 = 5 & not (PR . n) `2 = 6 & not (PR . n) `2 = 7 & not (PR . n) `2 = 8 holds (PR . n) `2 = 9 proof let Al be QC-alphabet ; ::_thesis: for PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] for n being Nat st 1 <= n & n <= len PR & not (PR . n) `2 = 0 & not (PR . n) `2 = 1 & not (PR . n) `2 = 2 & not (PR . n) `2 = 3 & not (PR . n) `2 = 4 & not (PR . n) `2 = 5 & not (PR . n) `2 = 6 & not (PR . n) `2 = 7 & not (PR . n) `2 = 8 holds (PR . n) `2 = 9 let PR be FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:]; ::_thesis: for n being Nat st 1 <= n & n <= len PR & not (PR . n) `2 = 0 & not (PR . n) `2 = 1 & not (PR . n) `2 = 2 & not (PR . n) `2 = 3 & not (PR . n) `2 = 4 & not (PR . n) `2 = 5 & not (PR . n) `2 = 6 & not (PR . n) `2 = 7 & not (PR . n) `2 = 8 holds (PR . n) `2 = 9 let n be Nat; ::_thesis: ( 1 <= n & n <= len PR & not (PR . n) `2 = 0 & not (PR . n) `2 = 1 & not (PR . n) `2 = 2 & not (PR . n) `2 = 3 & not (PR . n) `2 = 4 & not (PR . n) `2 = 5 & not (PR . n) `2 = 6 & not (PR . n) `2 = 7 & not (PR . n) `2 = 8 implies (PR . n) `2 = 9 ) assume ( 1 <= n & n <= len PR ) ; ::_thesis: ( (PR . n) `2 = 0 or (PR . n) `2 = 1 or (PR . n) `2 = 2 or (PR . n) `2 = 3 or (PR . n) `2 = 4 or (PR . n) `2 = 5 or (PR . n) `2 = 6 or (PR . n) `2 = 7 or (PR . n) `2 = 8 or (PR . n) `2 = 9 ) then n in dom PR by FINSEQ_3:25; then PR . n in rng PR by FUNCT_1:def_3; then (PR . n) `2 in { k where k is Element of NAT : k <= 9 } by CQC_THE1:def_3, MCART_1:10; then ex k being Element of NAT st ( k = (PR . n) `2 & k <= 9 ) ; hence ( (PR . n) `2 = 0 or (PR . n) `2 = 1 or (PR . n) `2 = 2 or (PR . n) `2 = 3 or (PR . n) `2 = 4 or (PR . n) `2 = 5 or (PR . n) `2 = 6 or (PR . n) `2 = 7 or (PR . n) `2 = 8 or (PR . n) `2 = 9 ) by NAT_1:33; ::_thesis: verum end; Lm3: for Al being QC-alphabet for n being Element of NAT for PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] st 1 <= n & n <= len PR holds (PR . n) `1 is FinSequence of CQC-WFF Al proof let Al be QC-alphabet ; ::_thesis: for n being Element of NAT for PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] st 1 <= n & n <= len PR holds (PR . n) `1 is FinSequence of CQC-WFF Al let n be Element of NAT ; ::_thesis: for PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] st 1 <= n & n <= len PR holds (PR . n) `1 is FinSequence of CQC-WFF Al let PR be FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:]; ::_thesis: ( 1 <= n & n <= len PR implies (PR . n) `1 is FinSequence of CQC-WFF Al ) assume ( 1 <= n & n <= len PR ) ; ::_thesis: (PR . n) `1 is FinSequence of CQC-WFF Al then n in dom PR by FINSEQ_3:25; then PR . n in rng PR by FUNCT_1:def_3; then (PR . n) `1 in set_of_CQC-WFF-seq Al by MCART_1:10; hence (PR . n) `1 is FinSequence of CQC-WFF Al by Def6; ::_thesis: verum end; theorem :: CALCUL_1:32 for Al being QC-alphabet for p being Element of CQC-WFF Al for X being Subset of (CQC-WFF Al) st p is_formal_provable_from X holds X |= p proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for X being Subset of (CQC-WFF Al) st p is_formal_provable_from X holds X |= p let p be Element of CQC-WFF Al; ::_thesis: for X being Subset of (CQC-WFF Al) st p is_formal_provable_from X holds X |= p let X be Subset of (CQC-WFF Al); ::_thesis: ( p is_formal_provable_from X implies X |= p ) assume p is_formal_provable_from X ; ::_thesis: X |= p then consider f being FinSequence of CQC-WFF Al such that A1: rng (Ant f) c= X and A2: Suc f = p and A3: |- f by Def10; consider PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A4: PR is a_proof and A5: f = (PR . (len PR)) `1 by A3, Def9; PR <> {} by A4, Def8; then A6: 1 <= len PR by FINSEQ_1:20; defpred S1[ Element of NAT ] means ( \$1 <= len PR implies for m being Element of NAT st 1 <= m & m <= \$1 holds ex g being FinSequence of CQC-WFF Al st ( g = (PR . m) `1 & Ant g |= Suc g ) ); A7: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A8: S1[k] ; ::_thesis: S1[k + 1] assume A9: k + 1 <= len PR ; ::_thesis: for m being Element of NAT st 1 <= m & m <= k + 1 holds ex g being FinSequence of CQC-WFF Al st ( g = (PR . m) `1 & Ant g |= Suc g ) A10: k <= k + 1 by NAT_1:11; let m be Element of NAT ; ::_thesis: ( 1 <= m & m <= k + 1 implies ex g being FinSequence of CQC-WFF Al st ( g = (PR . m) `1 & Ant g |= Suc g ) ) assume that A11: 1 <= m and A12: m <= k + 1 ; ::_thesis: ex g being FinSequence of CQC-WFF Al st ( g = (PR . m) `1 & Ant g |= Suc g ) A13: m <= len PR by A9, A12, XXREAL_0:2; A14: now__::_thesis:_(_m_=_k_+_1_implies_ex_g_being_set_st_ (_g_=_(PR_._m)_`1_&_ex_g_being_FinSequence_of_CQC-WFF_Al_st_ (_g_=_(PR_._m)_`1_&_Ant_g_|=_Suc_g_)_)_) assume A15: m = k + 1 ; ::_thesis: ex g being set st ( g = (PR . m) `1 & ex g being FinSequence of CQC-WFF Al st ( g = (PR . m) `1 & Ant g |= Suc g ) ) take g = (PR . m) `1 ; ::_thesis: ( g = (PR . m) `1 & ex g being FinSequence of CQC-WFF Al st ( g = (PR . m) `1 & Ant g |= Suc g ) ) thus g = (PR . m) `1 ; ::_thesis: ex g being FinSequence of CQC-WFF Al st ( g = (PR . m) `1 & Ant g |= Suc g ) reconsider g = (PR . m) `1 as FinSequence of CQC-WFF Al by A11, A13, Lm3; A16: PR,m is_a_correct_step by A4, A11, A13, Def8; now__::_thesis:_Ant_g_|=_Suc_g percases ( (PR . m) `2 = 0 or (PR . m) `2 = 1 or (PR . m) `2 = 2 or (PR . m) `2 = 3 or (PR . m) `2 = 4 or (PR . m) `2 = 5 or (PR . m) `2 = 6 or (PR . m) `2 = 7 or (PR . m) `2 = 8 or (PR . m) `2 = 9 ) by A11, A13, Th31; suppose (PR . m) `2 = 0 ; ::_thesis: Ant g |= Suc g then ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . m) `1 = f ) by A16, Def7; hence Ant g |= Suc g by Th15; ::_thesis: verum end; suppose (PR . m) `2 = 1 ; ::_thesis: Ant g |= Suc g then ex f being FinSequence of CQC-WFF Al st g = f ^ <*(VERUM Al)*> by A16, Def7; hence Ant g |= Suc g by Th30; ::_thesis: verum end; suppose (PR . m) `2 = 2 ; ::_thesis: Ant g |= Suc g then consider i being Element of NAT , f, f1 being FinSequence of CQC-WFF Al such that A17: 1 <= i and A18: i < m and A19: ( Ant f is_Subsequence_of Ant f1 & Suc f = Suc f1 & (PR . i) `1 = f & (PR . m) `1 = f1 ) by A16, Def7; i <= k by A15, A18, NAT_1:13; then ex h being FinSequence of CQC-WFF Al st ( h = (PR . i) `1 & Ant h |= Suc h ) by A8, A9, A10, A17, XXREAL_0:2; hence Ant g |= Suc g by A19, Th16; ::_thesis: verum end; suppose (PR . m) `2 = 3 ; ::_thesis: Ant g |= Suc g then consider i, j being Element of NAT , f, f1 being FinSequence of CQC-WFF Al such that A20: 1 <= i and A21: i < m and A22: 1 <= j and A23: j < i and A24: ( len f > 1 & len f1 > 1 & Ant (Ant f) = Ant (Ant f1) & 'not' (Suc (Ant f)) = Suc (Ant f1) & Suc f = Suc f1 & f = (PR . j) `1 & f1 = (PR . i) `1 ) and A25: (Ant (Ant f)) ^ <*(Suc f)*> = (PR . m) `1 by A16, Def7; A26: ( Ant g = Ant (Ant f) & Suc g = Suc f ) by A25, Th5; A27: i <= k by A15, A21, NAT_1:13; then j <= k by A23, XXREAL_0:2; then A28: ex h1 being FinSequence of CQC-WFF Al st ( h1 = (PR . j) `1 & Ant h1 |= Suc h1 ) by A8, A9, A10, A22, XXREAL_0:2; ex h being FinSequence of CQC-WFF Al st ( h = (PR . i) `1 & Ant h |= Suc h ) by A8, A9, A10, A20, A27, XXREAL_0:2; hence Ant g |= Suc g by A24, A28, A26, Th18; ::_thesis: verum end; suppose (PR . m) `2 = 4 ; ::_thesis: Ant g |= Suc g then consider i, j being Element of NAT , f, f1 being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al such that A29: 1 <= i and A30: i < m and A31: 1 <= j and A32: j < i and A33: ( len f > 1 & Ant f = Ant f1 & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc f1 & f = (PR . j) `1 & f1 = (PR . i) `1 ) and A34: (Ant (Ant f)) ^ <*p*> = (PR . m) `1 by A16, Def7; A35: ( Ant g = Ant (Ant f) & Suc g = p ) by A34, Th5; A36: i <= k by A15, A30, NAT_1:13; then j <= k by A32, XXREAL_0:2; then A37: ex h1 being FinSequence of CQC-WFF Al st ( h1 = (PR . j) `1 & Ant h1 |= Suc h1 ) by A8, A9, A10, A31, XXREAL_0:2; ex h being FinSequence of CQC-WFF Al st ( h = (PR . i) `1 & Ant h |= Suc h ) by A8, A9, A10, A29, A36, XXREAL_0:2; hence Ant g |= Suc g by A33, A37, A35, Th19; ::_thesis: verum end; suppose (PR . m) `2 = 5 ; ::_thesis: Ant g |= Suc g then consider i, j being Element of NAT , f, f1 being FinSequence of CQC-WFF Al such that A38: 1 <= i and A39: i < m and A40: 1 <= j and A41: j < i and A42: ( Ant f = Ant f1 & f = (PR . j) `1 & f1 = (PR . i) `1 ) and A43: (Ant f) ^ <*((Suc f) '&' (Suc f1))*> = (PR . m) `1 by A16, Def7; A44: ( Ant g = Ant f & Suc g = (Suc f) '&' (Suc f1) ) by A43, Th5; A45: i <= k by A15, A39, NAT_1:13; then j <= k by A41, XXREAL_0:2; then A46: ex h1 being FinSequence of CQC-WFF Al st ( h1 = (PR . j) `1 & Ant h1 |= Suc h1 ) by A8, A9, A10, A40, XXREAL_0:2; ex h being FinSequence of CQC-WFF Al st ( h = (PR . i) `1 & Ant h |= Suc h ) by A8, A9, A10, A38, A45, XXREAL_0:2; hence Ant g |= Suc g by A42, A46, A44, Th20; ::_thesis: verum end; suppose (PR . m) `2 = 6 ; ::_thesis: Ant g |= Suc g then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p, q being Element of CQC-WFF Al such that A47: 1 <= i and A48: i < m and A49: ( p '&' q = Suc f & f = (PR . i) `1 ) and A50: (Ant f) ^ <*p*> = (PR . m) `1 by A16, Def7; i <= k by A15, A48, NAT_1:13; then A51: ex h being FinSequence of CQC-WFF Al st ( h = (PR . i) `1 & Ant h |= Suc h ) by A8, A9, A10, A47, XXREAL_0:2; ( Ant g = Ant f & Suc g = p ) by A50, Th5; hence Ant g |= Suc g by A49, A51, Th21; ::_thesis: verum end; suppose (PR . m) `2 = 7 ; ::_thesis: Ant g |= Suc g then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p, q being Element of CQC-WFF Al such that A52: 1 <= i and A53: i < m and A54: ( p '&' q = Suc f & f = (PR . i) `1 ) and A55: (Ant f) ^ <*q*> = (PR . m) `1 by A16, Def7; i <= k by A15, A53, NAT_1:13; then A56: ex h being FinSequence of CQC-WFF Al st ( h = (PR . i) `1 & Ant h |= Suc h ) by A8, A9, A10, A52, XXREAL_0:2; ( Ant g = Ant f & Suc g = q ) by A55, Th5; hence Ant g |= Suc g by A54, A56, Th22; ::_thesis: verum end; suppose (PR . m) `2 = 8 ; ::_thesis: Ant g |= Suc g then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al, x, y being bound_QC-variable of Al such that A57: 1 <= i and A58: i < m and A59: ( Suc f = All (x,p) & f = (PR . i) `1 ) and A60: (Ant f) ^ <*(p . (x,y))*> = (PR . m) `1 by A16, Def7; i <= k by A15, A58, NAT_1:13; then A61: ex h being FinSequence of CQC-WFF Al st ( h = (PR . i) `1 & Ant h |= Suc h ) by A8, A9, A10, A57, XXREAL_0:2; ( Ant g = Ant f & Suc g = p . (x,y) ) by A60, Th5; hence Ant g |= Suc g by A59, A61, Th25; ::_thesis: verum end; suppose (PR . m) `2 = 9 ; ::_thesis: Ant g |= Suc g then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al, x, y being bound_QC-variable of Al such that A62: 1 <= i and A63: i < m and A64: ( Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 ) and A65: (Ant f) ^ <*(All (x,p))*> = (PR . m) `1 by A16, Def7; i <= k by A15, A63, NAT_1:13; then A66: ex h being FinSequence of CQC-WFF Al st ( h = (PR . i) `1 & Ant h |= Suc h ) by A8, A9, A10, A62, XXREAL_0:2; ( Ant g = Ant f & Suc g = All (x,p) ) by A65, Th5; hence Ant g |= Suc g by A64, A66, Th29; ::_thesis: verum end; end; end; hence ex g being FinSequence of CQC-WFF Al st ( g = (PR . m) `1 & Ant g |= Suc g ) ; ::_thesis: verum end; ( m <= k implies ex g being FinSequence of CQC-WFF Al st ( g = (PR . m) `1 & Ant g |= Suc g ) ) by A8, A9, A11, A10, XXREAL_0:2; hence ex g being FinSequence of CQC-WFF Al st ( g = (PR . m) `1 & Ant g |= Suc g ) by A12, A14, NAT_1:8; ::_thesis: verum end; A67: S1[ 0 ] ; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A67, A7); then consider g being FinSequence of CQC-WFF Al such that A68: g = (PR . (len PR)) `1 and A69: Ant g |= Suc g by A6; let A be non empty set ; :: according to CALCUL_1:def_12 ::_thesis: for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= X holds J,v |= p let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) st J,v |= X holds J,v |= p let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= X implies J,v |= p ) assume J,v |= X ; ::_thesis: J,v |= p then for p being Element of CQC-WFF Al st p in rng (Ant f) holds J,v |= p by A1, Def11; then J,v |= rng (Ant f) by Def11; then J,v |= Ant g by A5, A68, Def14; hence J,v |= p by A2, A5, A68, A69, Def15; ::_thesis: verum end; begin theorem Th33: :: CALCUL_1:33 for Al being QC-alphabet for f being FinSequence of CQC-WFF Al st Suc f is_tail_of Ant f holds |- f proof let Al be QC-alphabet ; ::_thesis: for f being FinSequence of CQC-WFF Al st Suc f is_tail_of Ant f holds |- f let f be FinSequence of CQC-WFF Al; ::_thesis: ( Suc f is_tail_of Ant f implies |- f ) set PR = <*[f,0]*>; A1: rng <*[f,0]*> = {[f,0]} by FINSEQ_1:38; now__::_thesis:_for_a_being_set_st_a_in_rng_<*[f,0]*>_holds_ a_in_[:(set_of_CQC-WFF-seq_Al),Proof_Step_Kinds:] let a be set ; ::_thesis: ( a in rng <*[f,0]*> implies a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ) assume a in rng <*[f,0]*> ; ::_thesis: a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] then A2: a = [f,0] by A1, TARSKI:def_1; f in set_of_CQC-WFF-seq Al by Def6; hence a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by A2, CQC_THE1:21, ZFMISC_1:87; ::_thesis: verum end; then rng <*[f,0]*> c= [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by TARSKI:def_3; then reconsider PR = <*[f,0]*> as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by FINSEQ_1:def_4; assume A3: Suc f is_tail_of Ant f ; ::_thesis: |- f now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_len_PR_holds_ PR,n_is_a_correct_step let n be Nat; ::_thesis: ( 1 <= n & n <= len PR implies PR,n is_a_correct_step ) assume that A4: 1 <= n and A5: n <= len PR ; ::_thesis: PR,n is_a_correct_step n <= 1 by A5, FINSEQ_1:39; then n = 1 by A4, XXREAL_0:1; then PR . n = [f,0] by FINSEQ_1:40; then ( (PR . n) `1 = f & (PR . n) `2 = 0 ) by MCART_1:7; hence PR,n is_a_correct_step by A3, Def7; ::_thesis: verum end; then A6: PR is a_proof by Def8; PR . 1 = [f,0] by FINSEQ_1:40; then PR . (len PR) = [f,0] by FINSEQ_1:40; then (PR . (len PR)) `1 = f by MCART_1:7; hence |- f by A6, Def9; ::_thesis: verum end; theorem Th34: :: CALCUL_1:34 for Al being QC-alphabet for PR, PR1 being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] for n being Nat st 1 <= n & n <= len PR holds ( PR,n is_a_correct_step iff PR ^ PR1,n is_a_correct_step ) proof let Al be QC-alphabet ; ::_thesis: for PR, PR1 being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] for n being Nat st 1 <= n & n <= len PR holds ( PR,n is_a_correct_step iff PR ^ PR1,n is_a_correct_step ) let PR, PR1 be FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:]; ::_thesis: for n being Nat st 1 <= n & n <= len PR holds ( PR,n is_a_correct_step iff PR ^ PR1,n is_a_correct_step ) let n be Nat; ::_thesis: ( 1 <= n & n <= len PR implies ( PR,n is_a_correct_step iff PR ^ PR1,n is_a_correct_step ) ) assume that A1: 1 <= n and A2: n <= len PR ; ::_thesis: ( PR,n is_a_correct_step iff PR ^ PR1,n is_a_correct_step ) n in dom PR by A1, A2, FINSEQ_3:25; then A3: (PR ^ PR1) . n = PR . n by FINSEQ_1:def_7; len (PR ^ PR1) = (len PR) + (len PR1) by FINSEQ_1:22; then len PR <= len (PR ^ PR1) by NAT_1:11; then A4: n <= len (PR ^ PR1) by A2, XXREAL_0:2; thus ( PR,n is_a_correct_step implies PR ^ PR1,n is_a_correct_step ) ::_thesis: ( PR ^ PR1,n is_a_correct_step implies PR,n is_a_correct_step ) proof assume A5: PR,n is_a_correct_step ; ::_thesis: PR ^ PR1,n is_a_correct_step percases ( ((PR ^ PR1) . n) `2 = 0 or ((PR ^ PR1) . n) `2 = 1 or ((PR ^ PR1) . n) `2 = 2 or ((PR ^ PR1) . n) `2 = 3 or ((PR ^ PR1) . n) `2 = 4 or ((PR ^ PR1) . n) `2 = 5 or ((PR ^ PR1) . n) `2 = 6 or ((PR ^ PR1) . n) `2 = 7 or ((PR ^ PR1) . n) `2 = 8 or ((PR ^ PR1) . n) `2 = 9 ) by A1, A4, Th31; :: according to CALCUL_1:def_7 case ((PR ^ PR1) . n) `2 = 0 ; ::_thesis: ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & ((PR ^ PR1) . n) `1 = f ) hence ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & ((PR ^ PR1) . n) `1 = f ) by A3, A5, Def7; ::_thesis: verum end; case ((PR ^ PR1) . n) `2 = 1 ; ::_thesis: ex f being FinSequence of CQC-WFF Al st ((PR ^ PR1) . n) `1 = f ^ <*(VERUM Al)*> hence ex f being FinSequence of CQC-WFF Al st ((PR ^ PR1) . n) `1 = f ^ <*(VERUM Al)*> by A3, A5, Def7; ::_thesis: verum end; case ((PR ^ PR1) . n) `2 = 2 ; ::_thesis: ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & ((PR ^ PR1) . i) `1 = f & ((PR ^ PR1) . n) `1 = g ) then consider i being Element of NAT , f, g being FinSequence of CQC-WFF Al such that A6: 1 <= i and A7: i < n and A8: ( Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) by A3, A5, Def7; i <= len PR by A2, A7, XXREAL_0:2; then i in dom PR by A6, FINSEQ_3:25; then PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; hence ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & ((PR ^ PR1) . i) `1 = f & ((PR ^ PR1) . n) `1 = g ) by A3, A6, A7, A8; ::_thesis: verum end; case ((PR ^ PR1) . n) `2 = 3 ; ::_thesis: ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = ((PR ^ PR1) . n) `1 ) then consider i, j being Element of NAT , f, g being FinSequence of CQC-WFF Al such that A9: 1 <= i and A10: i < n and A11: 1 <= j and A12: j < i and A13: ( len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) by A3, A5, Def7; A14: i <= len PR by A2, A10, XXREAL_0:2; then i in Seg (len PR) by A9, FINSEQ_1:1; then i in dom PR by FINSEQ_1:def_3; then A15: PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; j <= len PR by A12, A14, XXREAL_0:2; then j in dom PR by A11, FINSEQ_3:25; then PR . j = (PR ^ PR1) . j by FINSEQ_1:def_7; hence ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = ((PR ^ PR1) . n) `1 ) by A3, A9, A10, A11, A12, A13, A15; ::_thesis: verum end; case ((PR ^ PR1) . n) `2 = 4 ; ::_thesis: ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant (Ant f)) ^ <*p*> = ((PR ^ PR1) . n) `1 ) then consider i, j being Element of NAT , f, g being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al such that A16: 1 <= i and A17: i < n and A18: 1 <= j and A19: j < i and A20: ( len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) by A3, A5, Def7; A21: i <= len PR by A2, A17, XXREAL_0:2; then i in Seg (len PR) by A16, FINSEQ_1:1; then i in dom PR by FINSEQ_1:def_3; then A22: PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; j <= len PR by A19, A21, XXREAL_0:2; then j in dom PR by A18, FINSEQ_3:25; then PR . j = (PR ^ PR1) . j by FINSEQ_1:def_7; hence ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant (Ant f)) ^ <*p*> = ((PR ^ PR1) . n) `1 ) by A3, A16, A17, A18, A19, A20, A22; ::_thesis: verum end; case ((PR ^ PR1) . n) `2 = 5 ; ::_thesis: ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = ((PR ^ PR1) . n) `1 ) then consider i, j being Element of NAT , f, g being FinSequence of CQC-WFF Al such that A23: 1 <= i and A24: i < n and A25: 1 <= j and A26: j < i and A27: ( Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) by A3, A5, Def7; A28: i <= len PR by A2, A24, XXREAL_0:2; then i in Seg (len PR) by A23, FINSEQ_1:1; then i in dom PR by FINSEQ_1:def_3; then A29: PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; j <= len PR by A26, A28, XXREAL_0:2; then j in dom PR by A25, FINSEQ_3:25; then PR . j = (PR ^ PR1) . j by FINSEQ_1:def_7; hence ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = ((PR ^ PR1) . n) `1 ) by A3, A23, A24, A25, A26, A27, A29; ::_thesis: verum end; case ((PR ^ PR1) . n) `2 = 6 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*p*> = ((PR ^ PR1) . n) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p, q being Element of CQC-WFF Al such that A30: 1 <= i and A31: i < n and A32: ( p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) by A3, A5, Def7; i <= len PR by A2, A31, XXREAL_0:2; then i in dom PR by A30, FINSEQ_3:25; then PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*p*> = ((PR ^ PR1) . n) `1 ) by A3, A30, A31, A32; ::_thesis: verum end; case ((PR ^ PR1) . n) `2 = 7 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*q*> = ((PR ^ PR1) . n) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p, q being Element of CQC-WFF Al such that A33: 1 <= i and A34: i < n and A35: ( p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) by A3, A5, Def7; i <= len PR by A2, A34, XXREAL_0:2; then i in dom PR by A33, FINSEQ_3:25; then PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*q*> = ((PR ^ PR1) . n) `1 ) by A3, A33, A34, A35; ::_thesis: verum end; case ((PR ^ PR1) . n) `2 = 8 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*(p . (x,y))*> = ((PR ^ PR1) . n) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al, x, y being bound_QC-variable of Al such that A36: 1 <= i and A37: i < n and A38: ( Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) by A3, A5, Def7; i <= len PR by A2, A37, XXREAL_0:2; then i in dom PR by A36, FINSEQ_3:25; then PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*(p . (x,y))*> = ((PR ^ PR1) . n) `1 ) by A3, A36, A37, A38; ::_thesis: verum end; case ((PR ^ PR1) . n) `2 = 9 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*(All (x,p))*> = ((PR ^ PR1) . n) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al, x, y being bound_QC-variable of Al such that A39: 1 <= i and A40: i < n and A41: ( Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) by A3, A5, Def7; i <= len PR by A2, A40, XXREAL_0:2; then i in dom PR by A39, FINSEQ_3:25; then PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*(All (x,p))*> = ((PR ^ PR1) . n) `1 ) by A3, A39, A40, A41; ::_thesis: verum end; end; end; assume A42: PR ^ PR1,n is_a_correct_step ; ::_thesis: PR,n is_a_correct_step percases ( (PR . n) `2 = 0 or (PR . n) `2 = 1 or (PR . n) `2 = 2 or (PR . n) `2 = 3 or (PR . n) `2 = 4 or (PR . n) `2 = 5 or (PR . n) `2 = 6 or (PR . n) `2 = 7 or (PR . n) `2 = 8 or (PR . n) `2 = 9 ) by A1, A2, Th31; :: according to CALCUL_1:def_7 case (PR . n) `2 = 0 ; ::_thesis: ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) hence ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) by A3, A42, Def7; ::_thesis: verum end; case (PR . n) `2 = 1 ; ::_thesis: ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> hence ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> by A3, A42, Def7; ::_thesis: verum end; case (PR . n) `2 = 2 ; ::_thesis: ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) then consider i being Element of NAT , f, g being FinSequence of CQC-WFF Al such that A43: 1 <= i and A44: i < n and A45: ( Ant f is_Subsequence_of Ant g & Suc f = Suc g & ((PR ^ PR1) . i) `1 = f & ((PR ^ PR1) . n) `1 = g ) by A3, A42, Def7; i <= len PR by A2, A44, XXREAL_0:2; then i in dom PR by A43, FINSEQ_3:25; then PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; hence ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) by A3, A43, A44, A45; ::_thesis: verum end; case (PR . n) `2 = 3 ; ::_thesis: ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) then consider i, j being Element of NAT , f, f1 being FinSequence of CQC-WFF Al such that A46: 1 <= i and A47: i < n and A48: 1 <= j and A49: j < i and A50: ( len f > 1 & len f1 > 1 & Ant (Ant f) = Ant (Ant f1) & 'not' (Suc (Ant f)) = Suc (Ant f1) & Suc f = Suc f1 & f = ((PR ^ PR1) . j) `1 & f1 = ((PR ^ PR1) . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = ((PR ^ PR1) . n) `1 ) by A3, A42, Def7; A51: i <= len PR by A2, A47, XXREAL_0:2; then i in Seg (len PR) by A46, FINSEQ_1:1; then i in dom PR by FINSEQ_1:def_3; then A52: PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; j <= len PR by A49, A51, XXREAL_0:2; then j in dom PR by A48, FINSEQ_3:25; then PR . j = (PR ^ PR1) . j by FINSEQ_1:def_7; hence ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) by A3, A46, A47, A48, A49, A50, A52; ::_thesis: verum end; case (PR . n) `2 = 4 ; ::_thesis: ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) then consider i, j being Element of NAT , f, g being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al such that A53: 1 <= i and A54: i < n and A55: 1 <= j and A56: j < i and A57: ( len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant (Ant f)) ^ <*p*> = ((PR ^ PR1) . n) `1 ) by A3, A42, Def7; A58: i <= len PR by A2, A54, XXREAL_0:2; then i in Seg (len PR) by A53, FINSEQ_1:1; then i in dom PR by FINSEQ_1:def_3; then A59: PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; j <= len PR by A56, A58, XXREAL_0:2; then j in dom PR by A55, FINSEQ_3:25; then PR . j = (PR ^ PR1) . j by FINSEQ_1:def_7; hence ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) by A3, A53, A54, A55, A56, A57, A59; ::_thesis: verum end; case (PR . n) `2 = 5 ; ::_thesis: ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) then consider i, j being Element of NAT , f, g being FinSequence of CQC-WFF Al such that A60: 1 <= i and A61: i < n and A62: 1 <= j and A63: j < i and A64: ( Ant f = Ant g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = ((PR ^ PR1) . n) `1 ) by A3, A42, Def7; A65: i <= len PR by A2, A61, XXREAL_0:2; then i in Seg (len PR) by A60, FINSEQ_1:1; then i in dom PR by FINSEQ_1:def_3; then A66: PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; j <= len PR by A63, A65, XXREAL_0:2; then j in dom PR by A62, FINSEQ_3:25; then PR . j = (PR ^ PR1) . j by FINSEQ_1:def_7; hence ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) by A3, A60, A61, A62, A63, A64, A66; ::_thesis: verum end; case (PR . n) `2 = 6 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p, q being Element of CQC-WFF Al such that A67: 1 <= i and A68: i < n and A69: ( p '&' q = Suc f & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*p*> = ((PR ^ PR1) . n) `1 ) by A3, A42, Def7; i <= len PR by A2, A68, XXREAL_0:2; then i in dom PR by A67, FINSEQ_3:25; then PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) by A3, A67, A68, A69; ::_thesis: verum end; case (PR . n) `2 = 7 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p, q being Element of CQC-WFF Al such that A70: 1 <= i and A71: i < n and A72: ( p '&' q = Suc f & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*q*> = ((PR ^ PR1) . n) `1 ) by A3, A42, Def7; i <= len PR by A2, A71, XXREAL_0:2; then i in dom PR by A70, FINSEQ_3:25; then PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) by A3, A70, A71, A72; ::_thesis: verum end; case (PR . n) `2 = 8 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al, x, y being bound_QC-variable of Al such that A73: 1 <= i and A74: i < n and A75: ( Suc f = All (x,p) & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*(p . (x,y))*> = ((PR ^ PR1) . n) `1 ) by A3, A42, Def7; i <= len PR by A2, A74, XXREAL_0:2; then i in dom PR by A73, FINSEQ_3:25; then PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) by A3, A73, A74, A75; ::_thesis: verum end; case (PR . n) `2 = 9 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al, x, y being bound_QC-variable of Al such that A76: 1 <= i and A77: i < n and A78: ( Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*(All (x,p))*> = ((PR ^ PR1) . n) `1 ) by A3, A42, Def7; i <= len PR by A2, A77, XXREAL_0:2; then i in dom PR by A76, FINSEQ_3:25; then PR . i = (PR ^ PR1) . i by FINSEQ_1:def_7; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) by A3, A76, A77, A78; ::_thesis: verum end; end; end; theorem Th35: :: CALCUL_1:35 for Al being QC-alphabet for n being Element of NAT for PR1, PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] st 1 <= n & n <= len PR1 & PR1,n is_a_correct_step holds PR ^ PR1,n + (len PR) is_a_correct_step proof let Al be QC-alphabet ; ::_thesis: for n being Element of NAT for PR1, PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] st 1 <= n & n <= len PR1 & PR1,n is_a_correct_step holds PR ^ PR1,n + (len PR) is_a_correct_step let n be Element of NAT ; ::_thesis: for PR1, PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] st 1 <= n & n <= len PR1 & PR1,n is_a_correct_step holds PR ^ PR1,n + (len PR) is_a_correct_step let PR1, PR be FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:]; ::_thesis: ( 1 <= n & n <= len PR1 & PR1,n is_a_correct_step implies PR ^ PR1,n + (len PR) is_a_correct_step ) assume that A1: 1 <= n and A2: n <= len PR1 and A3: PR1,n is_a_correct_step ; ::_thesis: PR ^ PR1,n + (len PR) is_a_correct_step n in dom PR1 by A1, A2, FINSEQ_3:25; then A4: PR1 . n = (PR ^ PR1) . (n + (len PR)) by FINSEQ_1:def_7; n + (len PR) <= (len PR) + (len PR1) by A2, XREAL_1:6; then A5: n + (len PR) <= len (PR ^ PR1) by FINSEQ_1:22; percases ( ((PR ^ PR1) . (n + (len PR))) `2 = 0 or ((PR ^ PR1) . (n + (len PR))) `2 = 1 or ((PR ^ PR1) . (n + (len PR))) `2 = 2 or ((PR ^ PR1) . (n + (len PR))) `2 = 3 or ((PR ^ PR1) . (n + (len PR))) `2 = 4 or ((PR ^ PR1) . (n + (len PR))) `2 = 5 or ((PR ^ PR1) . (n + (len PR))) `2 = 6 or ((PR ^ PR1) . (n + (len PR))) `2 = 7 or ((PR ^ PR1) . (n + (len PR))) `2 = 8 or ((PR ^ PR1) . (n + (len PR))) `2 = 9 ) by A1, A5, Th31, NAT_1:12; :: according to CALCUL_1:def_7 case ((PR ^ PR1) . (n + (len PR))) `2 = 0 ; ::_thesis: ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & ((PR ^ PR1) . (n + (len PR))) `1 = f ) hence ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & ((PR ^ PR1) . (n + (len PR))) `1 = f ) by A3, A4, Def7; ::_thesis: verum end; case ((PR ^ PR1) . (n + (len PR))) `2 = 1 ; ::_thesis: ex f being FinSequence of CQC-WFF Al st ((PR ^ PR1) . (n + (len PR))) `1 = f ^ <*(VERUM Al)*> hence ex f being FinSequence of CQC-WFF Al st ((PR ^ PR1) . (n + (len PR))) `1 = f ^ <*(VERUM Al)*> by A3, A4, Def7; ::_thesis: verum end; case ((PR ^ PR1) . (n + (len PR))) `2 = 2 ; ::_thesis: ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & Ant f is_Subsequence_of Ant g & Suc f = Suc g & ((PR ^ PR1) . i) `1 = f & ((PR ^ PR1) . (n + (len PR))) `1 = g ) then consider i being Element of NAT , f, g being FinSequence of CQC-WFF Al such that A6: 1 <= i and A7: i < n and A8: ( Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR1 . i) `1 = f & (PR1 . n) `1 = g ) by A3, A4, Def7; i <= len PR1 by A2, A7, XXREAL_0:2; then i in dom PR1 by A6, FINSEQ_3:25; then A9: PR1 . i = (PR ^ PR1) . ((len PR) + i) by FINSEQ_1:def_7; ( 1 <= (len PR) + i & (len PR) + i < n + (len PR) ) by A6, A7, NAT_1:12, XREAL_1:6; hence ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & Ant f is_Subsequence_of Ant g & Suc f = Suc g & ((PR ^ PR1) . i) `1 = f & ((PR ^ PR1) . (n + (len PR))) `1 = g ) by A4, A8, A9; ::_thesis: verum end; case ((PR ^ PR1) . (n + (len PR))) `2 = 3 ; ::_thesis: ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = ((PR ^ PR1) . (n + (len PR))) `1 ) then consider i, j being Element of NAT , f, f1 being FinSequence of CQC-WFF Al such that A10: 1 <= i and A11: i < n and A12: 1 <= j and A13: j < i and A14: ( len f > 1 & len f1 > 1 & Ant (Ant f) = Ant (Ant f1) & 'not' (Suc (Ant f)) = Suc (Ant f1) & Suc f = Suc f1 & f = (PR1 . j) `1 & f1 = (PR1 . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR1 . n) `1 ) by A3, A4, Def7; A15: ( 1 <= (len PR) + j & (len PR) + j < i + (len PR) ) by A12, A13, NAT_1:12, XREAL_1:6; A16: i <= len PR1 by A2, A11, XXREAL_0:2; then i in dom PR1 by A10, FINSEQ_3:25; then A17: PR1 . i = (PR ^ PR1) . ((len PR) + i) by FINSEQ_1:def_7; j <= len PR1 by A13, A16, XXREAL_0:2; then j in dom PR1 by A12, FINSEQ_3:25; then A18: PR1 . j = (PR ^ PR1) . ((len PR) + j) by FINSEQ_1:def_7; ( 1 <= (len PR) + i & (len PR) + i < n + (len PR) ) by A10, A11, NAT_1:12, XREAL_1:6; hence ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = ((PR ^ PR1) . (n + (len PR))) `1 ) by A4, A14, A15, A17, A18; ::_thesis: verum end; case ((PR ^ PR1) . (n + (len PR))) `2 = 4 ; ::_thesis: ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant (Ant f)) ^ <*p*> = ((PR ^ PR1) . (n + (len PR))) `1 ) then consider i, j being Element of NAT , f, g being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al such that A19: 1 <= i and A20: i < n and A21: 1 <= j and A22: j < i and A23: ( len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR1 . j) `1 & g = (PR1 . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR1 . n) `1 ) by A3, A4, Def7; A24: ( 1 <= (len PR) + j & (len PR) + j < i + (len PR) ) by A21, A22, NAT_1:12, XREAL_1:6; A25: i <= len PR1 by A2, A20, XXREAL_0:2; then i in dom PR1 by A19, FINSEQ_3:25; then A26: PR1 . i = (PR ^ PR1) . ((len PR) + i) by FINSEQ_1:def_7; j <= len PR1 by A22, A25, XXREAL_0:2; then j in dom PR1 by A21, FINSEQ_3:25; then A27: PR1 . j = (PR ^ PR1) . ((len PR) + j) by FINSEQ_1:def_7; ( 1 <= (len PR) + i & (len PR) + i < n + (len PR) ) by A19, A20, NAT_1:12, XREAL_1:6; hence ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant (Ant f)) ^ <*p*> = ((PR ^ PR1) . (n + (len PR))) `1 ) by A4, A23, A24, A26, A27; ::_thesis: verum end; case ((PR ^ PR1) . (n + (len PR))) `2 = 5 ; ::_thesis: ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & 1 <= j & j < i & Ant f = Ant g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = ((PR ^ PR1) . (n + (len PR))) `1 ) then consider i, j being Element of NAT , f, g being FinSequence of CQC-WFF Al such that A28: 1 <= i and A29: i < n and A30: 1 <= j and A31: j < i and A32: ( Ant f = Ant g & f = (PR1 . j) `1 & g = (PR1 . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR1 . n) `1 ) by A3, A4, Def7; A33: ( 1 <= (len PR) + j & (len PR) + j < i + (len PR) ) by A30, A31, NAT_1:12, XREAL_1:6; A34: i <= len PR1 by A2, A29, XXREAL_0:2; then i in dom PR1 by A28, FINSEQ_3:25; then A35: PR1 . i = (PR ^ PR1) . ((len PR) + i) by FINSEQ_1:def_7; j <= len PR1 by A31, A34, XXREAL_0:2; then j in dom PR1 by A30, FINSEQ_3:25; then A36: PR1 . j = (PR ^ PR1) . ((len PR) + j) by FINSEQ_1:def_7; ( 1 <= (len PR) + i & (len PR) + i < n + (len PR) ) by A28, A29, NAT_1:12, XREAL_1:6; hence ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & 1 <= j & j < i & Ant f = Ant g & f = ((PR ^ PR1) . j) `1 & g = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = ((PR ^ PR1) . (n + (len PR))) `1 ) by A4, A32, A33, A35, A36; ::_thesis: verum end; case ((PR ^ PR1) . (n + (len PR))) `2 = 6 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & p '&' q = Suc f & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*p*> = ((PR ^ PR1) . (n + (len PR))) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p, q being Element of CQC-WFF Al such that A37: 1 <= i and A38: i < n and A39: ( p '&' q = Suc f & f = (PR1 . i) `1 & (Ant f) ^ <*p*> = (PR1 . n) `1 ) by A3, A4, Def7; i <= len PR1 by A2, A38, XXREAL_0:2; then i in dom PR1 by A37, FINSEQ_3:25; then A40: PR1 . i = (PR ^ PR1) . ((len PR) + i) by FINSEQ_1:def_7; ( 1 <= (len PR) + i & (len PR) + i < n + (len PR) ) by A37, A38, NAT_1:12, XREAL_1:6; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & p '&' q = Suc f & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*p*> = ((PR ^ PR1) . (n + (len PR))) `1 ) by A4, A39, A40; ::_thesis: verum end; case ((PR ^ PR1) . (n + (len PR))) `2 = 7 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & p '&' q = Suc f & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*q*> = ((PR ^ PR1) . (n + (len PR))) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p, q being Element of CQC-WFF Al such that A41: 1 <= i and A42: i < n and A43: ( p '&' q = Suc f & f = (PR1 . i) `1 & (Ant f) ^ <*q*> = (PR1 . n) `1 ) by A3, A4, Def7; i <= len PR1 by A2, A42, XXREAL_0:2; then i in dom PR1 by A41, FINSEQ_3:25; then A44: PR1 . i = (PR ^ PR1) . ((len PR) + i) by FINSEQ_1:def_7; ( 1 <= (len PR) + i & (len PR) + i < n + (len PR) ) by A41, A42, NAT_1:12, XREAL_1:6; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n + (len PR) & p '&' q = Suc f & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*q*> = ((PR ^ PR1) . (n + (len PR))) `1 ) by A4, A43, A44; ::_thesis: verum end; case ((PR ^ PR1) . (n + (len PR))) `2 = 8 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n + (len PR) & Suc f = All (x,p) & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*(p . (x,y))*> = ((PR ^ PR1) . (n + (len PR))) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al, x, y being bound_QC-variable of Al such that A45: 1 <= i and A46: i < n and A47: ( Suc f = All (x,p) & f = (PR1 . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR1 . n) `1 ) by A3, A4, Def7; i <= len PR1 by A2, A46, XXREAL_0:2; then i in dom PR1 by A45, FINSEQ_3:25; then A48: PR1 . i = (PR ^ PR1) . ((len PR) + i) by FINSEQ_1:def_7; ( 1 <= (len PR) + i & (len PR) + i < n + (len PR) ) by A45, A46, NAT_1:12, XREAL_1:6; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n + (len PR) & Suc f = All (x,p) & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*(p . (x,y))*> = ((PR ^ PR1) . (n + (len PR))) `1 ) by A4, A47, A48; ::_thesis: verum end; case ((PR ^ PR1) . (n + (len PR))) `2 = 9 ; ::_thesis: ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n + (len PR) & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*(All (x,p))*> = ((PR ^ PR1) . (n + (len PR))) `1 ) then consider i being Element of NAT , f being FinSequence of CQC-WFF Al, p being Element of CQC-WFF Al, x, y being bound_QC-variable of Al such that A49: 1 <= i and A50: i < n and A51: ( Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR1 . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR1 . n) `1 ) by A3, A4, Def7; i <= len PR1 by A2, A50, XXREAL_0:2; then i in dom PR1 by A49, FINSEQ_3:25; then A52: PR1 . i = (PR ^ PR1) . ((len PR) + i) by FINSEQ_1:def_7; ( 1 <= (len PR) + i & (len PR) + i < n + (len PR) ) by A49, A50, NAT_1:12, XREAL_1:6; hence ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n + (len PR) & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = ((PR ^ PR1) . i) `1 & (Ant f) ^ <*(All (x,p))*> = ((PR ^ PR1) . (n + (len PR))) `1 ) by A4, A51, A52; ::_thesis: verum end; end; end; theorem Th36: :: CALCUL_1:36 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al st Ant f is_Subsequence_of Ant g & Suc f = Suc g & |- f holds |- g proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al st Ant f is_Subsequence_of Ant g & Suc f = Suc g & |- f holds |- g let f, g be FinSequence of CQC-WFF Al; ::_thesis: ( Ant f is_Subsequence_of Ant g & Suc f = Suc g & |- f implies |- g ) assume that A1: ( Ant f is_Subsequence_of Ant g & Suc f = Suc g ) and A2: |- f ; ::_thesis: |- g consider PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A3: PR is a_proof and A4: (PR . (len PR)) `1 = f by A2, Def9; A5: g in set_of_CQC-WFF-seq Al by Def6; now__::_thesis:_for_a_being_set_st_a_in_rng_<*[g,2]*>_holds_ a_in_[:(set_of_CQC-WFF-seq_Al),Proof_Step_Kinds:] let a be set ; ::_thesis: ( a in rng <*[g,2]*> implies a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ) assume a in rng <*[g,2]*> ; ::_thesis: a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] then a in {[g,2]} by FINSEQ_1:38; then a = [g,2] by TARSKI:def_1; hence a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by A5, CQC_THE1:21, ZFMISC_1:87; ::_thesis: verum end; then rng <*[g,2]*> c= [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by TARSKI:def_3; then reconsider PR1 = <*[g,2]*> as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by FINSEQ_1:def_4; 1 in Seg 1 by FINSEQ_1:2, TARSKI:def_1; then A6: 1 in dom PR1 by FINSEQ_1:38; set PR2 = PR ^ PR1; reconsider PR2 = PR ^ PR1 as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ; A7: PR <> {} by A3, Def8; now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_len_PR2_holds_ PR2,n_is_a_correct_step let n be Nat; ::_thesis: ( 1 <= n & n <= len PR2 implies PR2,n is_a_correct_step ) assume ( 1 <= n & n <= len PR2 ) ; ::_thesis: PR2,n is_a_correct_step then A8: n in dom PR2 by FINSEQ_3:25; A9: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR1_&_n_=_(len_PR)_+_k_)_implies_PR2,n_is_a_correct_step_) A10: 1 <= len PR by A7, Th6; then len PR in dom PR by FINSEQ_3:25; then A11: f = (PR2 . (len PR)) `1 by A4, FINSEQ_1:def_7; given k being Nat such that A12: k in dom PR1 and A13: n = (len PR) + k ; ::_thesis: PR2,n is_a_correct_step k in Seg 1 by A12, FINSEQ_1:38; then A14: k = 1 by FINSEQ_1:2, TARSKI:def_1; then A15: PR1 . k = [g,2] by FINSEQ_1:40; then (PR1 . k) `1 = g by MCART_1:7; then A16: (PR2 . n) `1 = g by A12, A13, FINSEQ_1:def_7; (PR1 . k) `2 = 2 by A15, MCART_1:7; then A17: (PR2 . n) `2 = 2 by A12, A13, FINSEQ_1:def_7; len PR < n by A13, A14, NAT_1:13; hence PR2,n is_a_correct_step by A1, A16, A17, A10, A11, Def7; ::_thesis: verum end; now__::_thesis:_(_n_in_dom_PR_implies_PR2,n_is_a_correct_step_) assume n in dom PR ; ::_thesis: PR2,n is_a_correct_step then A18: ( 1 <= n & n <= len PR ) by FINSEQ_3:25; then PR,n is_a_correct_step by A3, Def8; hence PR2,n is_a_correct_step by A18, Th34; ::_thesis: verum end; hence PR2,n is_a_correct_step by A8, A9, FINSEQ_1:25; ::_thesis: verum end; then A19: PR2 is a_proof by Def8; PR2 . (len PR2) = PR2 . ((len PR) + (len PR1)) by FINSEQ_1:22; then PR2 . (len PR2) = PR2 . ((len PR) + 1) by FINSEQ_1:39; then PR2 . (len PR2) = PR1 . 1 by A6, FINSEQ_1:def_7; then PR2 . (len PR2) = [g,2] by FINSEQ_1:40; then (PR2 . (len PR2)) `1 = g by MCART_1:7; hence |- g by A19, Def9; ::_thesis: verum end; theorem Th37: :: CALCUL_1:37 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al st 1 < len f & 1 < len g & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & |- f & |- g holds |- (Ant (Ant f)) ^ <*(Suc f)*> proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al st 1 < len f & 1 < len g & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & |- f & |- g holds |- (Ant (Ant f)) ^ <*(Suc f)*> let f, g be FinSequence of CQC-WFF Al; ::_thesis: ( 1 < len f & 1 < len g & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & |- f & |- g implies |- (Ant (Ant f)) ^ <*(Suc f)*> ) assume that A1: ( 1 < len f & 1 < len g & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g ) and A2: |- f and A3: |- g ; ::_thesis: |- (Ant (Ant f)) ^ <*(Suc f)*> consider PR1 being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A4: PR1 is a_proof and A5: g = (PR1 . (len PR1)) `1 by A3, Def9; consider PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A6: PR is a_proof and A7: f = (PR . (len PR)) `1 by A2, Def9; A8: (Ant (Ant f)) ^ <*(Suc f)*> in set_of_CQC-WFF-seq Al by Def6; now__::_thesis:_for_a_being_set_st_a_in_rng_<*[((Ant_(Ant_f))_^_<*(Suc_f)*>),3]*>_holds_ a_in_[:(set_of_CQC-WFF-seq_Al),Proof_Step_Kinds:] let a be set ; ::_thesis: ( a in rng <*[((Ant (Ant f)) ^ <*(Suc f)*>),3]*> implies a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ) assume a in rng <*[((Ant (Ant f)) ^ <*(Suc f)*>),3]*> ; ::_thesis: a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] then a in {[((Ant (Ant f)) ^ <*(Suc f)*>),3]} by FINSEQ_1:38; then a = [((Ant (Ant f)) ^ <*(Suc f)*>),3] by TARSKI:def_1; hence a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by A8, CQC_THE1:21, ZFMISC_1:87; ::_thesis: verum end; then rng <*[((Ant (Ant f)) ^ <*(Suc f)*>),3]*> c= [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by TARSKI:def_3; then reconsider PR2 = <*[((Ant (Ant f)) ^ <*(Suc f)*>),3]*> as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by FINSEQ_1:def_4; 1 in Seg 1 by FINSEQ_1:2, TARSKI:def_1; then A9: 1 in dom PR2 by FINSEQ_1:38; set PR3 = (PR ^ PR1) ^ PR2; reconsider PR3 = (PR ^ PR1) ^ PR2 as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ; A10: PR <> {} by A6, Def8; now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_len_PR3_holds_ PR3,n_is_a_correct_step let n be Nat; ::_thesis: ( 1 <= n & n <= len PR3 implies PR3,n is_a_correct_step ) assume ( 1 <= n & n <= len PR3 ) ; ::_thesis: PR3,n is_a_correct_step then A11: n in dom PR3 by FINSEQ_3:25; A12: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR2_&_n_=_(len_(PR_^_PR1))_+_k_)_implies_PR3,n_is_a_correct_step_) given k being Nat such that A13: k in dom PR2 and A14: n = (len (PR ^ PR1)) + k ; ::_thesis: PR3,n is_a_correct_step k in Seg 1 by A13, FINSEQ_1:38; then A15: k = 1 by FINSEQ_1:2, TARSKI:def_1; then A16: PR2 . k = [((Ant (Ant f)) ^ <*(Suc f)*>),3] by FINSEQ_1:40; then (PR2 . k) `1 = (Ant (Ant f)) ^ <*(Suc f)*> by MCART_1:7; then A17: (PR3 . n) `1 = (Ant (Ant f)) ^ <*(Suc f)*> by A13, A14, FINSEQ_1:def_7; (PR2 . k) `2 = 3 by A16, MCART_1:7; then A18: (PR3 . n) `2 = 3 by A13, A14, FINSEQ_1:def_7; A19: len (PR ^ PR1) < n by A14, A15, NAT_1:13; A20: 1 <= len PR by A10, Th6; then len PR in dom PR by FINSEQ_3:25; then A21: f = ((PR ^ PR1) . (len PR)) `1 by A7, FINSEQ_1:def_7; A22: PR1 <> {} by A4, Def8; then A23: len PR < len (PR ^ PR1) by Th6; then len PR in dom (PR ^ PR1) by A20, FINSEQ_3:25; then A24: f = (PR3 . (len PR)) `1 by A21, FINSEQ_1:def_7; 1 <= len PR1 by A22, Th6; then len PR1 in dom PR1 by FINSEQ_3:25; then g = ((PR ^ PR1) . ((len PR) + (len PR1))) `1 by A5, FINSEQ_1:def_7; then A25: g = ((PR ^ PR1) . (len (PR ^ PR1))) `1 by FINSEQ_1:22; 1 <= len (PR ^ PR1) by A10, Th6; then len (PR ^ PR1) in dom (PR ^ PR1) by FINSEQ_3:25; then A26: g = (PR3 . (len (PR ^ PR1))) `1 by A25, FINSEQ_1:def_7; 1 <= len (PR ^ PR1) by A10, Th6; hence PR3,n is_a_correct_step by A1, A17, A18, A20, A23, A24, A19, A26, Def7; ::_thesis: verum end; now__::_thesis:_(_n_in_dom_(PR_^_PR1)_implies_PR3,n_is_a_correct_step_) A27: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR1_&_n_=_(len_PR)_+_k_)_implies_PR3,n_is_a_correct_step_) given k being Nat such that A28: k in dom PR1 and A29: n = (len PR) + k ; ::_thesis: PR3,n is_a_correct_step A30: 1 <= k by A28, FINSEQ_3:25; A31: k <= len PR1 by A28, FINSEQ_3:25; then n <= (len PR1) + (len PR) by A29, XREAL_1:6; then A32: n <= len (PR ^ PR1) by FINSEQ_1:22; k <= n by A29, NAT_1:11; then A33: 1 <= n by A30, XXREAL_0:2; PR1,k is_a_correct_step by A4, A30, A31, Def8; then PR ^ PR1,n is_a_correct_step by A28, A29, A30, A31, Th35; hence PR3,n is_a_correct_step by A33, A32, Th34; ::_thesis: verum end; A34: now__::_thesis:_(_n_in_dom_PR_implies_PR3,n_is_a_correct_step_) assume A35: n in dom PR ; ::_thesis: PR3,n is_a_correct_step then A36: 1 <= n by FINSEQ_3:25; A37: n <= len PR by A35, FINSEQ_3:25; len PR <= len (PR ^ PR1) by Th6; then A38: n <= len (PR ^ PR1) by A37, XXREAL_0:2; PR,n is_a_correct_step by A6, A36, A37, Def8; then PR ^ PR1,n is_a_correct_step by A36, A37, Th34; hence PR3,n is_a_correct_step by A36, A38, Th34; ::_thesis: verum end; assume n in dom (PR ^ PR1) ; ::_thesis: PR3,n is_a_correct_step hence PR3,n is_a_correct_step by A34, A27, FINSEQ_1:25; ::_thesis: verum end; hence PR3,n is_a_correct_step by A11, A12, FINSEQ_1:25; ::_thesis: verum end; then A39: PR3 is a_proof by Def8; PR3 . (len PR3) = PR3 . ((len (PR ^ PR1)) + (len PR2)) by FINSEQ_1:22; then PR3 . (len PR3) = PR3 . ((len (PR ^ PR1)) + 1) by FINSEQ_1:39; then PR3 . (len PR3) = PR2 . 1 by A9, FINSEQ_1:def_7; then PR3 . (len PR3) = [((Ant (Ant f)) ^ <*(Suc f)*>),3] by FINSEQ_1:40; then (PR3 . (len PR3)) `1 = (Ant (Ant f)) ^ <*(Suc f)*> by MCART_1:7; hence |- (Ant (Ant f)) ^ <*(Suc f)*> by A39, Def9; ::_thesis: verum end; theorem Th38: :: CALCUL_1:38 for Al being QC-alphabet for p being Element of CQC-WFF Al for f, g being FinSequence of CQC-WFF Al st len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & |- f & |- g holds |- (Ant (Ant f)) ^ <*p*> proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for f, g being FinSequence of CQC-WFF Al st len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & |- f & |- g holds |- (Ant (Ant f)) ^ <*p*> let p be Element of CQC-WFF Al; ::_thesis: for f, g being FinSequence of CQC-WFF Al st len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & |- f & |- g holds |- (Ant (Ant f)) ^ <*p*> let f, g be FinSequence of CQC-WFF Al; ::_thesis: ( len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & |- f & |- g implies |- (Ant (Ant f)) ^ <*p*> ) assume that A1: ( len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g ) and A2: |- f and A3: |- g ; ::_thesis: |- (Ant (Ant f)) ^ <*p*> consider PR1 being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A4: PR1 is a_proof and A5: g = (PR1 . (len PR1)) `1 by A3, Def9; consider PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A6: PR is a_proof and A7: f = (PR . (len PR)) `1 by A2, Def9; A8: (Ant (Ant f)) ^ <*p*> in set_of_CQC-WFF-seq Al by Def6; now__::_thesis:_for_a_being_set_st_a_in_rng_<*[((Ant_(Ant_f))_^_<*p*>),4]*>_holds_ a_in_[:(set_of_CQC-WFF-seq_Al),Proof_Step_Kinds:] let a be set ; ::_thesis: ( a in rng <*[((Ant (Ant f)) ^ <*p*>),4]*> implies a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ) assume a in rng <*[((Ant (Ant f)) ^ <*p*>),4]*> ; ::_thesis: a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] then a in {[((Ant (Ant f)) ^ <*p*>),4]} by FINSEQ_1:38; then a = [((Ant (Ant f)) ^ <*p*>),4] by TARSKI:def_1; hence a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by A8, CQC_THE1:21, ZFMISC_1:87; ::_thesis: verum end; then rng <*[((Ant (Ant f)) ^ <*p*>),4]*> c= [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by TARSKI:def_3; then reconsider PR2 = <*[((Ant (Ant f)) ^ <*p*>),4]*> as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by FINSEQ_1:def_4; 1 in Seg 1 by FINSEQ_1:2, TARSKI:def_1; then A9: 1 in dom PR2 by FINSEQ_1:38; set PR3 = (PR ^ PR1) ^ PR2; reconsider PR3 = (PR ^ PR1) ^ PR2 as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ; A10: PR <> {} by A6, Def8; now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_len_PR3_holds_ PR3,n_is_a_correct_step let n be Nat; ::_thesis: ( 1 <= n & n <= len PR3 implies PR3,n is_a_correct_step ) assume ( 1 <= n & n <= len PR3 ) ; ::_thesis: PR3,n is_a_correct_step then A11: n in dom PR3 by FINSEQ_3:25; A12: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR2_&_n_=_(len_(PR_^_PR1))_+_k_)_implies_PR3,n_is_a_correct_step_) given k being Nat such that A13: k in dom PR2 and A14: n = (len (PR ^ PR1)) + k ; ::_thesis: PR3,n is_a_correct_step k in Seg 1 by A13, FINSEQ_1:38; then A15: k = 1 by FINSEQ_1:2, TARSKI:def_1; then A16: PR2 . k = [((Ant (Ant f)) ^ <*p*>),4] by FINSEQ_1:40; then (PR2 . k) `1 = (Ant (Ant f)) ^ <*p*> by MCART_1:7; then A17: (PR3 . n) `1 = (Ant (Ant f)) ^ <*p*> by A13, A14, FINSEQ_1:def_7; (PR2 . k) `2 = 4 by A16, MCART_1:7; then A18: (PR3 . n) `2 = 4 by A13, A14, FINSEQ_1:def_7; A19: len (PR ^ PR1) < n by A14, A15, NAT_1:13; A20: 1 <= len PR by A10, Th6; then len PR in dom PR by FINSEQ_3:25; then A21: f = ((PR ^ PR1) . (len PR)) `1 by A7, FINSEQ_1:def_7; A22: PR1 <> {} by A4, Def8; then A23: len PR < len (PR ^ PR1) by Th6; then len PR in dom (PR ^ PR1) by A20, FINSEQ_3:25; then A24: f = (PR3 . (len PR)) `1 by A21, FINSEQ_1:def_7; 1 <= len PR1 by A22, Th6; then len PR1 in dom PR1 by FINSEQ_3:25; then g = ((PR ^ PR1) . ((len PR) + (len PR1))) `1 by A5, FINSEQ_1:def_7; then A25: g = ((PR ^ PR1) . (len (PR ^ PR1))) `1 by FINSEQ_1:22; 1 <= len (PR ^ PR1) by A10, Th6; then len (PR ^ PR1) in dom (PR ^ PR1) by FINSEQ_3:25; then A26: g = (PR3 . (len (PR ^ PR1))) `1 by A25, FINSEQ_1:def_7; 1 <= len (PR ^ PR1) by A10, Th6; hence PR3,n is_a_correct_step by A1, A17, A18, A20, A23, A24, A19, A26, Def7; ::_thesis: verum end; now__::_thesis:_(_n_in_dom_(PR_^_PR1)_implies_PR3,n_is_a_correct_step_) A27: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR1_&_n_=_(len_PR)_+_k_)_implies_PR3,n_is_a_correct_step_) given k being Nat such that A28: k in dom PR1 and A29: n = (len PR) + k ; ::_thesis: PR3,n is_a_correct_step A30: 1 <= k by A28, FINSEQ_3:25; A31: k <= len PR1 by A28, FINSEQ_3:25; then n <= (len PR1) + (len PR) by A29, XREAL_1:6; then A32: n <= len (PR ^ PR1) by FINSEQ_1:22; k <= n by A29, NAT_1:11; then A33: 1 <= n by A30, XXREAL_0:2; PR1,k is_a_correct_step by A4, A30, A31, Def8; then PR ^ PR1,n is_a_correct_step by A28, A29, A30, A31, Th35; hence PR3,n is_a_correct_step by A33, A32, Th34; ::_thesis: verum end; A34: now__::_thesis:_(_n_in_dom_PR_implies_PR3,n_is_a_correct_step_) assume A35: n in dom PR ; ::_thesis: PR3,n is_a_correct_step then A36: 1 <= n by FINSEQ_3:25; A37: n <= len PR by A35, FINSEQ_3:25; len PR <= len (PR ^ PR1) by Th6; then A38: n <= len (PR ^ PR1) by A37, XXREAL_0:2; PR,n is_a_correct_step by A6, A36, A37, Def8; then PR ^ PR1,n is_a_correct_step by A36, A37, Th34; hence PR3,n is_a_correct_step by A36, A38, Th34; ::_thesis: verum end; assume n in dom (PR ^ PR1) ; ::_thesis: PR3,n is_a_correct_step hence PR3,n is_a_correct_step by A34, A27, FINSEQ_1:25; ::_thesis: verum end; hence PR3,n is_a_correct_step by A11, A12, FINSEQ_1:25; ::_thesis: verum end; then A39: PR3 is a_proof by Def8; PR3 . (len PR3) = PR3 . ((len (PR ^ PR1)) + (len PR2)) by FINSEQ_1:22; then PR3 . (len PR3) = PR3 . ((len (PR ^ PR1)) + 1) by FINSEQ_1:39; then PR3 . (len PR3) = PR2 . 1 by A9, FINSEQ_1:def_7; then PR3 . (len PR3) = [((Ant (Ant f)) ^ <*p*>),4] by FINSEQ_1:40; then (PR3 . (len PR3)) `1 = (Ant (Ant f)) ^ <*p*> by MCART_1:7; hence |- (Ant (Ant f)) ^ <*p*> by A39, Def9; ::_thesis: verum end; theorem Th39: :: CALCUL_1:39 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al st Ant f = Ant g & |- f & |- g holds |- (Ant f) ^ <*((Suc f) '&' (Suc g))*> proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al st Ant f = Ant g & |- f & |- g holds |- (Ant f) ^ <*((Suc f) '&' (Suc g))*> let f, g be FinSequence of CQC-WFF Al; ::_thesis: ( Ant f = Ant g & |- f & |- g implies |- (Ant f) ^ <*((Suc f) '&' (Suc g))*> ) assume that A1: Ant f = Ant g and A2: |- f and A3: |- g ; ::_thesis: |- (Ant f) ^ <*((Suc f) '&' (Suc g))*> consider PR1 being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A4: PR1 is a_proof and A5: g = (PR1 . (len PR1)) `1 by A3, Def9; consider PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A6: PR is a_proof and A7: f = (PR . (len PR)) `1 by A2, Def9; A8: (Ant f) ^ <*((Suc f) '&' (Suc g))*> in set_of_CQC-WFF-seq Al by Def6; now__::_thesis:_for_a_being_set_st_a_in_rng_<*[((Ant_f)_^_<*((Suc_f)_'&'_(Suc_g))*>),5]*>_holds_ a_in_[:(set_of_CQC-WFF-seq_Al),Proof_Step_Kinds:] let a be set ; ::_thesis: ( a in rng <*[((Ant f) ^ <*((Suc f) '&' (Suc g))*>),5]*> implies a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ) assume a in rng <*[((Ant f) ^ <*((Suc f) '&' (Suc g))*>),5]*> ; ::_thesis: a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] then a in {[((Ant f) ^ <*((Suc f) '&' (Suc g))*>),5]} by FINSEQ_1:38; then a = [((Ant f) ^ <*((Suc f) '&' (Suc g))*>),5] by TARSKI:def_1; hence a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by A8, CQC_THE1:21, ZFMISC_1:87; ::_thesis: verum end; then rng <*[((Ant f) ^ <*((Suc f) '&' (Suc g))*>),5]*> c= [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by TARSKI:def_3; then reconsider PR2 = <*[((Ant f) ^ <*((Suc f) '&' (Suc g))*>),5]*> as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by FINSEQ_1:def_4; 1 in Seg 1 by FINSEQ_1:2, TARSKI:def_1; then A9: 1 in dom PR2 by FINSEQ_1:38; set PR3 = (PR ^ PR1) ^ PR2; reconsider PR3 = (PR ^ PR1) ^ PR2 as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ; A10: PR <> {} by A6, Def8; now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_len_PR3_holds_ PR3,n_is_a_correct_step let n be Nat; ::_thesis: ( 1 <= n & n <= len PR3 implies PR3,n is_a_correct_step ) assume ( 1 <= n & n <= len PR3 ) ; ::_thesis: PR3,n is_a_correct_step then A11: n in dom PR3 by FINSEQ_3:25; A12: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR2_&_n_=_(len_(PR_^_PR1))_+_k_)_implies_PR3,n_is_a_correct_step_) given k being Nat such that A13: k in dom PR2 and A14: n = (len (PR ^ PR1)) + k ; ::_thesis: PR3,n is_a_correct_step k in Seg 1 by A13, FINSEQ_1:38; then A15: k = 1 by FINSEQ_1:2, TARSKI:def_1; then A16: PR2 . k = [((Ant f) ^ <*((Suc f) '&' (Suc g))*>),5] by FINSEQ_1:40; then (PR2 . k) `1 = (Ant f) ^ <*((Suc f) '&' (Suc g))*> by MCART_1:7; then A17: (PR3 . n) `1 = (Ant f) ^ <*((Suc f) '&' (Suc g))*> by A13, A14, FINSEQ_1:def_7; (PR2 . k) `2 = 5 by A16, MCART_1:7; then A18: (PR3 . n) `2 = 5 by A13, A14, FINSEQ_1:def_7; A19: len (PR ^ PR1) < n by A14, A15, NAT_1:13; A20: 1 <= len PR by A10, Th6; then len PR in dom PR by FINSEQ_3:25; then A21: f = ((PR ^ PR1) . (len PR)) `1 by A7, FINSEQ_1:def_7; A22: PR1 <> {} by A4, Def8; then A23: len PR < len (PR ^ PR1) by Th6; then len PR in dom (PR ^ PR1) by A20, FINSEQ_3:25; then A24: f = (PR3 . (len PR)) `1 by A21, FINSEQ_1:def_7; 1 <= len PR1 by A22, Th6; then len PR1 in dom PR1 by FINSEQ_3:25; then g = ((PR ^ PR1) . ((len PR) + (len PR1))) `1 by A5, FINSEQ_1:def_7; then A25: g = ((PR ^ PR1) . (len (PR ^ PR1))) `1 by FINSEQ_1:22; 1 <= len (PR ^ PR1) by A10, Th6; then len (PR ^ PR1) in dom (PR ^ PR1) by FINSEQ_3:25; then A26: g = (PR3 . (len (PR ^ PR1))) `1 by A25, FINSEQ_1:def_7; 1 <= len (PR ^ PR1) by A10, Th6; hence PR3,n is_a_correct_step by A1, A17, A18, A20, A23, A24, A19, A26, Def7; ::_thesis: verum end; now__::_thesis:_(_n_in_dom_(PR_^_PR1)_implies_PR3,n_is_a_correct_step_) A27: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR1_&_n_=_(len_PR)_+_k_)_implies_PR3,n_is_a_correct_step_) given k being Nat such that A28: k in dom PR1 and A29: n = (len PR) + k ; ::_thesis: PR3,n is_a_correct_step A30: 1 <= k by A28, FINSEQ_3:25; A31: k <= len PR1 by A28, FINSEQ_3:25; then n <= (len PR1) + (len PR) by A29, XREAL_1:6; then A32: n <= len (PR ^ PR1) by FINSEQ_1:22; k <= n by A29, NAT_1:11; then A33: 1 <= n by A30, XXREAL_0:2; PR1,k is_a_correct_step by A4, A30, A31, Def8; then PR ^ PR1,n is_a_correct_step by A28, A29, A30, A31, Th35; hence PR3,n is_a_correct_step by A33, A32, Th34; ::_thesis: verum end; A34: now__::_thesis:_(_n_in_dom_PR_implies_PR3,n_is_a_correct_step_) assume A35: n in dom PR ; ::_thesis: PR3,n is_a_correct_step then A36: 1 <= n by FINSEQ_3:25; A37: n <= len PR by A35, FINSEQ_3:25; len PR <= len (PR ^ PR1) by Th6; then A38: n <= len (PR ^ PR1) by A37, XXREAL_0:2; PR,n is_a_correct_step by A6, A36, A37, Def8; then PR ^ PR1,n is_a_correct_step by A36, A37, Th34; hence PR3,n is_a_correct_step by A36, A38, Th34; ::_thesis: verum end; assume n in dom (PR ^ PR1) ; ::_thesis: PR3,n is_a_correct_step hence PR3,n is_a_correct_step by A34, A27, FINSEQ_1:25; ::_thesis: verum end; hence PR3,n is_a_correct_step by A11, A12, FINSEQ_1:25; ::_thesis: verum end; then A39: PR3 is a_proof by Def8; PR3 . (len PR3) = PR3 . ((len (PR ^ PR1)) + (len PR2)) by FINSEQ_1:22; then PR3 . (len PR3) = PR3 . ((len (PR ^ PR1)) + 1) by FINSEQ_1:39; then PR3 . (len PR3) = PR2 . 1 by A9, FINSEQ_1:def_7; then PR3 . (len PR3) = [((Ant f) ^ <*((Suc f) '&' (Suc g))*>),5] by FINSEQ_1:40; then (PR3 . (len PR3)) `1 = (Ant f) ^ <*((Suc f) '&' (Suc g))*> by MCART_1:7; hence |- (Ant f) ^ <*((Suc f) '&' (Suc g))*> by A39, Def9; ::_thesis: verum end; theorem Th40: :: CALCUL_1:40 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st p '&' q = Suc f & |- f holds |- (Ant f) ^ <*p*> proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st p '&' q = Suc f & |- f holds |- (Ant f) ^ <*p*> let p, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st p '&' q = Suc f & |- f holds |- (Ant f) ^ <*p*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( p '&' q = Suc f & |- f implies |- (Ant f) ^ <*p*> ) assume that A1: p '&' q = Suc f and A2: |- f ; ::_thesis: |- (Ant f) ^ <*p*> consider PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A3: PR is a_proof and A4: (PR . (len PR)) `1 = f by A2, Def9; A5: (Ant f) ^ <*p*> in set_of_CQC-WFF-seq Al by Def6; now__::_thesis:_for_a_being_set_st_a_in_rng_<*[((Ant_f)_^_<*p*>),6]*>_holds_ a_in_[:(set_of_CQC-WFF-seq_Al),Proof_Step_Kinds:] let a be set ; ::_thesis: ( a in rng <*[((Ant f) ^ <*p*>),6]*> implies a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ) assume a in rng <*[((Ant f) ^ <*p*>),6]*> ; ::_thesis: a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] then a in {[((Ant f) ^ <*p*>),6]} by FINSEQ_1:38; then a = [((Ant f) ^ <*p*>),6] by TARSKI:def_1; hence a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by A5, CQC_THE1:21, ZFMISC_1:87; ::_thesis: verum end; then rng <*[((Ant f) ^ <*p*>),6]*> c= [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by TARSKI:def_3; then reconsider PR1 = <*[((Ant f) ^ <*p*>),6]*> as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by FINSEQ_1:def_4; 1 in Seg 1 by FINSEQ_1:2, TARSKI:def_1; then A6: 1 in dom PR1 by FINSEQ_1:38; set PR2 = PR ^ PR1; reconsider PR2 = PR ^ PR1 as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ; A7: PR <> {} by A3, Def8; now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_len_PR2_holds_ PR2,n_is_a_correct_step let n be Nat; ::_thesis: ( 1 <= n & n <= len PR2 implies PR2,n is_a_correct_step ) assume ( 1 <= n & n <= len PR2 ) ; ::_thesis: PR2,n is_a_correct_step then A8: n in dom PR2 by FINSEQ_3:25; A9: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR1_&_n_=_(len_PR)_+_k_)_implies_PR2,n_is_a_correct_step_) A10: 1 <= len PR by A7, Th6; then len PR in dom PR by FINSEQ_3:25; then A11: f = (PR2 . (len PR)) `1 by A4, FINSEQ_1:def_7; given k being Nat such that A12: k in dom PR1 and A13: n = (len PR) + k ; ::_thesis: PR2,n is_a_correct_step k in Seg 1 by A12, FINSEQ_1:38; then A14: k = 1 by FINSEQ_1:2, TARSKI:def_1; then A15: PR1 . k = [((Ant f) ^ <*p*>),6] by FINSEQ_1:40; then (PR1 . k) `1 = (Ant f) ^ <*p*> by MCART_1:7; then A16: (PR2 . n) `1 = (Ant f) ^ <*p*> by A12, A13, FINSEQ_1:def_7; (PR1 . k) `2 = 6 by A15, MCART_1:7; then A17: (PR2 . n) `2 = 6 by A12, A13, FINSEQ_1:def_7; len PR < n by A13, A14, NAT_1:13; hence PR2,n is_a_correct_step by A1, A16, A17, A10, A11, Def7; ::_thesis: verum end; now__::_thesis:_(_n_in_dom_PR_implies_PR2,n_is_a_correct_step_) assume n in dom PR ; ::_thesis: PR2,n is_a_correct_step then A18: ( 1 <= n & n <= len PR ) by FINSEQ_3:25; then PR,n is_a_correct_step by A3, Def8; hence PR2,n is_a_correct_step by A18, Th34; ::_thesis: verum end; hence PR2,n is_a_correct_step by A8, A9, FINSEQ_1:25; ::_thesis: verum end; then A19: PR2 is a_proof by Def8; PR2 . (len PR2) = PR2 . ((len PR) + (len PR1)) by FINSEQ_1:22; then PR2 . (len PR2) = PR2 . ((len PR) + 1) by FINSEQ_1:39; then PR2 . (len PR2) = PR1 . 1 by A6, FINSEQ_1:def_7; then PR2 . (len PR2) = [((Ant f) ^ <*p*>),6] by FINSEQ_1:40; then (PR2 . (len PR2)) `1 = (Ant f) ^ <*p*> by MCART_1:7; hence |- (Ant f) ^ <*p*> by A19, Def9; ::_thesis: verum end; theorem Th41: :: CALCUL_1:41 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st p '&' q = Suc f & |- f holds |- (Ant f) ^ <*q*> proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st p '&' q = Suc f & |- f holds |- (Ant f) ^ <*q*> let p, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st p '&' q = Suc f & |- f holds |- (Ant f) ^ <*q*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( p '&' q = Suc f & |- f implies |- (Ant f) ^ <*q*> ) assume that A1: p '&' q = Suc f and A2: |- f ; ::_thesis: |- (Ant f) ^ <*q*> consider PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A3: PR is a_proof and A4: (PR . (len PR)) `1 = f by A2, Def9; A5: (Ant f) ^ <*q*> in set_of_CQC-WFF-seq Al by Def6; now__::_thesis:_for_a_being_set_st_a_in_rng_<*[((Ant_f)_^_<*q*>),7]*>_holds_ a_in_[:(set_of_CQC-WFF-seq_Al),Proof_Step_Kinds:] let a be set ; ::_thesis: ( a in rng <*[((Ant f) ^ <*q*>),7]*> implies a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ) assume a in rng <*[((Ant f) ^ <*q*>),7]*> ; ::_thesis: a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] then a in {[((Ant f) ^ <*q*>),7]} by FINSEQ_1:38; then a = [((Ant f) ^ <*q*>),7] by TARSKI:def_1; hence a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by A5, CQC_THE1:21, ZFMISC_1:87; ::_thesis: verum end; then rng <*[((Ant f) ^ <*q*>),7]*> c= [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by TARSKI:def_3; then reconsider PR1 = <*[((Ant f) ^ <*q*>),7]*> as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by FINSEQ_1:def_4; 1 in Seg 1 by FINSEQ_1:2, TARSKI:def_1; then A6: 1 in dom PR1 by FINSEQ_1:38; set PR2 = PR ^ PR1; reconsider PR2 = PR ^ PR1 as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ; A7: PR <> {} by A3, Def8; now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_len_PR2_holds_ PR2,n_is_a_correct_step let n be Nat; ::_thesis: ( 1 <= n & n <= len PR2 implies PR2,n is_a_correct_step ) assume ( 1 <= n & n <= len PR2 ) ; ::_thesis: PR2,n is_a_correct_step then A8: n in dom PR2 by FINSEQ_3:25; A9: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR1_&_n_=_(len_PR)_+_k_)_implies_PR2,n_is_a_correct_step_) A10: 1 <= len PR by A7, Th6; then len PR in dom PR by FINSEQ_3:25; then A11: f = (PR2 . (len PR)) `1 by A4, FINSEQ_1:def_7; given k being Nat such that A12: k in dom PR1 and A13: n = (len PR) + k ; ::_thesis: PR2,n is_a_correct_step k in Seg 1 by A12, FINSEQ_1:38; then A14: k = 1 by FINSEQ_1:2, TARSKI:def_1; then A15: PR1 . k = [((Ant f) ^ <*q*>),7] by FINSEQ_1:40; then (PR1 . k) `1 = (Ant f) ^ <*q*> by MCART_1:7; then A16: (PR2 . n) `1 = (Ant f) ^ <*q*> by A12, A13, FINSEQ_1:def_7; (PR1 . k) `2 = 7 by A15, MCART_1:7; then A17: (PR2 . n) `2 = 7 by A12, A13, FINSEQ_1:def_7; len PR < n by A13, A14, NAT_1:13; hence PR2,n is_a_correct_step by A1, A16, A17, A10, A11, Def7; ::_thesis: verum end; now__::_thesis:_(_n_in_dom_PR_implies_PR2,n_is_a_correct_step_) assume n in dom PR ; ::_thesis: PR2,n is_a_correct_step then A18: ( 1 <= n & n <= len PR ) by FINSEQ_3:25; then PR,n is_a_correct_step by A3, Def8; hence PR2,n is_a_correct_step by A18, Th34; ::_thesis: verum end; hence PR2,n is_a_correct_step by A8, A9, FINSEQ_1:25; ::_thesis: verum end; then A19: PR2 is a_proof by Def8; PR2 . (len PR2) = PR2 . ((len PR) + (len PR1)) by FINSEQ_1:22; then PR2 . (len PR2) = PR2 . ((len PR) + 1) by FINSEQ_1:39; then PR2 . (len PR2) = PR1 . 1 by A6, FINSEQ_1:def_7; then PR2 . (len PR2) = [((Ant f) ^ <*q*>),7] by FINSEQ_1:40; then (PR2 . (len PR2)) `1 = (Ant f) ^ <*q*> by MCART_1:7; hence |- (Ant f) ^ <*q*> by A19, Def9; ::_thesis: verum end; theorem Th42: :: CALCUL_1:42 for Al being QC-alphabet for p being Element of CQC-WFF Al for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & |- f holds |- (Ant f) ^ <*(p . (x,y))*> proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & |- f holds |- (Ant f) ^ <*(p . (x,y))*> let p be Element of CQC-WFF Al; ::_thesis: for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & |- f holds |- (Ant f) ^ <*(p . (x,y))*> let x, y be bound_QC-variable of Al; ::_thesis: for f being FinSequence of CQC-WFF Al st Suc f = All (x,p) & |- f holds |- (Ant f) ^ <*(p . (x,y))*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( Suc f = All (x,p) & |- f implies |- (Ant f) ^ <*(p . (x,y))*> ) assume that A1: Suc f = All (x,p) and A2: |- f ; ::_thesis: |- (Ant f) ^ <*(p . (x,y))*> consider PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A3: PR is a_proof and A4: (PR . (len PR)) `1 = f by A2, Def9; A5: (Ant f) ^ <*(p . (x,y))*> in set_of_CQC-WFF-seq Al by Def6; now__::_thesis:_for_a_being_set_st_a_in_rng_<*[((Ant_f)_^_<*(p_._(x,y))*>),8]*>_holds_ a_in_[:(set_of_CQC-WFF-seq_Al),Proof_Step_Kinds:] let a be set ; ::_thesis: ( a in rng <*[((Ant f) ^ <*(p . (x,y))*>),8]*> implies a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ) assume a in rng <*[((Ant f) ^ <*(p . (x,y))*>),8]*> ; ::_thesis: a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] then a in {[((Ant f) ^ <*(p . (x,y))*>),8]} by FINSEQ_1:38; then a = [((Ant f) ^ <*(p . (x,y))*>),8] by TARSKI:def_1; hence a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by A5, CQC_THE1:21, ZFMISC_1:87; ::_thesis: verum end; then rng <*[((Ant f) ^ <*(p . (x,y))*>),8]*> c= [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by TARSKI:def_3; then reconsider PR1 = <*[((Ant f) ^ <*(p . (x,y))*>),8]*> as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by FINSEQ_1:def_4; 1 in Seg 1 by FINSEQ_1:2, TARSKI:def_1; then A6: 1 in dom PR1 by FINSEQ_1:38; set PR2 = PR ^ PR1; reconsider PR2 = PR ^ PR1 as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ; A7: PR <> {} by A3, Def8; now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_len_PR2_holds_ PR2,n_is_a_correct_step let n be Nat; ::_thesis: ( 1 <= n & n <= len PR2 implies PR2,n is_a_correct_step ) assume ( 1 <= n & n <= len PR2 ) ; ::_thesis: PR2,n is_a_correct_step then A8: n in dom PR2 by FINSEQ_3:25; A9: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR1_&_n_=_(len_PR)_+_k_)_implies_PR2,n_is_a_correct_step_) A10: 1 <= len PR by A7, Th6; then len PR in dom PR by FINSEQ_3:25; then A11: f = (PR2 . (len PR)) `1 by A4, FINSEQ_1:def_7; given k being Nat such that A12: k in dom PR1 and A13: n = (len PR) + k ; ::_thesis: PR2,n is_a_correct_step k in Seg 1 by A12, FINSEQ_1:38; then A14: k = 1 by FINSEQ_1:2, TARSKI:def_1; then A15: PR1 . k = [((Ant f) ^ <*(p . (x,y))*>),8] by FINSEQ_1:40; then (PR1 . k) `1 = (Ant f) ^ <*(p . (x,y))*> by MCART_1:7; then A16: (PR2 . n) `1 = (Ant f) ^ <*(p . (x,y))*> by A12, A13, FINSEQ_1:def_7; (PR1 . k) `2 = 8 by A15, MCART_1:7; then A17: (PR2 . n) `2 = 8 by A12, A13, FINSEQ_1:def_7; len PR < n by A13, A14, NAT_1:13; hence PR2,n is_a_correct_step by A1, A16, A17, A10, A11, Def7; ::_thesis: verum end; now__::_thesis:_(_n_in_dom_PR_implies_PR2,n_is_a_correct_step_) assume n in dom PR ; ::_thesis: PR2,n is_a_correct_step then A18: ( 1 <= n & n <= len PR ) by FINSEQ_3:25; then PR,n is_a_correct_step by A3, Def8; hence PR2,n is_a_correct_step by A18, Th34; ::_thesis: verum end; hence PR2,n is_a_correct_step by A8, A9, FINSEQ_1:25; ::_thesis: verum end; then A19: PR2 is a_proof by Def8; PR2 . (len PR2) = PR2 . ((len PR) + (len PR1)) by FINSEQ_1:22; then PR2 . (len PR2) = PR2 . ((len PR) + 1) by FINSEQ_1:39; then PR2 . (len PR2) = PR1 . 1 by A6, FINSEQ_1:def_7; then PR2 . (len PR2) = [((Ant f) ^ <*(p . (x,y))*>),8] by FINSEQ_1:40; then (PR2 . (len PR2)) `1 = (Ant f) ^ <*(p . (x,y))*> by MCART_1:7; hence |- (Ant f) ^ <*(p . (x,y))*> by A19, Def9; ::_thesis: verum end; theorem Th43: :: CALCUL_1:43 for Al being QC-alphabet for p being Element of CQC-WFF Al for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & |- f holds |- (Ant f) ^ <*(All (x,p))*> proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & |- f holds |- (Ant f) ^ <*(All (x,p))*> let p be Element of CQC-WFF Al; ::_thesis: for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & |- f holds |- (Ant f) ^ <*(All (x,p))*> let x, y be bound_QC-variable of Al; ::_thesis: for f being FinSequence of CQC-WFF Al st Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & |- f holds |- (Ant f) ^ <*(All (x,p))*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & |- f implies |- (Ant f) ^ <*(All (x,p))*> ) assume that A1: ( Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) ) and A2: |- f ; ::_thesis: |- (Ant f) ^ <*(All (x,p))*> consider PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] such that A3: PR is a_proof and A4: (PR . (len PR)) `1 = f by A2, Def9; A5: (Ant f) ^ <*(All (x,p))*> in set_of_CQC-WFF-seq Al by Def6; now__::_thesis:_for_a_being_set_st_a_in_rng_<*[((Ant_f)_^_<*(All_(x,p))*>),9]*>_holds_ a_in_[:(set_of_CQC-WFF-seq_Al),Proof_Step_Kinds:] let a be set ; ::_thesis: ( a in rng <*[((Ant f) ^ <*(All (x,p))*>),9]*> implies a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ) assume a in rng <*[((Ant f) ^ <*(All (x,p))*>),9]*> ; ::_thesis: a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] then a in {[((Ant f) ^ <*(All (x,p))*>),9]} by FINSEQ_1:38; then a = [((Ant f) ^ <*(All (x,p))*>),9] by TARSKI:def_1; hence a in [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by A5, CQC_THE1:21, ZFMISC_1:87; ::_thesis: verum end; then rng <*[((Ant f) ^ <*(All (x,p))*>),9]*> c= [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by TARSKI:def_3; then reconsider PR1 = <*[((Ant f) ^ <*(All (x,p))*>),9]*> as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] by FINSEQ_1:def_4; 1 in Seg 1 by FINSEQ_1:2, TARSKI:def_1; then A6: 1 in dom PR1 by FINSEQ_1:38; set PR2 = PR ^ PR1; reconsider PR2 = PR ^ PR1 as FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] ; A7: PR <> {} by A3, Def8; now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_len_PR2_holds_ PR2,n_is_a_correct_step let n be Nat; ::_thesis: ( 1 <= n & n <= len PR2 implies PR2,n is_a_correct_step ) assume ( 1 <= n & n <= len PR2 ) ; ::_thesis: PR2,n is_a_correct_step then A8: n in dom PR2 by FINSEQ_3:25; A9: now__::_thesis:_(_ex_k_being_Nat_st_ (_k_in_dom_PR1_&_n_=_(len_PR)_+_k_)_implies_PR2,n_is_a_correct_step_) A10: 1 <= len PR by A7, Th6; then len PR in dom PR by FINSEQ_3:25; then A11: f = (PR2 . (len PR)) `1 by A4, FINSEQ_1:def_7; given k being Nat such that A12: k in dom PR1 and A13: n = (len PR) + k ; ::_thesis: PR2,n is_a_correct_step k in Seg 1 by A12, FINSEQ_1:38; then A14: k = 1 by FINSEQ_1:2, TARSKI:def_1; then A15: PR1 . k = [((Ant f) ^ <*(All (x,p))*>),9] by FINSEQ_1:40; then (PR1 . k) `1 = (Ant f) ^ <*(All (x,p))*> by MCART_1:7; then A16: (PR2 . n) `1 = (Ant f) ^ <*(All (x,p))*> by A12, A13, FINSEQ_1:def_7; (PR1 . k) `2 = 9 by A15, MCART_1:7; then A17: (PR2 . n) `2 = 9 by A12, A13, FINSEQ_1:def_7; len PR < n by A13, A14, NAT_1:13; hence PR2,n is_a_correct_step by A1, A16, A17, A10, A11, Def7; ::_thesis: verum end; now__::_thesis:_(_n_in_dom_PR_implies_PR2,n_is_a_correct_step_) assume n in dom PR ; ::_thesis: PR2,n is_a_correct_step then A18: ( 1 <= n & n <= len PR ) by FINSEQ_3:25; then PR,n is_a_correct_step by A3, Def8; hence PR2,n is_a_correct_step by A18, Th34; ::_thesis: verum end; hence PR2,n is_a_correct_step by A8, A9, FINSEQ_1:25; ::_thesis: verum end; then A19: PR2 is a_proof by Def8; PR2 . (len PR2) = PR2 . ((len PR) + (len PR1)) by FINSEQ_1:22; then PR2 . (len PR2) = PR2 . ((len PR) + 1) by FINSEQ_1:39; then PR2 . (len PR2) = PR1 . 1 by A6, FINSEQ_1:def_7; then PR2 . (len PR2) = [((Ant f) ^ <*(All (x,p))*>),9] by FINSEQ_1:40; then (PR2 . (len PR2)) `1 = (Ant f) ^ <*(All (x,p))*> by MCART_1:7; hence |- (Ant f) ^ <*(All (x,p))*> by A19, Def9; ::_thesis: verum end; theorem Th44: :: CALCUL_1:44 for Al being QC-alphabet for p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f & |- (Ant f) ^ <*('not' (Suc f))*> holds |- (Ant f) ^ <*p*> proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f & |- (Ant f) ^ <*('not' (Suc f))*> holds |- (Ant f) ^ <*p*> let p be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- f & |- (Ant f) ^ <*('not' (Suc f))*> holds |- (Ant f) ^ <*p*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- f & |- (Ant f) ^ <*('not' (Suc f))*> implies |- (Ant f) ^ <*p*> ) assume that A1: |- f and A2: |- (Ant f) ^ <*('not' (Suc f))*> ; ::_thesis: |- (Ant f) ^ <*p*> set f1 = ((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*>; ( Ant (((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*>) = (Ant f) ^ <*('not' p)*> & Suc f = Suc (((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*>) ) by Th5; then A3: |- ((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*> by A1, Th8, Th36; set f3 = ((Ant f) ^ <*('not' p)*>) ^ <*('not' (Suc f))*>; set f2 = (Ant f) ^ <*('not' (Suc f))*>; Suc ((Ant f) ^ <*('not' (Suc f))*>) = 'not' (Suc f) by Th5; then A4: Suc ((Ant f) ^ <*('not' (Suc f))*>) = Suc (((Ant f) ^ <*('not' p)*>) ^ <*('not' (Suc f))*>) by Th5; ( Ant (((Ant f) ^ <*('not' p)*>) ^ <*('not' (Suc f))*>) = (Ant f) ^ <*('not' p)*> & Ant ((Ant f) ^ <*('not' (Suc f))*>) = Ant f ) by Th5; then A5: |- ((Ant f) ^ <*('not' p)*>) ^ <*('not' (Suc f))*> by A2, A4, Th8, Th36; Suc (((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*>) = Suc f by Th5; then A6: 'not' (Suc (((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*>)) = Suc (((Ant f) ^ <*('not' p)*>) ^ <*('not' (Suc f))*>) by Th5; A7: 1 < len (((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*>) by Th9; A8: Ant (((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*>) = (Ant f) ^ <*('not' p)*> by Th5; then ( Ant (((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*>) = Ant (((Ant f) ^ <*('not' p)*>) ^ <*('not' (Suc f))*>) & Suc (Ant (((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*>)) = 'not' p ) by Th5; then |- (Ant (Ant (((Ant f) ^ <*('not' p)*>) ^ <*(Suc f)*>))) ^ <*p*> by A3, A5, A6, A7, Th38; hence |- (Ant f) ^ <*p*> by A8, Th5; ::_thesis: verum end; theorem Th45: :: CALCUL_1:45 for Al being QC-alphabet for p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st 1 <= len f & |- f & |- f ^ <*p*> holds |- (Ant f) ^ <*p*> proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st 1 <= len f & |- f & |- f ^ <*p*> holds |- (Ant f) ^ <*p*> let p be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st 1 <= len f & |- f & |- f ^ <*p*> holds |- (Ant f) ^ <*p*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( 1 <= len f & |- f & |- f ^ <*p*> implies |- (Ant f) ^ <*p*> ) assume that A1: 1 <= len f and A2: |- f and A3: |- f ^ <*p*> ; ::_thesis: |- (Ant f) ^ <*p*> set f2 = ((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*>; set f1 = ((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>; A4: Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*>) = (Ant f) ^ <*('not' (Suc f))*> by Th5; then A5: len (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*>)) in dom (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*>)) by Th10; ((Ant f) ^ <*('not' (Suc f))*>) . ((len (Ant f)) + 1) = 'not' (Suc f) by FINSEQ_1:42; then (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*>)) . ((len (Ant f)) + 1) = 'not' (Suc f) by Th5; then (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*>)) . ((len (Ant f)) + 1) = Suc (((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*>) by Th5; then (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*>)) . (len (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*>))) = Suc (((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*>) by A4, FINSEQ_2:16; then A6: |- ((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*> by A5, Lm2, Th33; set f4 = (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*p*>; ((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*> = (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*('not' (Suc f))*> by Th5; then A7: ((Ant f) ^ <*('not' (Suc f))*>) ^ <*('not' (Suc f))*> = (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*('not' (Suc (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)))*> by Th5; ( Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>) = (Ant f) ^ <*('not' (Suc f))*> & Suc (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>) = Suc f ) by Th5; then |- ((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*> by A2, Th8, Th36; then A8: |- (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*p*> by A6, A7, Th44; set f3 = f ^ <*p*>; 1 + 1 <= (len f) + 1 by A1, XREAL_1:6; then 1 + 1 <= (len f) + (len <*p*>) by FINSEQ_1:39; then 1 + 1 <= len (f ^ <*p*>) by FINSEQ_1:22; then A9: 1 < len (f ^ <*p*>) by NAT_1:13; A10: Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>) = (Ant f) ^ <*('not' (Suc f))*> by Th5; then Suc (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) = 'not' (Suc f) by Th5; then Suc (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) = 'not' (Suc (Ant (f ^ <*p*>))) by Th5; then A11: Suc (Ant ((Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*p*>)) = 'not' (Suc (Ant (f ^ <*p*>))) by Th5; 1 <= len (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) by A10, Th10; then 1 + 1 <= (len (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>))) + 1 by XREAL_1:6; then 1 + 1 <= (len (Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>))) + (len <*p*>) by FINSEQ_1:39; then 1 + 1 <= len ((Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*p*>) by FINSEQ_1:22; then A12: 1 < len ((Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*p*>) by NAT_1:13; Ant ((Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*p*>) = Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>) by Th5; then Ant (Ant ((Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*p*>)) = Ant f by A10, Th5; then A13: Ant (Ant ((Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*p*>)) = Ant (Ant (f ^ <*p*>)) by Th5; Suc ((Ant (((Ant f) ^ <*('not' (Suc f))*>) ^ <*(Suc f)*>)) ^ <*p*>) = p by Th5; then |- (Ant (Ant (f ^ <*p*>))) ^ <*(Suc (f ^ <*p*>))*> by A3, A8, A9, A12, A13, A11, Th5, Th37; then |- (Ant f) ^ <*(Suc (f ^ <*p*>))*> by Th5; hence |- (Ant f) ^ <*p*> by Th5; ::_thesis: verum end; theorem Th46: :: CALCUL_1:46 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*q*> holds |- (f ^ <*('not' q)*>) ^ <*('not' p)*> proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*q*> holds |- (f ^ <*('not' q)*>) ^ <*('not' p)*> let p, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*q*> holds |- (f ^ <*('not' q)*>) ^ <*('not' p)*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- (f ^ <*p*>) ^ <*q*> implies |- (f ^ <*('not' q)*>) ^ <*('not' p)*> ) set f1 = (f ^ <*p*>) ^ <*q*>; set f2 = (((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>; assume A1: |- (f ^ <*p*>) ^ <*q*> ; ::_thesis: |- (f ^ <*('not' q)*>) ^ <*('not' p)*> A2: Ant ((f ^ <*p*>) ^ <*q*>) = f ^ <*p*> by Th5; then A3: Ant (Ant ((f ^ <*p*>) ^ <*q*>)) = f by Th5; set f4 = (((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>; set f3 = (((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*('not' q)*>; A4: 1 < len ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>) by Th9; A5: Suc ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>) = 'not' p by Th5; then A6: Suc ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>) = Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>) by Th5; Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*('not' q)*>) = ((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*> by Th5; then Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*('not' q)*>) = (Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ (<*('not' q)*> ^ <*p*>) by FINSEQ_1:32; then A7: Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*('not' q)*>) = (Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q),p*> by FINSEQ_1:def_9; then (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*('not' q)*>)) . ((len f) + 1) = 'not' q by A3, Th14; then A8: (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*('not' q)*>)) . ((len f) + 1) = Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*('not' q)*>) by Th5; Suc ((f ^ <*p*>) ^ <*q*>) = q by Th5; then A9: Suc ((f ^ <*p*>) ^ <*q*>) = Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>) by Th5; A10: Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>) = ((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*> by Th5; then A11: 1 < len ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>) by Th9; ( Suc (Ant ((f ^ <*p*>) ^ <*q*>)) = p & 0 + 1 <= len (Ant ((f ^ <*p*>) ^ <*q*>)) ) by A2, Th5, Th10; then A12: |- (((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*> by A1, A10, A9, Th13, Th36; 1 in dom <*('not' q),p*> by Th14; then A13: (len f) + 1 in dom (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*('not' q)*>)) by A3, A7, FINSEQ_1:28; ( Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>) = q & Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>) = ((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*> ) by Th5; then |- (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' (Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)))*> by A8, A13, Lm2, Th33; then A14: |- (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*> by A12, Th44; A15: Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = ((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*> by Th5; then Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = (Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ (<*('not' q)*> ^ <*('not' p)*>) by FINSEQ_1:32; then A16: Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = (Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q),('not' p)*> by FINSEQ_1:def_9; then (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) . ((len f) + 2) = 'not' p by A3, Th14; then A17: (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) . ((len f) + 2) = Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>) by Th5; 2 in dom <*('not' q),('not' p)*> by Th14; then (len f) + 2 in dom (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) by A3, A16, FINSEQ_1:28; then A18: |- (((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*> by A17, Lm2, Th33; A19: Ant ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>) = ((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*> by A10, Th5; then Suc (Ant ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>)) = p by Th5; then A20: 'not' (Suc (Ant ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>))) = Suc (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) by A15, Th5; A21: Ant (Ant ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>)) = (Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*> by A19, Th5; then Ant (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) = Ant (Ant ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>)) by A15, Th5; hence |- (f ^ <*('not' q)*>) ^ <*('not' p)*> by A3, A14, A18, A11, A4, A21, A20, A5, A6, Th37; ::_thesis: verum end; theorem :: CALCUL_1:47 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*('not' p)*>) ^ <*('not' q)*> holds |- (f ^ <*q*>) ^ <*p*> proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*('not' p)*>) ^ <*('not' q)*> holds |- (f ^ <*q*>) ^ <*p*> let p, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- (f ^ <*('not' p)*>) ^ <*('not' q)*> holds |- (f ^ <*q*>) ^ <*p*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- (f ^ <*('not' p)*>) ^ <*('not' q)*> implies |- (f ^ <*q*>) ^ <*p*> ) set f1 = (f ^ <*('not' p)*>) ^ <*('not' q)*>; set f2 = (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' q)*>; assume A1: |- (f ^ <*('not' p)*>) ^ <*('not' q)*> ; ::_thesis: |- (f ^ <*q*>) ^ <*p*> A2: Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' q)*>) = ((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*> by Th5; Suc ((f ^ <*('not' p)*>) ^ <*('not' q)*>) = 'not' q by Th5; then A3: Suc ((f ^ <*('not' p)*>) ^ <*('not' q)*>) = Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' q)*>) by Th5; A4: Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>) = f ^ <*('not' p)*> by Th5; then A5: Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>)) = f by Th5; ( Suc (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>)) = 'not' p & 0 + 1 <= len (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>)) ) by A4, Th5, Th10; then A6: |- (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' q)*> by A1, A2, A3, Th13, Th36; set f4 = (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>; set f3 = (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>; A7: 1 < len ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>) by Th9; A8: Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>) = ((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*> by Th5; then A9: 1 < len ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>) by Th9; Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>) = (Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ (<*q*> ^ <*('not' p)*>) by A8, FINSEQ_1:32; then A10: Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>) = (Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q,('not' p)*> by FINSEQ_1:def_9; then (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) . ((len f) + 1) = q by A5, Th14; then A11: (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) . ((len f) + 1) = Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>) by Th5; 1 in dom <*q,('not' p)*> by Th14; then (len f) + 1 in dom (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) by A5, A10, FINSEQ_1:28; then A12: |- (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*> by A11, Lm2, Th33; A13: Suc ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>) = p by Th5; then A14: Suc ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>) = Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>) by Th5; Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' q)*>) = ((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*> by Th5; then A15: Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' q)*>) = Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>) by Th5; Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' q)*>) = 'not' q by Th5; then A16: Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' q)*>) = 'not' (Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) by Th5; A17: Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>) = ((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*> by Th5; then Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>) = (Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ (<*q*> ^ <*p*>) by FINSEQ_1:32; then A18: Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>) = (Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q,p*> by FINSEQ_1:def_9; then (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>)) . ((len f) + 2) = p by A5, Th14; then A19: (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>)) . ((len f) + 2) = Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>) by Th5; 2 in dom <*q,p*> by Th14; then (len f) + 2 in dom (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>)) by A5, A18, FINSEQ_1:28; then A20: |- (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*> by A19, Lm2, Th33; A21: Ant ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>) = ((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*> by A8, Th5; then Suc (Ant ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>)) = 'not' p by Th5; then A22: Suc (Ant ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>)) = 'not' (Suc (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>))) by A17, Th5; 0 + 1 <= len ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' q)*>) by Th10; then |- (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*('not' (Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)))*> by A6, A16, A15, Th3; then A23: |- (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*> by A12, Th44; A24: Ant (Ant ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>)) = (Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*> by A21, Th5; then Ant (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*p*>)) = Ant (Ant ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>)) by A17, Th5; hence |- (f ^ <*q*>) ^ <*p*> by A5, A23, A20, A9, A7, A24, A22, A13, A14, Th37; ::_thesis: verum end; theorem Th48: :: CALCUL_1:48 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*('not' p)*>) ^ <*q*> holds |- (f ^ <*('not' q)*>) ^ <*p*> proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*('not' p)*>) ^ <*q*> holds |- (f ^ <*('not' q)*>) ^ <*p*> let p, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- (f ^ <*('not' p)*>) ^ <*q*> holds |- (f ^ <*('not' q)*>) ^ <*p*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- (f ^ <*('not' p)*>) ^ <*q*> implies |- (f ^ <*('not' q)*>) ^ <*p*> ) set f1 = (f ^ <*('not' p)*>) ^ <*q*>; set f2 = (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>; assume A1: |- (f ^ <*('not' p)*>) ^ <*q*> ; ::_thesis: |- (f ^ <*('not' q)*>) ^ <*p*> A2: Ant ((f ^ <*('not' p)*>) ^ <*q*>) = f ^ <*('not' p)*> by Th5; then A3: Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>)) = f by Th5; set f4 = (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>; set f3 = (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' q)*>; A4: 1 < len ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>) by Th9; Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' q)*>) = ((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*> by Th5; then Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' q)*>) = (Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ (<*('not' q)*> ^ <*('not' p)*>) by FINSEQ_1:32; then A5: Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' q)*>) = (Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q),('not' p)*> by FINSEQ_1:def_9; then (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' q)*>)) . ((len f) + 1) = 'not' q by A3, Th14; then A6: (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' q)*>)) . ((len f) + 1) = Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' q)*>) by Th5; 1 in dom <*('not' q),('not' p)*> by Th14; then A7: (len f) + 1 in dom (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*('not' q)*>)) by A3, A5, FINSEQ_1:28; Suc ((f ^ <*('not' p)*>) ^ <*q*>) = q by Th5; then A8: Suc ((f ^ <*('not' p)*>) ^ <*q*>) = Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>) by Th5; A9: Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>) = ((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*> by Th5; then A10: 1 < len ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>) by Th9; ( Suc (Ant ((f ^ <*('not' p)*>) ^ <*q*>)) = 'not' p & 0 + 1 <= len (Ant ((f ^ <*('not' p)*>) ^ <*q*>)) ) by A2, Th5, Th10; then A11: |- (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*> by A1, A9, A8, Th13, Th36; ( Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>) = q & Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>) = ((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*> ) by Th5; then |- (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*('not' (Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>)))*> by A6, A7, Lm2, Th33; then A12: |- (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*> by A11, Th44; A13: Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>) = ((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*> by Th5; then Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>) = (Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ (<*('not' q)*> ^ <*p*>) by FINSEQ_1:32; then A14: Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>) = (Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q),p*> by FINSEQ_1:def_9; then (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>)) . ((len f) + 2) = p by A3, Th14; then A15: (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>)) . ((len f) + 2) = Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>) by Th5; 2 in dom <*('not' q),p*> by Th14; then (len f) + 2 in dom (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>)) by A3, A14, FINSEQ_1:28; then A16: |- (((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*> by A15, Lm2, Th33; A17: Ant ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>) = ((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*> by A9, Th5; then Ant (Ant ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>)) = (Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*> by Th5; then A18: Ant (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>)) = Ant (Ant ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>)) by A13, Th5; Suc (Ant ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>)) = 'not' p by A17, Th5; then A19: Suc (Ant ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>)) = 'not' (Suc (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>))) by A13, Th5; A20: Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>) = p by Th5; then Suc ((Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*('not' p)*>) ^ <*q*>)) ^ <*p*>) = Suc ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>) by Th5; then |- (Ant (Ant ((((Ant (Ant ((f ^ <*('not' p)*>) ^ <*q*>))) ^ <*('not' q)*>) ^ <*p*>) ^ <*p*>))) ^ <*p*> by A12, A16, A10, A4, A18, A19, A20, Th37; hence |- (f ^ <*('not' q)*>) ^ <*p*> by A3, A13, Th5; ::_thesis: verum end; theorem Th49: :: CALCUL_1:49 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*('not' q)*> holds |- (f ^ <*q*>) ^ <*('not' p)*> proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*('not' q)*> holds |- (f ^ <*q*>) ^ <*('not' p)*> let p, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*('not' q)*> holds |- (f ^ <*q*>) ^ <*('not' p)*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- (f ^ <*p*>) ^ <*('not' q)*> implies |- (f ^ <*q*>) ^ <*('not' p)*> ) set f1 = (f ^ <*p*>) ^ <*('not' q)*>; set f2 = (((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*('not' q)*>; assume A1: |- (f ^ <*p*>) ^ <*('not' q)*> ; ::_thesis: |- (f ^ <*q*>) ^ <*('not' p)*> A2: Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*('not' q)*>) = ((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*> by Th5; Suc ((f ^ <*p*>) ^ <*('not' q)*>) = 'not' q by Th5; then A3: Suc ((f ^ <*p*>) ^ <*('not' q)*>) = Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*('not' q)*>) by Th5; A4: Ant ((f ^ <*p*>) ^ <*('not' q)*>) = f ^ <*p*> by Th5; then A5: Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>)) = f by Th5; ( Suc (Ant ((f ^ <*p*>) ^ <*('not' q)*>)) = p & 0 + 1 <= len (Ant ((f ^ <*p*>) ^ <*('not' q)*>)) ) by A4, Th5, Th10; then A6: |- (((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*('not' q)*> by A1, A2, A3, Th13, Th36; set f4 = (((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>; set f3 = (((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>; A7: 1 < len ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>) by Th9; A8: Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>) = ((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*> by Th5; then A9: 1 < len ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>) by Th9; Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>) = (Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ (<*q*> ^ <*p*>) by A8, FINSEQ_1:32; then A10: Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>) = (Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q,p*> by FINSEQ_1:def_9; then (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) . ((len f) + 1) = q by A5, Th14; then A11: (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) . ((len f) + 1) = Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>) by Th5; 1 in dom <*q,p*> by Th14; then (len f) + 1 in dom (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) by A5, A10, FINSEQ_1:28; then A12: |- (((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*> by A11, Lm2, Th33; A13: Suc ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>) = 'not' p by Th5; then A14: Suc ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>) = Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>) by Th5; Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*('not' q)*>) = ((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*> by Th5; then A15: Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*('not' q)*>) = Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>) by Th5; Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*('not' q)*>) = 'not' q by Th5; then A16: Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*('not' q)*>) = 'not' (Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) by Th5; A17: Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = ((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*> by Th5; then Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = (Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ (<*q*> ^ <*('not' p)*>) by FINSEQ_1:32; then A18: Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = (Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q,('not' p)*> by FINSEQ_1:def_9; then (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) . ((len f) + 2) = 'not' p by A5, Th14; then A19: (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) . ((len f) + 2) = Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>) by Th5; 2 in dom <*q,('not' p)*> by Th14; then (len f) + 2 in dom (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) by A5, A18, FINSEQ_1:28; then A20: |- (((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*> by A19, Lm2, Th33; A21: Ant ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>) = ((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*> by A8, Th5; then Suc (Ant ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>)) = p by Th5; then A22: 'not' (Suc (Ant ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>))) = Suc (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) by A17, Th5; 0 + 1 <= len ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*('not' q)*>) by Th10; then |- (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' (Suc ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)))*> by A6, A16, A15, Th3; then A23: |- (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*> by A12, Th44; A24: Ant (Ant ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>)) = (Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*> by A21, Th5; then Ant (Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) = Ant (Ant ((Ant ((((Ant (Ant ((f ^ <*p*>) ^ <*('not' q)*>))) ^ <*q*>) ^ <*p*>) ^ <*q*>)) ^ <*('not' p)*>)) by A17, Th5; hence |- (f ^ <*q*>) ^ <*('not' p)*> by A5, A23, A20, A9, A7, A24, A22, A13, A14, Th37; ::_thesis: verum end; theorem :: CALCUL_1:50 for Al being QC-alphabet for p, r, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> holds |- (f ^ <*(p 'or' q)*>) ^ <*r*> proof let Al be QC-alphabet ; ::_thesis: for p, r, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> holds |- (f ^ <*(p 'or' q)*>) ^ <*r*> let p, r, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> holds |- (f ^ <*(p 'or' q)*>) ^ <*r*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> implies |- (f ^ <*(p 'or' q)*>) ^ <*r*> ) set f1 = (f ^ <*('not' r)*>) ^ <*('not' p)*>; set f2 = (f ^ <*('not' r)*>) ^ <*('not' q)*>; A1: ( Suc ((f ^ <*('not' r)*>) ^ <*('not' p)*>) = 'not' p & Suc ((f ^ <*('not' r)*>) ^ <*('not' q)*>) = 'not' q ) by Th5; assume ( |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> ) ; ::_thesis: |- (f ^ <*(p 'or' q)*>) ^ <*r*> then A2: ( |- (f ^ <*('not' r)*>) ^ <*('not' p)*> & |- (f ^ <*('not' r)*>) ^ <*('not' q)*> ) by Th46; A3: Ant ((f ^ <*('not' r)*>) ^ <*('not' p)*>) = f ^ <*('not' r)*> by Th5; then Ant ((f ^ <*('not' r)*>) ^ <*('not' p)*>) = Ant ((f ^ <*('not' r)*>) ^ <*('not' q)*>) by Th5; then |- (f ^ <*('not' r)*>) ^ <*(('not' p) '&' ('not' q))*> by A2, A1, A3, Th39; then |- (f ^ <*('not' (('not' p) '&' ('not' q)))*>) ^ <*r*> by Th48; hence |- (f ^ <*(p 'or' q)*>) ^ <*r*> by QC_LANG2:def_3; ::_thesis: verum end; theorem :: CALCUL_1:51 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f ^ <*p*> holds |- f ^ <*(p 'or' q)*> proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f ^ <*p*> holds |- f ^ <*(p 'or' q)*> let p, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- f ^ <*p*> holds |- f ^ <*(p 'or' q)*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- f ^ <*p*> implies |- f ^ <*(p 'or' q)*> ) set f1 = (f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>; assume A1: |- f ^ <*p*> ; ::_thesis: |- f ^ <*(p 'or' q)*> A2: Ant ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>) = f ^ <*(('not' p) '&' ('not' q))*> by Th5; (len f) + 1 = (len f) + (len <*(('not' p) '&' ('not' q))*>) by FINSEQ_1:39; then (len f) + 1 = len (Ant ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>)) by A2, FINSEQ_1:22; then A3: (len f) + 1 in dom (Ant ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>)) by A2, Th10; A4: Suc ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>) = ('not' p) '&' ('not' q) by Th5; (Ant ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>)) . ((len f) + 1) = ('not' p) '&' ('not' q) by A2, FINSEQ_1:42; then Suc ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>) is_tail_of Ant ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>) by A4, A3, Lm2; then |- (f ^ <*(('not' p) '&' ('not' q))*>) ^ <*('not' p)*> by A2, A4, Th33, Th40; then |- (f ^ <*p*>) ^ <*('not' (('not' p) '&' ('not' q)))*> by Th49; then A5: |- (f ^ <*p*>) ^ <*(p 'or' q)*> by QC_LANG2:def_3; 1 <= len (f ^ <*p*>) by Th10; then |- (Ant (f ^ <*p*>)) ^ <*(p 'or' q)*> by A1, A5, Th45; hence |- f ^ <*(p 'or' q)*> by Th5; ::_thesis: verum end; theorem :: CALCUL_1:52 for Al being QC-alphabet for q, p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f ^ <*q*> holds |- f ^ <*(p 'or' q)*> proof let Al be QC-alphabet ; ::_thesis: for q, p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f ^ <*q*> holds |- f ^ <*(p 'or' q)*> let q, p be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- f ^ <*q*> holds |- f ^ <*(p 'or' q)*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- f ^ <*q*> implies |- f ^ <*(p 'or' q)*> ) set f1 = (f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>; assume A1: |- f ^ <*q*> ; ::_thesis: |- f ^ <*(p 'or' q)*> A2: Ant ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>) = f ^ <*(('not' p) '&' ('not' q))*> by Th5; (len f) + 1 = (len f) + (len <*(('not' p) '&' ('not' q))*>) by FINSEQ_1:39; then (len f) + 1 = len (Ant ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>)) by A2, FINSEQ_1:22; then A3: (len f) + 1 in dom (Ant ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>)) by A2, Th10; A4: Suc ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>) = ('not' p) '&' ('not' q) by Th5; (Ant ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>)) . ((len f) + 1) = ('not' p) '&' ('not' q) by A2, FINSEQ_1:42; then Suc ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>) is_tail_of Ant ((f ^ <*(('not' p) '&' ('not' q))*>) ^ <*(('not' p) '&' ('not' q))*>) by A4, A3, Lm2; then |- (f ^ <*(('not' p) '&' ('not' q))*>) ^ <*('not' q)*> by A2, A4, Th33, Th41; then |- (f ^ <*q*>) ^ <*('not' (('not' p) '&' ('not' q)))*> by Th49; then A5: |- (f ^ <*q*>) ^ <*(p 'or' q)*> by QC_LANG2:def_3; 1 <= len (f ^ <*q*>) by Th10; then |- (Ant (f ^ <*q*>)) ^ <*(p 'or' q)*> by A1, A5, Th45; hence |- f ^ <*(p 'or' q)*> by Th5; ::_thesis: verum end; theorem Th53: :: CALCUL_1:53 for Al being QC-alphabet for p, r, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> holds |- (f ^ <*(p 'or' q)*>) ^ <*r*> proof let Al be QC-alphabet ; ::_thesis: for p, r, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> holds |- (f ^ <*(p 'or' q)*>) ^ <*r*> let p, r, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> holds |- (f ^ <*(p 'or' q)*>) ^ <*r*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> implies |- (f ^ <*(p 'or' q)*>) ^ <*r*> ) set f1 = (f ^ <*('not' r)*>) ^ <*('not' p)*>; set f2 = (f ^ <*('not' r)*>) ^ <*('not' q)*>; A1: Ant ((f ^ <*('not' r)*>) ^ <*('not' p)*>) = f ^ <*('not' r)*> by Th5; A2: Suc ((f ^ <*('not' r)*>) ^ <*('not' q)*>) = 'not' q by Th5; assume ( |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> ) ; ::_thesis: |- (f ^ <*(p 'or' q)*>) ^ <*r*> then A3: ( |- (f ^ <*('not' r)*>) ^ <*('not' p)*> & |- (f ^ <*('not' r)*>) ^ <*('not' q)*> ) by Th46; ( Suc ((f ^ <*('not' r)*>) ^ <*('not' p)*>) = 'not' p & Ant ((f ^ <*('not' r)*>) ^ <*('not' q)*>) = f ^ <*('not' r)*> ) by Th5; then |- (Ant ((f ^ <*('not' r)*>) ^ <*('not' p)*>)) ^ <*(('not' p) '&' ('not' q))*> by A3, A2, Th5, Th39; then |- (f ^ <*('not' (('not' p) '&' ('not' q)))*>) ^ <*r*> by A1, Th48; hence |- (f ^ <*(p 'or' q)*>) ^ <*r*> by QC_LANG2:def_3; ::_thesis: verum end; theorem Th54: :: CALCUL_1:54 for Al being QC-alphabet for p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f ^ <*p*> holds |- f ^ <*('not' ('not' p))*> proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f ^ <*p*> holds |- f ^ <*('not' ('not' p))*> let p be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- f ^ <*p*> holds |- f ^ <*('not' ('not' p))*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- f ^ <*p*> implies |- f ^ <*('not' ('not' p))*> ) assume A1: |- f ^ <*p*> ; ::_thesis: |- f ^ <*('not' ('not' p))*> set f5 = f ^ <*p*>; set f4 = ((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>; set f2 = ((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>; set f1 = ((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>; set f3 = (Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) ^ <*('not' ('not' p))*>; A2: Suc (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>) = p by Th5; A3: Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>) = (f ^ <*p*>) ^ <*('not' p)*> by Th5; then A4: 1 < len ((Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) ^ <*('not' ('not' p))*>) by Th9; A5: Ant (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>) = (f ^ <*p*>) ^ <*('not' ('not' p))*> by Th5; then Ant (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>) = f ^ (<*p*> ^ <*('not' ('not' p))*>) by FINSEQ_1:32; then A6: Ant (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>) = f ^ <*p,('not' ('not' p))*> by FINSEQ_1:def_9; A7: Ant ((Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) ^ <*('not' ('not' p))*>) = Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>) by Th5; then Suc (Ant ((Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) ^ <*('not' ('not' p))*>)) = 'not' p by A3, Th5; then A8: 'not' (Suc (Ant ((Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) ^ <*('not' ('not' p))*>))) = Suc (Ant (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>)) by A5, Th5; Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>) = f ^ (<*p*> ^ <*('not' p)*>) by A3, FINSEQ_1:32; then A9: Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>) = f ^ <*p,('not' p)*> by FINSEQ_1:def_9; (len f) + 2 = (len f) + (len <*p,('not' p)*>) by FINSEQ_1:44; then (len f) + 2 = len (Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) by A9, FINSEQ_1:22; then ( 1 <= (len f) + 1 & (len f) + 1 <= len (Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) ) by NAT_1:11, XREAL_1:6; then A10: (len f) + 1 in dom (Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) by FINSEQ_3:25; (Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) . ((len f) + 1) = p by A9, Th14; then A11: |- ((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*> by A2, A10, Lm2, Th33; A12: 1 < len (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>) by Th9; 0 + 1 <= len (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>) by Th10; then A13: ((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*> = (Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) ^ <*(Suc (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>))*> by Th3; A14: 1 <= len (f ^ <*p*>) by Th10; (len f) + 2 = (len f) + (len <*p,('not' ('not' p))*>) by FINSEQ_1:44; then (len f) + 2 = len (Ant (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>)) by A6, FINSEQ_1:22; then A15: (len f) + 2 in dom (Ant (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>)) by A5, Th10; A16: Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = (f ^ <*p*>) ^ <*('not' p)*> by Th5; then Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = f ^ (<*p*> ^ <*('not' p)*>) by FINSEQ_1:32; then A17: Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = f ^ <*p,('not' p)*> by FINSEQ_1:def_9; (len f) + 2 = (len f) + (len <*p,('not' p)*>) by FINSEQ_1:44; then (len f) + 2 = len (Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) by A17, FINSEQ_1:22; then A18: (len f) + 2 in dom (Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) by A16, Th10; A19: Suc (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = 'not' p by Th5; then A20: 'not' (Suc (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) = Suc (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>) by Th5; (Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) . ((len f) + 2) = 'not' p by A17, Th14; then A21: |- ((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*> by A18, A19, Lm2, Th33; A22: Suc (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>) = 'not' ('not' p) by Th5; then A23: Suc ((Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) ^ <*('not' ('not' p))*>) = Suc (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>) by Th5; (Ant (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>)) . ((len f) + 2) = 'not' ('not' p) by A6, Th14; then A24: |- ((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*> by A15, A22, Lm2, Th33; Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>) = Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*('not' p)*>) by A16, Th5; then A25: |- (Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) ^ <*('not' ('not' p))*> by A11, A21, A13, A20, Th44; A26: Ant (Ant ((Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) ^ <*('not' ('not' p))*>)) = f ^ <*p*> by A3, A7, Th5; then Ant (Ant ((Ant (((f ^ <*p*>) ^ <*('not' p)*>) ^ <*p*>)) ^ <*('not' ('not' p))*>)) = Ant (Ant (((f ^ <*p*>) ^ <*('not' ('not' p))*>) ^ <*('not' ('not' p))*>)) by A5, Th5; then |- (f ^ <*p*>) ^ <*('not' ('not' p))*> by A25, A22, A24, A23, A26, A8, A4, A12, Th37; then |- (Ant (f ^ <*p*>)) ^ <*('not' ('not' p))*> by A1, A14, Th45; hence |- f ^ <*('not' ('not' p))*> by Th5; ::_thesis: verum end; theorem Th55: :: CALCUL_1:55 for Al being QC-alphabet for p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f ^ <*('not' ('not' p))*> holds |- f ^ <*p*> proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f ^ <*('not' ('not' p))*> holds |- f ^ <*p*> let p be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- f ^ <*('not' ('not' p))*> holds |- f ^ <*p*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- f ^ <*('not' ('not' p))*> implies |- f ^ <*p*> ) assume A1: |- f ^ <*('not' ('not' p))*> ; ::_thesis: |- f ^ <*p*> set f5 = f ^ <*('not' ('not' p))*>; set f4 = ((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>; set f2 = ((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>; set f1 = ((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>; set f3 = (Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) ^ <*p*>; A2: Suc (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = 'not' p by Th5; A3: Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = (f ^ <*('not' ('not' p))*>) ^ <*('not' p)*> by Th5; then A4: 1 < len ((Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) ^ <*p*>) by Th9; Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = f ^ (<*('not' ('not' p))*> ^ <*('not' p)*>) by A3, FINSEQ_1:32; then A5: Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = f ^ <*('not' ('not' p)),('not' p)*> by FINSEQ_1:def_9; (len f) + 2 = (len f) + (len <*('not' ('not' p)),('not' p)*>) by FINSEQ_1:44; then (len f) + 2 = len (Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) by A5, FINSEQ_1:22; then A6: (len f) + 2 in dom (Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) by A3, Th10; (Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) . ((len f) + 2) = 'not' p by A5, Th14; then A7: |- ((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*> by A2, A6, Lm2, Th33; A8: 1 < len (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>) by Th9; A9: Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>) = (f ^ <*('not' ('not' p))*>) ^ <*('not' p)*> by Th5; then Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>) = f ^ (<*('not' ('not' p))*> ^ <*('not' p)*>) by FINSEQ_1:32; then A10: Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>) = f ^ <*('not' ('not' p)),('not' p)*> by FINSEQ_1:def_9; (len f) + 2 = (len f) + (len <*('not' ('not' p)),('not' p)*>) by FINSEQ_1:44; then (len f) + 2 = len (Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>)) by A10, FINSEQ_1:22; then ( 1 <= (len f) + 1 & (len f) + 1 <= len (Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>)) ) by NAT_1:11, XREAL_1:6; then A11: (len f) + 1 in dom (Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>)) by FINSEQ_3:25; 0 + 1 <= len (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>) by Th10; then A12: ((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*> = (Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>)) ^ <*(Suc (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>))*> by Th3; A13: 1 <= len (f ^ <*('not' ('not' p))*>) by Th10; A14: Ant (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>) = (f ^ <*('not' ('not' p))*>) ^ <*p*> by Th5; then Ant (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>) = f ^ (<*('not' ('not' p))*> ^ <*p*>) by FINSEQ_1:32; then A15: Ant (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>) = f ^ <*('not' ('not' p)),p*> by FINSEQ_1:def_9; A16: Ant ((Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) ^ <*p*>) = Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>) by Th5; then Suc (Ant ((Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) ^ <*p*>)) = 'not' p by A3, Th5; then A17: Suc (Ant ((Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) ^ <*p*>)) = 'not' (Suc (Ant (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>))) by A14, Th5; (len f) + 2 = (len f) + (len <*('not' ('not' p)),p*>) by FINSEQ_1:44; then (len f) + 2 = len (Ant (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>)) by A15, FINSEQ_1:22; then A18: (len f) + 2 in dom (Ant (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>)) by A14, Th10; A19: Suc (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>) = 'not' ('not' p) by Th5; then A20: 'not' (Suc (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) = Suc (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>) by Th5; (Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>)) . ((len f) + 1) = 'not' ('not' p) by A10, Th14; then A21: |- ((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*> by A11, A19, Lm2, Th33; A22: Suc (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>) = p by Th5; then A23: Suc ((Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) ^ <*p*>) = Suc (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>) by Th5; (Ant (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>)) . ((len f) + 2) = p by A15, Th14; then A24: |- ((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*> by A18, A22, Lm2, Th33; Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>) = Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' ('not' p))*>) by A9, Th5; then A25: |- (Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) ^ <*p*> by A7, A21, A12, A20, Th44; A26: Ant (Ant ((Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) ^ <*p*>)) = f ^ <*('not' ('not' p))*> by A3, A16, Th5; then Ant (Ant ((Ant (((f ^ <*('not' ('not' p))*>) ^ <*('not' p)*>) ^ <*('not' p)*>)) ^ <*p*>)) = Ant (Ant (((f ^ <*('not' ('not' p))*>) ^ <*p*>) ^ <*p*>)) by A14, Th5; then |- (f ^ <*('not' ('not' p))*>) ^ <*p*> by A25, A22, A24, A23, A26, A17, A4, A8, Th37; then |- (Ant (f ^ <*('not' ('not' p))*>)) ^ <*p*> by A1, A13, Th45; hence |- f ^ <*p*> by Th5; ::_thesis: verum end; theorem :: CALCUL_1:56 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f ^ <*(p => q)*> & |- f ^ <*p*> holds |- f ^ <*q*> proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for f being FinSequence of CQC-WFF Al st |- f ^ <*(p => q)*> & |- f ^ <*p*> holds |- f ^ <*q*> let p, q be Element of CQC-WFF Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- f ^ <*(p => q)*> & |- f ^ <*p*> holds |- f ^ <*q*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- f ^ <*(p => q)*> & |- f ^ <*p*> implies |- f ^ <*q*> ) assume that A1: |- f ^ <*(p => q)*> and A2: |- f ^ <*p*> ; ::_thesis: |- f ^ <*q*> set f3 = (f ^ <*q*>) ^ <*q*>; set f2 = (f ^ <*('not' p)*>) ^ <*('not' p)*>; set f1 = (f ^ <*('not' p)*>) ^ <*p*>; A3: Ant ((f ^ <*('not' p)*>) ^ <*p*>) = f ^ <*('not' p)*> by Th5; Suc (f ^ <*p*>) = p by Th5; then A4: Suc (f ^ <*p*>) = Suc ((f ^ <*('not' p)*>) ^ <*p*>) by Th5; A5: 0 + 1 <= len ((f ^ <*('not' p)*>) ^ <*('not' p)*>) by Th10; A6: Ant ((f ^ <*q*>) ^ <*q*>) = f ^ <*q*> by Th5; then A7: (Ant ((f ^ <*q*>) ^ <*q*>)) . ((len f) + 1) = q by FINSEQ_1:42; (len f) + 1 = (len f) + (len <*q*>) by FINSEQ_1:39; then (len f) + 1 = len (Ant ((f ^ <*q*>) ^ <*q*>)) by A6, FINSEQ_1:22; then A8: (len f) + 1 in dom (Ant ((f ^ <*q*>) ^ <*q*>)) by A6, Th10; Suc ((f ^ <*q*>) ^ <*q*>) = q by Th5; then A9: |- (f ^ <*q*>) ^ <*q*> by A7, A8, Lm2, Th33; A10: Ant ((f ^ <*('not' p)*>) ^ <*('not' p)*>) = f ^ <*('not' p)*> by Th5; (len f) + 1 = (len f) + (len <*('not' p)*>) by FINSEQ_1:39; then (len f) + 1 = len (Ant ((f ^ <*('not' p)*>) ^ <*('not' p)*>)) by A10, FINSEQ_1:22; then A11: (len f) + 1 in dom (Ant ((f ^ <*('not' p)*>) ^ <*('not' p)*>)) by A10, Th10; A12: Suc ((f ^ <*('not' p)*>) ^ <*('not' p)*>) = 'not' p by Th5; then A13: 'not' (Suc ((f ^ <*('not' p)*>) ^ <*p*>)) = Suc ((f ^ <*('not' p)*>) ^ <*('not' p)*>) by Th5; (Ant ((f ^ <*('not' p)*>) ^ <*('not' p)*>)) . ((len f) + 1) = 'not' p by A10, FINSEQ_1:42; then A14: |- (f ^ <*('not' p)*>) ^ <*('not' p)*> by A11, A12, Lm2, Th33; Ant ((f ^ <*('not' p)*>) ^ <*p*>) = Ant ((f ^ <*('not' p)*>) ^ <*('not' p)*>) by A10, Th5; then A15: |- (Ant ((f ^ <*('not' p)*>) ^ <*p*>)) ^ <*('not' (Suc ((f ^ <*('not' p)*>) ^ <*p*>)))*> by A14, A5, A13, Th3; Ant (f ^ <*p*>) = f by Th5; then |- (f ^ <*('not' p)*>) ^ <*p*> by A2, A3, A4, Th8, Th36; then |- (Ant ((f ^ <*('not' p)*>) ^ <*p*>)) ^ <*q*> by A15, Th44; then |- (f ^ <*(('not' p) 'or' q)*>) ^ <*q*> by A3, A9, Th53; then A16: |- (f ^ <*('not' (('not' ('not' p)) '&' ('not' q)))*>) ^ <*q*> by QC_LANG2:def_3; set f4 = (f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>; set f5 = (Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>)) ^ <*p*>; set f6 = (Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>)) ^ <*('not' q)*>; A17: ( Ant ((Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>)) ^ <*p*>) = Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>) & Suc ((Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>)) ^ <*p*>) = p ) by Th5; A18: ( Ant ((Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>)) ^ <*('not' q)*>) = Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>) & Suc ((Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>)) ^ <*('not' q)*>) = 'not' q ) by Th5; A19: Suc ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>) = ('not' ('not' p)) '&' ('not' q) by Th5; then |- (Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>)) ^ <*('not' ('not' p))*> by A16, Th40, Th48; then A20: |- (Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>)) ^ <*p*> by Th55; |- (Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>)) ^ <*('not' q)*> by A16, A19, Th41, Th48; then |- (Ant ((f ^ <*('not' q)*>) ^ <*(('not' ('not' p)) '&' ('not' q))*>)) ^ <*(p '&' ('not' q))*> by A20, A17, A18, Th39; then |- (f ^ <*('not' q)*>) ^ <*(p '&' ('not' q))*> by Th5; then |- (f ^ <*('not' (p '&' ('not' q)))*>) ^ <*q*> by Th48; then A21: |- (f ^ <*(p => q)*>) ^ <*q*> by QC_LANG2:def_2; 1 <= len (f ^ <*(p => q)*>) by Th10; then |- (Ant (f ^ <*(p => q)*>)) ^ <*q*> by A1, A21, Th45; hence |- f ^ <*q*> by Th5; ::_thesis: verum end; theorem Th57: :: CALCUL_1:57 for Al being QC-alphabet for p being Element of CQC-WFF Al for x, y being bound_QC-variable of Al holds ('not' p) . (x,y) = 'not' (p . (x,y)) proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for x, y being bound_QC-variable of Al holds ('not' p) . (x,y) = 'not' (p . (x,y)) let p be Element of CQC-WFF Al; ::_thesis: for x, y being bound_QC-variable of Al holds ('not' p) . (x,y) = 'not' (p . (x,y)) let x, y be bound_QC-variable of Al; ::_thesis: ('not' p) . (x,y) = 'not' (p . (x,y)) set S = [p,(Sbst (x,y))]; ( [p,(Sbst (x,y))] `1 = p & [p,(Sbst (x,y))] `2 = Sbst (x,y) ) ; then ( ('not' p) . (x,y) = CQC_Sub [('not' p),(Sbst (x,y))] & Sub_not [p,(Sbst (x,y))] = [('not' p),(Sbst (x,y))] ) by SUBSTUT1:def_20, SUBSTUT2:def_1; then ('not' p) . (x,y) = 'not' (CQC_Sub [p,(Sbst (x,y))]) by SUBSTUT1:29; hence ('not' p) . (x,y) = 'not' (p . (x,y)) by SUBSTUT2:def_1; ::_thesis: verum end; theorem :: CALCUL_1:58 for Al being QC-alphabet for p being Element of CQC-WFF Al for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st ex y being bound_QC-variable of Al st |- f ^ <*(p . (x,y))*> holds |- f ^ <*(Ex (x,p))*> proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st ex y being bound_QC-variable of Al st |- f ^ <*(p . (x,y))*> holds |- f ^ <*(Ex (x,p))*> let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st ex y being bound_QC-variable of Al st |- f ^ <*(p . (x,y))*> holds |- f ^ <*(Ex (x,p))*> let x be bound_QC-variable of Al; ::_thesis: for f being FinSequence of CQC-WFF Al st ex y being bound_QC-variable of Al st |- f ^ <*(p . (x,y))*> holds |- f ^ <*(Ex (x,p))*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( ex y being bound_QC-variable of Al st |- f ^ <*(p . (x,y))*> implies |- f ^ <*(Ex (x,p))*> ) given y being bound_QC-variable of Al such that A1: |- f ^ <*(p . (x,y))*> ; ::_thesis: |- f ^ <*(Ex (x,p))*> set f1 = (f ^ <*(All (x,('not' p)))*>) ^ <*(All (x,('not' p)))*>; A2: Ant ((f ^ <*(All (x,('not' p)))*>) ^ <*(All (x,('not' p)))*>) = f ^ <*(All (x,('not' p)))*> by Th5; (len f) + 1 = (len f) + (len <*(All (x,('not' p)))*>) by FINSEQ_1:39; then (len f) + 1 = len (Ant ((f ^ <*(All (x,('not' p)))*>) ^ <*(All (x,('not' p)))*>)) by A2, FINSEQ_1:22; then A3: (len f) + 1 in dom (Ant ((f ^ <*(All (x,('not' p)))*>) ^ <*(All (x,('not' p)))*>)) by A2, Th10; A4: Suc ((f ^ <*(All (x,('not' p)))*>) ^ <*(All (x,('not' p)))*>) = All (x,('not' p)) by Th5; (Ant ((f ^ <*(All (x,('not' p)))*>) ^ <*(All (x,('not' p)))*>)) . ((len f) + 1) = All (x,('not' p)) by A2, FINSEQ_1:42; then Suc ((f ^ <*(All (x,('not' p)))*>) ^ <*(All (x,('not' p)))*>) is_tail_of Ant ((f ^ <*(All (x,('not' p)))*>) ^ <*(All (x,('not' p)))*>) by A4, A3, Lm2; then |- (f ^ <*(All (x,('not' p)))*>) ^ <*(('not' p) . (x,y))*> by A2, A4, Th33, Th42; then |- (f ^ <*(All (x,('not' p)))*>) ^ <*('not' (p . (x,y)))*> by Th57; then |- (f ^ <*(p . (x,y))*>) ^ <*('not' (All (x,('not' p))))*> by Th49; then A5: |- (f ^ <*(p . (x,y))*>) ^ <*(Ex (x,p))*> by QC_LANG2:def_5; 1 <= len (f ^ <*(p . (x,y))*>) by Th10; then |- (Ant (f ^ <*(p . (x,y))*>)) ^ <*(Ex (x,p))*> by A1, A5, Th45; hence |- f ^ <*(Ex (x,p))*> by Th5; ::_thesis: verum end; theorem Th59: :: CALCUL_1:59 for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al holds still_not-bound_in (f ^ g) = (still_not-bound_in f) \/ (still_not-bound_in g) proof let Al be QC-alphabet ; ::_thesis: for f, g being FinSequence of CQC-WFF Al holds still_not-bound_in (f ^ g) = (still_not-bound_in f) \/ (still_not-bound_in g) let f, g be FinSequence of CQC-WFF Al; ::_thesis: still_not-bound_in (f ^ g) = (still_not-bound_in f) \/ (still_not-bound_in g) thus still_not-bound_in (f ^ g) c= (still_not-bound_in f) \/ (still_not-bound_in g) :: according to XBOOLE_0:def_10 ::_thesis: (still_not-bound_in f) \/ (still_not-bound_in g) c= still_not-bound_in (f ^ g) proof let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in still_not-bound_in (f ^ g) or b in (still_not-bound_in f) \/ (still_not-bound_in g) ) assume b in still_not-bound_in (f ^ g) ; ::_thesis: b in (still_not-bound_in f) \/ (still_not-bound_in g) then consider i being Element of NAT , p being Element of CQC-WFF Al such that A1: i in dom (f ^ g) and A2: ( p = (f ^ g) . i & b in still_not-bound_in p ) by Def5; A3: now__::_thesis:_(_ex_n_being_Nat_st_ (_n_in_dom_g_&_i_=_(len_f)_+_n_)_implies_b_in_(still_not-bound_in_f)_\/_(still_not-bound_in_g)_) given n being Nat such that A4: n in dom g and A5: i = (len f) + n ; ::_thesis: b in (still_not-bound_in f) \/ (still_not-bound_in g) (f ^ g) . i = g . n by A4, A5, FINSEQ_1:def_7; then A6: b in still_not-bound_in g by A2, A4, Def5; still_not-bound_in g c= (still_not-bound_in f) \/ (still_not-bound_in g) by XBOOLE_1:7; hence b in (still_not-bound_in f) \/ (still_not-bound_in g) by A6; ::_thesis: verum end; now__::_thesis:_(_i_in_dom_f_implies_b_in_(still_not-bound_in_f)_\/_(still_not-bound_in_g)_) assume A7: i in dom f ; ::_thesis: b in (still_not-bound_in f) \/ (still_not-bound_in g) then (f ^ g) . i = f . i by FINSEQ_1:def_7; then A8: b in still_not-bound_in f by A2, A7, Def5; still_not-bound_in f c= (still_not-bound_in f) \/ (still_not-bound_in g) by XBOOLE_1:7; hence b in (still_not-bound_in f) \/ (still_not-bound_in g) by A8; ::_thesis: verum end; hence b in (still_not-bound_in f) \/ (still_not-bound_in g) by A1, A3, FINSEQ_1:25; ::_thesis: verum end; thus (still_not-bound_in f) \/ (still_not-bound_in g) c= still_not-bound_in (f ^ g) ::_thesis: verum proof let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in (still_not-bound_in f) \/ (still_not-bound_in g) or b in still_not-bound_in (f ^ g) ) assume A9: b in (still_not-bound_in f) \/ (still_not-bound_in g) ; ::_thesis: b in still_not-bound_in (f ^ g) A10: now__::_thesis:_(_b_in_still_not-bound_in_g_implies_b_in_still_not-bound_in_(f_^_g)_) assume b in still_not-bound_in g ; ::_thesis: b in still_not-bound_in (f ^ g) then consider i being Element of NAT , p being Element of CQC-WFF Al such that A11: ( i in dom g & p = g . i ) and A12: b in still_not-bound_in p by Def5; ( (len f) + i in dom (f ^ g) & p = (f ^ g) . ((len f) + i) ) by A11, FINSEQ_1:28, FINSEQ_1:def_7; hence b in still_not-bound_in (f ^ g) by A12, Def5; ::_thesis: verum end; now__::_thesis:_(_b_in_still_not-bound_in_f_implies_b_in_still_not-bound_in_(f_^_g)_) assume b in still_not-bound_in f ; ::_thesis: b in still_not-bound_in (f ^ g) then consider i being Element of NAT , p being Element of CQC-WFF Al such that A13: i in dom f and A14: p = f . i and A15: b in still_not-bound_in p by Def5; ( dom f c= dom (f ^ g) & p = (f ^ g) . i ) by A13, A14, FINSEQ_1:26, FINSEQ_1:def_7; hence b in still_not-bound_in (f ^ g) by A13, A15, Def5; ::_thesis: verum end; hence b in still_not-bound_in (f ^ g) by A9, A10, XBOOLE_0:def_3; ::_thesis: verum end; end; theorem Th60: :: CALCUL_1:60 for Al being QC-alphabet for p being Element of CQC-WFF Al holds still_not-bound_in <*p*> = still_not-bound_in p proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al holds still_not-bound_in <*p*> = still_not-bound_in p let p be Element of CQC-WFF Al; ::_thesis: still_not-bound_in <*p*> = still_not-bound_in p A1: now__::_thesis:_for_b_being_set_st_b_in_still_not-bound_in_p_holds_ b_in_still_not-bound_in_<*p*> 1 in Seg 1 by FINSEQ_1:2, TARSKI:def_1; then A2: 1 in dom <*p*> by FINSEQ_1:38; A3: p = <*p*> . 1 by FINSEQ_1:40; let b be set ; ::_thesis: ( b in still_not-bound_in p implies b in still_not-bound_in <*p*> ) assume b in still_not-bound_in p ; ::_thesis: b in still_not-bound_in <*p*> hence b in still_not-bound_in <*p*> by A2, A3, Def5; ::_thesis: verum end; now__::_thesis:_for_b_being_set_st_b_in_still_not-bound_in_<*p*>_holds_ b_in_still_not-bound_in_p let b be set ; ::_thesis: ( b in still_not-bound_in <*p*> implies b in still_not-bound_in p ) assume b in still_not-bound_in <*p*> ; ::_thesis: b in still_not-bound_in p then consider i being Element of NAT , q being Element of CQC-WFF Al such that A4: i in dom <*p*> and A5: ( q = <*p*> . i & b in still_not-bound_in q ) by Def5; i in Seg 1 by A4, FINSEQ_1:38; then i = 1 by FINSEQ_1:2, TARSKI:def_1; hence b in still_not-bound_in p by A5, FINSEQ_1:40; ::_thesis: verum end; hence still_not-bound_in <*p*> = still_not-bound_in p by A1, TARSKI:1; ::_thesis: verum end; theorem :: CALCUL_1:61 for Al being QC-alphabet for p, q being Element of CQC-WFF Al for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*(p . (x,y))*>) ^ <*q*> & not y in still_not-bound_in ((f ^ <*(Ex (x,p))*>) ^ <*q*>) holds |- (f ^ <*(Ex (x,p))*>) ^ <*q*> proof let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*(p . (x,y))*>) ^ <*q*> & not y in still_not-bound_in ((f ^ <*(Ex (x,p))*>) ^ <*q*>) holds |- (f ^ <*(Ex (x,p))*>) ^ <*q*> let p, q be Element of CQC-WFF Al; ::_thesis: for x, y being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st |- (f ^ <*(p . (x,y))*>) ^ <*q*> & not y in still_not-bound_in ((f ^ <*(Ex (x,p))*>) ^ <*q*>) holds |- (f ^ <*(Ex (x,p))*>) ^ <*q*> let x, y be bound_QC-variable of Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- (f ^ <*(p . (x,y))*>) ^ <*q*> & not y in still_not-bound_in ((f ^ <*(Ex (x,p))*>) ^ <*q*>) holds |- (f ^ <*(Ex (x,p))*>) ^ <*q*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- (f ^ <*(p . (x,y))*>) ^ <*q*> & not y in still_not-bound_in ((f ^ <*(Ex (x,p))*>) ^ <*q*>) implies |- (f ^ <*(Ex (x,p))*>) ^ <*q*> ) assume that A1: |- (f ^ <*(p . (x,y))*>) ^ <*q*> and A2: not y in still_not-bound_in ((f ^ <*(Ex (x,p))*>) ^ <*q*>) ; ::_thesis: |- (f ^ <*(Ex (x,p))*>) ^ <*q*> set f1 = (f ^ <*('not' q)*>) ^ <*(('not' p) . (x,y))*>; |- (f ^ <*('not' q)*>) ^ <*('not' (p . (x,y)))*> by A1, Th46; then A3: |- (f ^ <*('not' q)*>) ^ <*(('not' p) . (x,y))*> by Th57; A4: not y in (still_not-bound_in (f ^ <*(Ex (x,p))*>)) \/ (still_not-bound_in <*q*>) by A2, Th59; then not y in still_not-bound_in (f ^ <*(Ex (x,p))*>) by XBOOLE_0:def_3; then A5: not y in (still_not-bound_in f) \/ (still_not-bound_in <*(Ex (x,p))*>) by Th59; then not y in still_not-bound_in <*(Ex (x,p))*> by XBOOLE_0:def_3; then not y in still_not-bound_in (Ex (x,p)) by Th60; then not y in (still_not-bound_in p) \ {x} by QC_LANG3:19; then not y in (still_not-bound_in ('not' p)) \ {x} by QC_LANG3:7; then A6: not y in still_not-bound_in (All (x,('not' p))) by QC_LANG3:12; not y in still_not-bound_in <*q*> by A4, XBOOLE_0:def_3; then not y in still_not-bound_in q by Th60; then not y in still_not-bound_in ('not' q) by QC_LANG3:7; then A7: not y in still_not-bound_in <*('not' q)*> by Th60; not y in still_not-bound_in f by A5, XBOOLE_0:def_3; then not y in (still_not-bound_in f) \/ (still_not-bound_in <*('not' q)*>) by A7, XBOOLE_0:def_3; then not y in still_not-bound_in (f ^ <*('not' q)*>) by Th59; then A8: not y in still_not-bound_in (Ant ((f ^ <*('not' q)*>) ^ <*(('not' p) . (x,y))*>)) by Th5; Suc ((f ^ <*('not' q)*>) ^ <*(('not' p) . (x,y))*>) = ('not' p) . (x,y) by Th5; then |- (Ant ((f ^ <*('not' q)*>) ^ <*(('not' p) . (x,y))*>)) ^ <*(All (x,('not' p)))*> by A3, A8, A6, Th43; then |- (f ^ <*('not' q)*>) ^ <*(All (x,('not' p)))*> by Th5; then |- (f ^ <*('not' (All (x,('not' p))))*>) ^ <*q*> by Th48; hence |- (f ^ <*(Ex (x,p))*>) ^ <*q*> by QC_LANG2:def_5; ::_thesis: verum end; theorem Th62: :: CALCUL_1:62 for Al being QC-alphabet for f being FinSequence of CQC-WFF Al holds still_not-bound_in f = union { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } proof let Al be QC-alphabet ; ::_thesis: for f being FinSequence of CQC-WFF Al holds still_not-bound_in f = union { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } let f be FinSequence of CQC-WFF Al; ::_thesis: still_not-bound_in f = union { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } defpred S1[ set ] means ex p being Element of CQC-WFF Al st ( \$1 = still_not-bound_in p & ex i being Element of NAT st ( i in dom f & p = f . i ) ); set X = { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } ; A1: now__::_thesis:_for_b_being_set_st_b_in_union__{__(still_not-bound_in_p)_where_p_is_Element_of_CQC-WFF_Al_:_ex_i_being_Element_of_NAT_st_ (_i_in_dom_f_&_p_=_f_._i_)__}__holds_ b_in_still_not-bound_in_f let b be set ; ::_thesis: ( b in union { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } implies b in still_not-bound_in f ) assume b in union { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } ; ::_thesis: b in still_not-bound_in f then consider Y being set such that A2: b in Y and A3: Y in { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } by TARSKI:def_4; S1[Y] by A3; hence b in still_not-bound_in f by A2, Def5; ::_thesis: verum end; now__::_thesis:_for_b_being_set_st_b_in_still_not-bound_in_f_holds_ b_in_union__{__(still_not-bound_in_p)_where_p_is_Element_of_CQC-WFF_Al_:_ex_i_being_Element_of_NAT_st_ (_i_in_dom_f_&_p_=_f_._i_)__}_ let b be set ; ::_thesis: ( b in still_not-bound_in f implies b in union { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } ) assume b in still_not-bound_in f ; ::_thesis: b in union { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } then consider i being Element of NAT , p being Element of CQC-WFF Al such that A4: ( i in dom f & p = f . i ) and A5: b in still_not-bound_in p by Def5; still_not-bound_in p in { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } by A4; hence b in union { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } by A5, TARSKI:def_4; ::_thesis: verum end; hence still_not-bound_in f = union { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } by A1, TARSKI:1; ::_thesis: verum end; theorem Th63: :: CALCUL_1:63 for Al being QC-alphabet for f being FinSequence of CQC-WFF Al holds still_not-bound_in f is finite proof let Al be QC-alphabet ; ::_thesis: for f being FinSequence of CQC-WFF Al holds still_not-bound_in f is finite let f be FinSequence of CQC-WFF Al; ::_thesis: still_not-bound_in f is finite defpred S1[ set , set ] means ex p being Element of CQC-WFF Al st ( \$2 = still_not-bound_in p & p = f . \$1 ); consider n being Nat such that A1: dom f = Seg n by FINSEQ_1:def_2; set X = { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } ; consider F1 being Function such that A2: rng F1 = Seg n and A3: dom F1 in omega by FINSET_1:def_1; A4: now__::_thesis:_for_b_being_set_st_b_in__{__(still_not-bound_in_p)_where_p_is_Element_of_CQC-WFF_Al_:_ex_i_being_Element_of_NAT_st_ (_i_in_dom_f_&_p_=_f_._i_)__}__holds_ b_is_finite let b be set ; ::_thesis: ( b in { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } implies b is finite ) assume b in { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } ; ::_thesis: b is finite then ex p being Element of CQC-WFF Al st ( b = still_not-bound_in p & ex i being Element of NAT st ( i in dom f & p = f . i ) ) ; hence b is finite by CQC_SIM1:19; ::_thesis: verum end; A5: for a being set st a in dom f holds ex b being set st S1[a,b] proof let a be set ; ::_thesis: ( a in dom f implies ex b being set st S1[a,b] ) assume a in dom f ; ::_thesis: ex b being set st S1[a,b] then f . a in rng f by FUNCT_1:3; then reconsider p = f . a as Element of CQC-WFF Al ; take still_not-bound_in p ; ::_thesis: S1[a, still_not-bound_in p] thus S1[a, still_not-bound_in p] ; ::_thesis: verum end; consider F2 being Function such that A6: ( dom F2 = dom f & ( for b being set st b in dom f holds S1[b,F2 . b] ) ) from CLASSES1:sch_1(A5); set F = F2 * F1; A7: now__::_thesis:_for_b_being_set_st_b_in__{__(still_not-bound_in_p)_where_p_is_Element_of_CQC-WFF_Al_:_ex_i_being_Element_of_NAT_st_ (_i_in_dom_f_&_p_=_f_._i_)__}__holds_ b_in_rng_(F2_*_F1) let b be set ; ::_thesis: ( b in { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } implies b in rng (F2 * F1) ) assume b in { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } ; ::_thesis: b in rng (F2 * F1) then consider p being Element of CQC-WFF Al such that A8: b = still_not-bound_in p and A9: ex i being Element of NAT st ( i in dom f & p = f . i ) ; consider i being Element of NAT such that A10: i in dom f and A11: p = f . i by A9; S1[i,F2 . i] by A6, A10; then b in rng F2 by A6, A8, A10, A11, FUNCT_1:3; hence b in rng (F2 * F1) by A6, A1, A2, RELAT_1:28; ::_thesis: verum end; now__::_thesis:_for_b_being_set_st_b_in_rng_(F2_*_F1)_holds_ b_in__{__(still_not-bound_in_p)_where_p_is_Element_of_CQC-WFF_Al_:_ex_i_being_Element_of_NAT_st_ (_i_in_dom_f_&_p_=_f_._i_)__}_ let b be set ; ::_thesis: ( b in rng (F2 * F1) implies b in { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } ) assume b in rng (F2 * F1) ; ::_thesis: b in { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } then b in rng F2 by A6, A1, A2, RELAT_1:28; then consider a being set such that A12: a in dom F2 and A13: b = F2 . a by FUNCT_1:def_3; reconsider a = a as Element of NAT by A6, A12; S1[a,F2 . a] by A6, A12; hence b in { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } by A6, A12, A13; ::_thesis: verum end; then A14: rng (F2 * F1) = { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } by A7, TARSKI:1; dom (F2 * F1) in omega by A6, A1, A2, A3, RELAT_1:27; then { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } is finite by A14, FINSET_1:def_1; then union { (still_not-bound_in p) where p is Element of CQC-WFF Al : ex i being Element of NAT st ( i in dom f & p = f . i ) } is finite by A4, FINSET_1:7; hence still_not-bound_in f is finite by Th62; ::_thesis: verum end; theorem Th64: :: CALCUL_1:64 for Al being QC-alphabet holds ( card (bound_QC-variables Al) = card (QC-symbols Al) & not bound_QC-variables Al is finite ) proof let Al be QC-alphabet ; ::_thesis: ( card (bound_QC-variables Al) = card (QC-symbols Al) & not bound_QC-variables Al is finite ) NAT c= QC-symbols Al by QC_LANG1:3; then A1: not QC-symbols Al is finite ; bound_QC-variables Al = [:{4},(QC-symbols Al):] by QC_LANG1:def_4; then card (bound_QC-variables Al) = card [:(QC-symbols Al),{4}:] by CARD_2:4; hence ( card (bound_QC-variables Al) = card (QC-symbols Al) & not bound_QC-variables Al is finite ) by A1, CARD_4:19; ::_thesis: verum end; theorem Th65: :: CALCUL_1:65 for Al being QC-alphabet for f being FinSequence of CQC-WFF Al holds not for x being bound_QC-variable of Al holds x in still_not-bound_in f proof let Al be QC-alphabet ; ::_thesis: for f being FinSequence of CQC-WFF Al holds not for x being bound_QC-variable of Al holds x in still_not-bound_in f let f be FinSequence of CQC-WFF Al; ::_thesis: not for x being bound_QC-variable of Al holds x in still_not-bound_in f still_not-bound_in f is finite by Th63; then still_not-bound_in f <> bound_QC-variables Al by Th64; then not for b being set holds ( b in still_not-bound_in f iff b in bound_QC-variables Al ) by TARSKI:1; then consider b being set such that A1: not b in still_not-bound_in f and A2: b in bound_QC-variables Al ; reconsider x = b as bound_QC-variable of Al by A2; take x ; ::_thesis: not x in still_not-bound_in f thus not x in still_not-bound_in f by A1; ::_thesis: verum end; theorem Th66: :: CALCUL_1:66 for Al being QC-alphabet for p being Element of CQC-WFF Al for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st |- f ^ <*(All (x,p))*> holds |- f ^ <*(All (x,('not' ('not' p))))*> proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st |- f ^ <*(All (x,p))*> holds |- f ^ <*(All (x,('not' ('not' p))))*> let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st |- f ^ <*(All (x,p))*> holds |- f ^ <*(All (x,('not' ('not' p))))*> let x be bound_QC-variable of Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- f ^ <*(All (x,p))*> holds |- f ^ <*(All (x,('not' ('not' p))))*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- f ^ <*(All (x,p))*> implies |- f ^ <*(All (x,('not' ('not' p))))*> ) assume A1: |- f ^ <*(All (x,p))*> ; ::_thesis: |- f ^ <*(All (x,('not' ('not' p))))*> consider y0 being bound_QC-variable of Al such that A2: not y0 in still_not-bound_in (f ^ <*(All (x,p))*>) by Th65; ( Ant (f ^ <*(All (x,p))*>) = f & Suc (f ^ <*(All (x,p))*>) = All (x,p) ) by Th5; then |- f ^ <*(p . (x,y0))*> by A1, Th42; then |- f ^ <*('not' ('not' (p . (x,y0))))*> by Th54; then |- f ^ <*('not' (('not' p) . (x,y0)))*> by Th57; then A3: |- f ^ <*(('not' ('not' p)) . (x,y0))*> by Th57; set f1 = f ^ <*(('not' ('not' p)) . (x,y0))*>; A4: not y0 in (still_not-bound_in f) \/ (still_not-bound_in <*(All (x,p))*>) by A2, Th59; then not y0 in still_not-bound_in f by XBOOLE_0:def_3; then A5: not y0 in still_not-bound_in (Ant (f ^ <*(('not' ('not' p)) . (x,y0))*>)) by Th5; not y0 in still_not-bound_in <*(All (x,p))*> by A4, XBOOLE_0:def_3; then not y0 in still_not-bound_in (All (x,p)) by Th60; then not y0 in (still_not-bound_in p) \ {x} by QC_LANG3:12; then not y0 in (still_not-bound_in ('not' p)) \ {x} by QC_LANG3:7; then not y0 in (still_not-bound_in ('not' ('not' p))) \ {x} by QC_LANG3:7; then A6: not y0 in still_not-bound_in (All (x,('not' ('not' p)))) by QC_LANG3:12; Suc (f ^ <*(('not' ('not' p)) . (x,y0))*>) = ('not' ('not' p)) . (x,y0) by Th5; then |- (Ant (f ^ <*(('not' ('not' p)) . (x,y0))*>)) ^ <*(All (x,('not' ('not' p))))*> by A3, A5, A6, Th43; hence |- f ^ <*(All (x,('not' ('not' p))))*> by Th5; ::_thesis: verum end; theorem Th67: :: CALCUL_1:67 for Al being QC-alphabet for p being Element of CQC-WFF Al for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st |- f ^ <*(All (x,('not' ('not' p))))*> holds |- f ^ <*(All (x,p))*> proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st |- f ^ <*(All (x,('not' ('not' p))))*> holds |- f ^ <*(All (x,p))*> let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al st |- f ^ <*(All (x,('not' ('not' p))))*> holds |- f ^ <*(All (x,p))*> let x be bound_QC-variable of Al; ::_thesis: for f being FinSequence of CQC-WFF Al st |- f ^ <*(All (x,('not' ('not' p))))*> holds |- f ^ <*(All (x,p))*> let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- f ^ <*(All (x,('not' ('not' p))))*> implies |- f ^ <*(All (x,p))*> ) assume A1: |- f ^ <*(All (x,('not' ('not' p))))*> ; ::_thesis: |- f ^ <*(All (x,p))*> consider y0 being bound_QC-variable of Al such that A2: not y0 in still_not-bound_in (f ^ <*(All (x,p))*>) by Th65; ( Ant (f ^ <*(All (x,('not' ('not' p))))*>) = f & Suc (f ^ <*(All (x,('not' ('not' p))))*>) = All (x,('not' ('not' p))) ) by Th5; then |- f ^ <*(('not' ('not' p)) . (x,y0))*> by A1, Th42; then |- f ^ <*('not' (('not' p) . (x,y0)))*> by Th57; then A3: |- f ^ <*('not' ('not' (p . (x,y0))))*> by Th57; set f1 = f ^ <*(p . (x,y0))*>; A4: not y0 in (still_not-bound_in f) \/ (still_not-bound_in <*(All (x,p))*>) by A2, Th59; then not y0 in still_not-bound_in f by XBOOLE_0:def_3; then A5: not y0 in still_not-bound_in (Ant (f ^ <*(p . (x,y0))*>)) by Th5; not y0 in still_not-bound_in <*(All (x,p))*> by A4, XBOOLE_0:def_3; then A6: not y0 in still_not-bound_in (All (x,p)) by Th60; Suc (f ^ <*(p . (x,y0))*>) = p . (x,y0) by Th5; then |- (Ant (f ^ <*(p . (x,y0))*>)) ^ <*(All (x,p))*> by A3, A5, A6, Th43, Th55; hence |- f ^ <*(All (x,p))*> by Th5; ::_thesis: verum end; theorem :: CALCUL_1:68 for Al being QC-alphabet for p being Element of CQC-WFF Al for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al holds ( |- f ^ <*(All (x,p))*> iff |- f ^ <*('not' (Ex (x,('not' p))))*> ) proof let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al holds ( |- f ^ <*(All (x,p))*> iff |- f ^ <*('not' (Ex (x,('not' p))))*> ) let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al for f being FinSequence of CQC-WFF Al holds ( |- f ^ <*(All (x,p))*> iff |- f ^ <*('not' (Ex (x,('not' p))))*> ) let x be bound_QC-variable of Al; ::_thesis: for f being FinSequence of CQC-WFF Al holds ( |- f ^ <*(All (x,p))*> iff |- f ^ <*('not' (Ex (x,('not' p))))*> ) let f be FinSequence of CQC-WFF Al; ::_thesis: ( |- f ^ <*(All (x,p))*> iff |- f ^ <*('not' (Ex (x,('not' p))))*> ) thus ( |- f ^ <*(All (x,p))*> implies |- f ^ <*('not' (Ex (x,('not' p))))*> ) ::_thesis: ( |- f ^ <*('not' (Ex (x,('not' p))))*> implies |- f ^ <*(All (x,p))*> ) proof assume |- f ^ <*(All (x,p))*> ; ::_thesis: |- f ^ <*('not' (Ex (x,('not' p))))*> then |- f ^ <*(All (x,('not' ('not' p))))*> by Th66; then |- f ^ <*('not' ('not' (All (x,('not' ('not' p))))))*> by Th54; hence |- f ^ <*('not' (Ex (x,('not' p))))*> by QC_LANG2:def_5; ::_thesis: verum end; assume |- f ^ <*('not' (Ex (x,('not' p))))*> ; ::_thesis: |- f ^ <*(All (x,p))*> then |- f ^ <*('not' ('not' (All (x,('not' ('not' p))))))*> by QC_LANG2:def_5; then |- f ^ <*(All (x,('not' ('not' p))))*> by Th55; hence |- f ^ <*(All (x,p))*> by Th67; ::_thesis: verum end; definition let f be FinSequence; let p be set ; redefine pred p is_tail_of f means :: CALCUL_1:def 16 ex i being Element of NAT st ( i in dom f & f . i = p ); compatibility ( p is_tail_of f iff ex i being Element of NAT st ( i in dom f & f . i = p ) ) by Lm1, Lm2; end; :: deftheorem defines is_tail_of CALCUL_1:def_16_:_ for f being FinSequence for p being set holds ( p is_tail_of f iff ex i being Element of NAT st ( i in dom f & f . i = p ) );