:: 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 ) );