:: FOMODEL4 semantic presentation

begin

                                                    Lm1:  for X, A, B being set st X is Subset of (Funcs (A,B)) holds
X is Subset-Family of [:A,B:]
proof
let X, A, B be set ; ::_thesis: ( X is Subset of (Funcs (A,B)) implies X is Subset-Family of [:A,B:] )
A1: Funcs (A,B) c= bool [:A,B:] by FRAENKEL:2;
assume  X is Subset of (Funcs (A,B)) ; ::_thesis: X is Subset-Family of [:A,B:]
hence  X is Subset-Family of [:A,B:] by A1, XBOOLE_1:1; ::_thesis: verum
end;

Lm2:  for S being Language
for t1, t2 being termal string of S holds (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is 0wff string of S
proof
let S be Language; ::_thesis: for t1, t2 being termal string of S holds (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is 0wff string of S
let t1, t2 be termal string of S; ::_thesis: (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is 0wff string of S
set E = TheEqSymbOf S;
set AT = AllTermsOf S;
set C = S -multiCat ;
set SS = AllSymbolsOf S;
reconsider tt1 = t1, tt2 = t2 as Element of AllTermsOf S by FOMODEL1:def_32;
reconsider T = <*tt1*> ^ <*tt2*> as 2 -element Element of (AllTermsOf S) * by FINSEQ_1:def_11;
reconsider ATT = AllTermsOf S as Subset of ((AllSymbolsOf S) *) by XBOOLE_1:1;
reconsider TT = T as Element of ATT * ;
reconsider TTT = TT as Element of ((AllSymbolsOf S) *) * ;
reconsider EE = TheEqSymbOf S as ofAtomicFormula Element of S ;
A1: abs (- 2) = - (- 2) by ABSVALUE:def_1
.= 2 ;
A2: abs (ar EE) = 2 by A1, FOMODEL1:def_23;
 (S -multiCat) . TT is Element of (AllSymbolsOf S) * ;
then reconsider tailer = (S -multiCat) . T as FinSequence of AllSymbolsOf S ;
reconsider SSS = AllSymbolsOf S as non empty set ;
reconsider EEE = EE as Element of SSS ;
reconsider header = <*EEE*> as FinSequence of AllSymbolsOf S ;
reconsider IT = header ^ tailer as non empty FinSequence of AllSymbolsOf S ;
reconsider phi = IT as string of S by FOMODEL1:13;
tailer = ((AllSymbolsOf S) -multiCat) . <*tt1,tt2*>
.= tt1 ^ tt2 by FOMODEL0:15 ;
then  <*(TheEqSymbOf S)*> ^ (t1 ^ t2) is 0wff string of S by A2, FOMODEL1:def_35;
hence  (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is 0wff string of S by FINSEQ_1:32; ::_thesis: verum
end;

definition
let S be Language;
funcS -sequents  ->  set  equals :: FOMODEL4:def 1
 {  [premises,conclusion] where premises is Subset of (AllFormulasOf S), conclusion is wff string of S : premises is finite  }  ;
coherence
  {  [premises,conclusion] where premises is Subset of (AllFormulasOf S), conclusion is wff string of S : premises is finite  }  is set ;
end;

:: deftheorem  defines -sequents FOMODEL4:def_1_:_
 for S being Language holds S -sequents =  {  [premises,conclusion] where premises is Subset of (AllFormulasOf S), conclusion is wff string of S : premises is finite  }  ;

registration
let S be Language;
clusterS -sequents  ->  non empty ;
coherence
 not S -sequents is empty
proof
set AF = AllFormulasOf S;
set premises = the finite Subset of (AllFormulasOf S);
set conclusion = the wff string of S;
 [ the finite Subset of (AllFormulasOf S), the wff string of S] in S -sequents ;
hence  not S -sequents is empty ; ::_thesis: verum
end;
end;

registration
let S be Language;
clusterS -sequents  ->  Relation-like ;
coherence
 S -sequents is Relation-like
proof
set SS = AllSymbolsOf S;
set strings = ((AllSymbolsOf S) *) \ {{}};
set FF = AllFormulasOf S;
now__::_thesis:_for_z_being_set_st_z_in_S_-sequents_holds_
z_in_[:(bool_(AllFormulasOf_S)),(((AllSymbolsOf_S)_*)_\_{{}}):]
let z be set ; ::_thesis: ( z in S -sequents implies z in [:(bool (AllFormulasOf S)),(((AllSymbolsOf S) *) \ {{}}):] )
assume  z in S -sequents ; ::_thesis: z in [:(bool (AllFormulasOf S)),(((AllSymbolsOf S) *) \ {{}}):]
then consider x being Subset of (AllFormulasOf S), y being wff string of S such that
A1: ( z = [x,y] & x is finite ) ;
thus  z in [:(bool (AllFormulasOf S)),(((AllSymbolsOf S) *) \ {{}}):] by A1; ::_thesis: verum
end;
then  S -sequents is Subset of [:(bool (AllFormulasOf S)),(((AllSymbolsOf S) *) \ {{}}):] by TARSKI:def_3;
hence  S -sequents is Relation-like ; ::_thesis: verum
end;
end;

definition
let S be Language;
let x be set ;
attrx is S -sequent-like  means :Def2: :: FOMODEL4:def 2
x in S -sequents ;
end;

:: deftheorem Def2 defines -sequent-like FOMODEL4:def_2_:_
 for S being Language
for x being set holds
( x is S -sequent-like iff x in S -sequents );

definition
let S be Language;
let X be set ;
attrX is S -sequents-like  means :Def3: :: FOMODEL4:def 3
X c= S -sequents ;
end;

:: deftheorem Def3 defines -sequents-like FOMODEL4:def_3_:_
 for S being Language
for X being set holds
( X is S -sequents-like iff X c= S -sequents );

registration
let S be Language;
cluster  -> S -sequents-like for Element of bool (S -sequents);
coherence
 for b1 being Subset of (S -sequents) holds b1 is S -sequents-like by Def3;
cluster  -> S -sequent-like for Element of S -sequents ;
coherence
 for b1 being Element of S -sequents holds b1 is S -sequent-like by Def2;
end;

registration
let S be Language;
clusterS -sequent-like for Element of S -sequents ;
existence
 ex b1 being Element of S -sequents st b1 is S -sequent-like
proof
take  the Element of S -sequents ; ::_thesis: the Element of S -sequents is S -sequent-like
thus  the Element of S -sequents is S -sequent-like ; ::_thesis: verum
end;
cluster Relation-like S -sequents-like for Element of bool (S -sequents);
existence
 ex b1 being Subset of (S -sequents) st b1 is S -sequents-like
proof
take  the Subset of (S -sequents) ; ::_thesis: the Subset of (S -sequents) is S -sequents-like
thus  the Subset of (S -sequents) is S -sequents-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
clusterS -sequent-like for set ;
existence
 ex b1 being set st b1 is S -sequent-like
proof
take  the Element of S -sequents ; ::_thesis: the Element of S -sequents is S -sequent-like
thus  the Element of S -sequents is S -sequent-like ; ::_thesis: verum
end;
clusterS -sequents-like for set ;
existence
 ex b1 being set st b1 is S -sequents-like
proof
take  the Subset of (S -sequents) ; ::_thesis: the Subset of (S -sequents) is S -sequents-like
thus  the Subset of (S -sequents) is S -sequents-like ; ::_thesis: verum
end;
end;

definition
let S be Language;
mode Rule of S is Element of Funcs ((bool (S -sequents)),(bool (S -sequents)));
end;

definition
let S be Language;
mode RuleSet of S is Subset of (Funcs ((bool (S -sequents)),(bool (S -sequents))));
end;


registration
let A, B be set ;
let X be Subset of (Funcs (A,B));
cluster union X ->  Relation-like ;
coherence
 union X is Relation-like ;
end;

registration
let S be Language;
let D be RuleSet of S;
cluster union D ->  Relation-like ;
coherence
 union D is Relation-like ;
end;

definition
let S be Language;
let D be RuleSet of S;
func OneStep D ->  Rule of S means :Def4: :: FOMODEL4:def 4
 for Seqs being Element of bool (S -sequents) holds it . Seqs = union ((union D) .: {Seqs});
existence
 ex b1 being Rule of S st
for Seqs being Element of bool (S -sequents) holds b1 . Seqs = union ((union D) .: {Seqs})
proof
set Q = S -sequents ;
set F = Funcs ((bool (S -sequents)),(bool (S -sequents)));
reconsider RR = union D as Relation of (bool (S -sequents)) by FOMODEL0:19;
deffunc H1( Element of bool (S -sequents)) ->  Element of bool (S -sequents) = union (RR .: {$1});
consider f being Function of (bool (S -sequents)),(bool (S -sequents)) such that
A1: for x being Element of bool (S -sequents) holds f . x = H1(x) from FUNCT_2:sch_4();
reconsider ff = f as Element of Funcs ((bool (S -sequents)),(bool (S -sequents))) by FUNCT_2:8;
take  ff ; ::_thesis: for Seqs being Element of bool (S -sequents) holds ff . Seqs = union ((union D) .: {Seqs})
thus  for Seqs being Element of bool (S -sequents) holds ff . Seqs = union ((union D) .: {Seqs}) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Rule of S st ( for Seqs being Element of bool (S -sequents) holds b1 . Seqs = union ((union D) .: {Seqs}) ) & ( for Seqs being Element of bool (S -sequents) holds b2 . Seqs = union ((union D) .: {Seqs}) ) holds
b1 = b2
proof
set Q = S -sequents ;
set F = Funcs ((bool (S -sequents)),(bool (S -sequents)));
let IT1, IT2 be Rule of S; ::_thesis: ( ( for Seqs being Element of bool (S -sequents) holds IT1 . Seqs = union ((union D) .: {Seqs}) ) & ( for Seqs being Element of bool (S -sequents) holds IT2 . Seqs = union ((union D) .: {Seqs}) ) implies IT1 = IT2 )
reconsider IT11 = IT1, IT22 = IT2 as Function of (bool (S -sequents)),(bool (S -sequents)) ;
deffunc H1( set ) ->  set = union ((union D) .: {$1});
assume A2: for Seqs being Element of bool (S -sequents) holds IT1 . Seqs = H1(Seqs) ; ::_thesis: ( ex Seqs being Element of bool (S -sequents) st not IT2 . Seqs = union ((union D) .: {Seqs}) or IT1 = IT2 )
assume A3: for Seqs being Element of bool (S -sequents) holds IT2 . Seqs = H1(Seqs) ; ::_thesis: IT1 = IT2
now__::_thesis:_for_x_being_Element_of_bool_(S_-sequents)_holds_IT11_._x_=_IT22_._x
let x be Element of bool (S -sequents); ::_thesis: IT11 . x = IT22 . x
thus IT11 . x = H1(x) by A2
.= IT22 . x by A3 ; ::_thesis: verum
end;
hence  IT1 = IT2 by FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem Def4 defines OneStep FOMODEL4:def_4_:_
 for S being Language
for D being RuleSet of S
for b3 being Rule of S holds
( b3 = OneStep D iff for Seqs being Element of bool (S -sequents) holds b3 . Seqs = union ((union D) .: {Seqs}) );

Lm3:  for m being Nat
for S being Language
for D being RuleSet of S
for R being Rule of S
for Seqts being Subset of (S -sequents)
for SQ being b2 -sequents-like set holds
( dom (OneStep D) = bool (S -sequents) & rng (OneStep D) c= dom (OneStep D) & iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) & dom (iter ((OneStep D),m)) = bool (S -sequents) & (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
proof
let m be Nat; ::_thesis: for S being Language
for D being RuleSet of S
for R being Rule of S
for Seqts being Subset of (S -sequents)
for SQ being b1 -sequents-like set holds
( dom (OneStep D) = bool (S -sequents) & rng (OneStep D) c= dom (OneStep D) & iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) & dom (iter ((OneStep D),m)) = bool (S -sequents) & (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
let S be Language; ::_thesis: for D being RuleSet of S
for R being Rule of S
for Seqts being Subset of (S -sequents)
for SQ being S -sequents-like set holds
( dom (OneStep D) = bool (S -sequents) & rng (OneStep D) c= dom (OneStep D) & iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) & dom (iter ((OneStep D),m)) = bool (S -sequents) & (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
let D be RuleSet of S; ::_thesis: for R being Rule of S
for Seqts being Subset of (S -sequents)
for SQ being S -sequents-like set holds
( dom (OneStep D) = bool (S -sequents) & rng (OneStep D) c= dom (OneStep D) & iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) & dom (iter ((OneStep D),m)) = bool (S -sequents) & (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
let R be Rule of S; ::_thesis: for Seqts being Subset of (S -sequents)
for SQ being S -sequents-like set holds
( dom (OneStep D) = bool (S -sequents) & rng (OneStep D) c= dom (OneStep D) & iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) & dom (iter ((OneStep D),m)) = bool (S -sequents) & (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
let Seqts be Subset of (S -sequents); ::_thesis: for SQ being S -sequents-like set holds
( dom (OneStep D) = bool (S -sequents) & rng (OneStep D) c= dom (OneStep D) & iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) & dom (iter ((OneStep D),m)) = bool (S -sequents) & (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
let SQ be S -sequents-like set ; ::_thesis: ( dom (OneStep D) = bool (S -sequents) & rng (OneStep D) c= dom (OneStep D) & iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) & dom (iter ((OneStep D),m)) = bool (S -sequents) & (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
set O = OneStep D;
set Q = S -sequents ;
thus A1: dom (OneStep D) = bool (S -sequents) by FUNCT_2:def_1; ::_thesis: ( rng (OneStep D) c= dom (OneStep D) & iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) & dom (iter ((OneStep D),m)) = bool (S -sequents) & (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
hence A2: rng (OneStep D) c= dom (OneStep D) by RELAT_1:def_19; ::_thesis: ( iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) & dom (iter ((OneStep D),m)) = bool (S -sequents) & (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
thus  iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) by FUNCT_7:83; ::_thesis: ( dom (iter ((OneStep D),m)) = bool (S -sequents) & (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
hence  dom (iter ((OneStep D),m)) = bool (S -sequents) by FUNCT_2:def_1; ::_thesis: ( (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) & Seqts in dom R & SQ in dom (iter (R,m)) )
thus  (dom (OneStep D)) \/ (rng (OneStep D)) = bool (S -sequents) by A1, A2, XBOOLE_1:12; ::_thesis: ( Seqts in dom R & SQ in dom (iter (R,m)) )
 dom R = bool (S -sequents) by FUNCT_2:def_1;
hence  Seqts in dom R ; ::_thesis: SQ in dom (iter (R,m))
 iter (R,m) is Function of (bool (S -sequents)),(bool (S -sequents)) by FUNCT_7:83;
then  ( dom (iter (R,m)) = bool (S -sequents) & SQ c= S -sequents ) by Def3, FUNCT_2:def_1;
hence  SQ in dom (iter (R,m)) ; ::_thesis: verum
end;

definition
let S be Language;
let D be RuleSet of S;
let m be Nat;
func(m,D) -derivables  ->  Rule of S equals :: FOMODEL4:def 5
 iter ((OneStep D),m);
coherence
 iter ((OneStep D),m) is Rule of S
proof
set Q = S -sequents ;
set O = OneStep D;
set IT = iter ((OneStep D),m);
 iter ((OneStep D),m) is Function of (bool (S -sequents)),(bool (S -sequents)) by FUNCT_7:83;
hence  iter ((OneStep D),m) is Rule of S by FUNCT_2:8; ::_thesis: verum
end;
end;

:: deftheorem  defines -derivables FOMODEL4:def_5_:_
 for S being Language
for D being RuleSet of S
for m being Nat holds (m,D) -derivables = iter ((OneStep D),m);

definition
let S be Language;
let D be RuleSet of S;
let Seqs1, Seqs2 be set ;
attrSeqs2 is Seqs1,D -derivable  means :Def6: :: FOMODEL4:def 6
Seqs2 c= union (((OneStep D) [*]) .: {Seqs1});
end;

:: deftheorem Def6 defines -derivable FOMODEL4:def_6_:_
 for S being Language
for D being RuleSet of S
for Seqs1, Seqs2 being set holds
( Seqs2 is Seqs1,D -derivable iff Seqs2 c= union (((OneStep D) [*]) .: {Seqs1}) );

definition
let m be Nat;
let S be Language;
let D be RuleSet of S;
let Seqts, seqt be set ;
attrseqt is m,Seqts,D -derivable  means :Def7: :: FOMODEL4:def 7
seqt in ((m,D) -derivables) . Seqts;
end;

:: deftheorem Def7 defines -derivable FOMODEL4:def_7_:_
 for m being Nat
for S being Language
for D being RuleSet of S
for Seqts, seqt being set holds
( seqt is m,Seqts,D -derivable iff seqt in ((m,D) -derivables) . Seqts );

definition
let S be Language;
let D be RuleSet of S;
funcD -iterators  ->  Subset-Family of [:(bool (S -sequents)),(bool (S -sequents)):] equals :: FOMODEL4:def 8
 {  (iter ((OneStep D),mm)) where mm is Element of NAT : verum  }  ;
coherence
  {  (iter ((OneStep D),mm)) where mm is Element of NAT : verum  }  is Subset-Family of [:(bool (S -sequents)),(bool (S -sequents)):]
proof
set O = OneStep D;
set Q = S -sequents ;
set IT =  {  (iter ((OneStep D),mm)) where mm is Element of NAT : verum  }  ;
now__::_thesis:_for_x_being_set_st_x_in__{__(iter_((OneStep_D),mm))_where_mm_is_Element_of_NAT_:_verum__}__holds_
x_in_bool_[:(bool_(S_-sequents)),(bool_(S_-sequents)):]
let x be set ; ::_thesis: ( x in  {  (iter ((OneStep D),mm)) where mm is Element of NAT : verum  }  implies x in bool [:(bool (S -sequents)),(bool (S -sequents)):] )
assume  x in  {  (iter ((OneStep D),mm)) where mm is Element of NAT : verum  }  ; ::_thesis: x in bool [:(bool (S -sequents)),(bool (S -sequents)):]
then consider mm being Element of NAT  such that
A1: x = iter ((OneStep D),mm) ;
 x is Relation of (bool (S -sequents)),(bool (S -sequents)) by Lm3, A1;
hence  x in bool [:(bool (S -sequents)),(bool (S -sequents)):] ; ::_thesis: verum
end;
hence   {  (iter ((OneStep D),mm)) where mm is Element of NAT : verum  }  is Subset-Family of [:(bool (S -sequents)),(bool (S -sequents)):] by TARSKI:def_3; ::_thesis: verum
end;
end;

:: deftheorem  defines -iterators FOMODEL4:def_8_:_
 for S being Language
for D being RuleSet of S holds D -iterators =  {  (iter ((OneStep D),mm)) where mm is Element of NAT : verum  }  ;

definition
let S be Language;
let R be Rule of S;
attrR is isotone  means :Def9: :: FOMODEL4:def 9
 for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 holds
R . Seqts1 c= R . Seqts2;
end;

:: deftheorem Def9 defines isotone FOMODEL4:def_9_:_
 for S being Language
for R being Rule of S holds
( R is isotone iff for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 holds
R . Seqts1 c= R . Seqts2 );

Lm4:  for S being Language
for R being Rule of S holds
( id (bool (S -sequents)) is Rule of S & ( R = id (bool (S -sequents)) implies R is isotone ) )
proof
let S be Language; ::_thesis: for R being Rule of S holds
( id (bool (S -sequents)) is Rule of S & ( R = id (bool (S -sequents)) implies R is isotone ) )
let R be Rule of S; ::_thesis: ( id (bool (S -sequents)) is Rule of S & ( R = id (bool (S -sequents)) implies R is isotone ) )
set Q = S -sequents ;
reconsider I = id (bool (S -sequents)) as Rule of S by FUNCT_2:8;
 id (bool (S -sequents)) = I ;
hence  id (bool (S -sequents)) is Rule of S ; ::_thesis: ( R = id (bool (S -sequents)) implies R is isotone )
A1: now__::_thesis:_for_Seqts1,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts1_c=_Seqts2_holds_
I_._Seqts1_c=_I_._Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts1 c= Seqts2 implies I . Seqts1 c= I . Seqts2 )
A2: ( I . Seqts1 = Seqts1 & I . Seqts2 = Seqts2 ) by FUNCT_1:17;
assume  Seqts1 c= Seqts2 ; ::_thesis: I . Seqts1 c= I . Seqts2
hence  I . Seqts1 c= I . Seqts2 by A2; ::_thesis: verum
end;
assume  R = id (bool (S -sequents)) ; ::_thesis: R is isotone
hence  R is isotone by A1, Def9; ::_thesis: verum
end;

registration
let S be Language;
cluster Relation-like bool (S -sequents) -defined bool (S -sequents) -valued Function-like total quasi_total isotone for Element of Funcs ((bool (S -sequents)),(bool (S -sequents)));
existence
 ex b1 being Rule of S st b1 is isotone
proof
set Q = S -sequents ;
reconsider I = id (bool (S -sequents)) as Rule of S by Lm4;
take  I ; ::_thesis: I is isotone
thus  I is isotone by Lm4; ::_thesis: verum
end;
end;

definition
let S be Language;
let D be RuleSet of S;
attrD is isotone  means :Def10: :: FOMODEL4:def 10
 for Seqts1, Seqts2 being Subset of (S -sequents)
for f being Function st Seqts1 c= Seqts2 & f in D holds
ex g being Function st
( g in D & f . Seqts1 c= g . Seqts2 );
end;

:: deftheorem Def10 defines isotone FOMODEL4:def_10_:_
 for S being Language
for D being RuleSet of S holds
( D is isotone iff for Seqts1, Seqts2 being Subset of (S -sequents)
for f being Function st Seqts1 c= Seqts2 & f in D holds
ex g being Function st
( g in D & f . Seqts1 c= g . Seqts2 ) );

registration
let S be Language;
let M be isotone Rule of S;
cluster{M} ->  isotone for RuleSet of S;
coherence
 for b1 being RuleSet of S st b1 = {M} holds
b1 is isotone
proof
set Q = S -sequents ;
now__::_thesis:_for_Seqts1,_Seqts2_being_Subset_of_(S_-sequents)
for_f_being_Function_st_Seqts1_c=_Seqts2_&_f_in_{M}_holds_
ex_g_being_Function_st_
(_g_in_{M}_&_f_._Seqts1_c=_g_._Seqts2_)
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: for f being Function st Seqts1 c= Seqts2 & f in {M} holds
ex g being Function st
( g in {M} & f . Seqts1 c= g . Seqts2 )
let f be Function; ::_thesis: ( Seqts1 c= Seqts2 & f in {M} implies ex g being Function st
( g in {M} & f . Seqts1 c= g . Seqts2 ) )
assume A1: ( Seqts1 c= Seqts2 & f in {M} ) ; ::_thesis: ex g being Function st
( g in {M} & f . Seqts1 c= g . Seqts2 )
then reconsider F = f as isotone Rule of S by TARSKI:def_1;
take g = f; ::_thesis: ( g in {M} & f . Seqts1 c= g . Seqts2 )
thus  g in {M} by A1; ::_thesis: f . Seqts1 c= g . Seqts2
 F . Seqts1 c= F . Seqts2 by A1, Def9;
hence  f . Seqts1 c= g . Seqts2 ; ::_thesis: verum
end;
hence  for b1 being RuleSet of S st b1 = {M} holds
b1 is isotone by Def10; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster functional isotone for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
existence
 ex b1 being RuleSet of S st b1 is isotone
proof
set M = the isotone Rule of S;
take D = { the isotone Rule of S}; ::_thesis: D is isotone
thus  D is isotone ; ::_thesis: verum
end;
end;


definition
let S be Language;
let D be RuleSet of S;
let Seqts be set ;
attrSeqts is D -derivable  means :Def11: :: FOMODEL4:def 11
Seqts is {} ,D -derivable ;
end;

:: deftheorem Def11 defines -derivable FOMODEL4:def_11_:_
 for S being Language
for D being RuleSet of S
for Seqts being set holds
( Seqts is D -derivable iff Seqts is {} ,D -derivable );

registration
let S be Language;
let D be RuleSet of S;
clusterD -derivable  ->  {} ,D -derivable for set ;
coherence
 for b1 being set st b1 is D -derivable holds
b1 is {} ,D -derivable by Def11;
cluster {} ,D -derivable  -> D -derivable for set ;
coherence
 for b1 being set st b1 is {} ,D -derivable holds
b1 is D -derivable by Def11;
end;

registration
let S be Language;
let D be RuleSet of S;
let Seqts be empty set ;
clusterSeqts,D -derivable  -> D -derivable for set ;
coherence
 for b1 being set st b1 is Seqts,D -derivable holds
b1 is D -derivable ;
end;

definition
let S be Language;
let D be RuleSet of S;
let X, phi be set ;
attrphi is X,D -provable  means :Def12: :: FOMODEL4:def 12
( {[X,phi]} is D -derivable or ex seqt being set st
( seqt `1 c= X & seqt `2 = phi & {seqt} is D -derivable ) );
end;

:: deftheorem Def12 defines -provable FOMODEL4:def_12_:_
 for S being Language
for D being RuleSet of S
for X, phi being set holds
( phi is X,D -provable iff ( {[X,phi]} is D -derivable or ex seqt being set st
( seqt `1 c= X & seqt `2 = phi & {seqt} is D -derivable ) ) );

definition
let S be Language;
let D be RuleSet of S;
let X, x be set ;
redefine attr x is X,D -provable  means :Def13: :: FOMODEL4:def 13
 ex seqt being set st
( seqt `1 c= X & seqt `2 = x & {seqt} is D -derivable );
compatibility
 ( x is X,D -provable iff ex seqt being set st
( seqt `1 c= X & seqt `2 = x & {seqt} is D -derivable ) )
proof
defpred S1[] means  ex seqt being set st
( seqt `1 c= X & seqt `2 = x & {seqt} is D -derivable );
thus  ( x is X,D -provable implies S1[] ) ::_thesis: ( ex seqt being set st
( seqt `1 c= X & seqt `2 = x & {seqt} is D -derivable ) implies x is X,D -provable )
proof
assume A1: x is X,D -provable ; ::_thesis: S1[]
percases ( {[X,x]} is D -derivable or not {[X,x]} is D -derivable ) ;
supposeA2: {[X,x]} is D -derivable ; ::_thesis: S1[]
take seqt = [X,x]; ::_thesis: ( seqt `1 c= X & seqt `2 = x & {seqt} is D -derivable )
thus  ( seqt `1 c= X & seqt `2 = x ) by MCART_1:7; ::_thesis: {seqt} is D -derivable
thus  {seqt} is D -derivable by A2; ::_thesis: verum
end;
suppose not {[X,x]} is D -derivable ; ::_thesis: S1[]
hence  S1[] by Def12, A1; ::_thesis: verum
end;
end;
end;
assume  S1[] ; ::_thesis: x is X,D -provable
hence  x is X,D -provable by Def12; ::_thesis: verum
end;
end;

:: deftheorem Def13 defines -provable FOMODEL4:def_13_:_
 for S being Language
for D being RuleSet of S
for X, x being set holds
( x is X,D -provable iff ex seqt being set st
( seqt `1 c= X & seqt `2 = x & {seqt} is D -derivable ) );

definition
let S be Language;
let D be RuleSet of S;
let R be Rule of S;
attrR is D -macro  means :: FOMODEL4:def 14
 for Seqs being Subset of (S -sequents) holds R . Seqs is Seqs,D -derivable ;
end;

:: deftheorem  defines -macro FOMODEL4:def_14_:_
 for S being Language
for D being RuleSet of S
for R being Rule of S holds
( R is D -macro iff for Seqs being Subset of (S -sequents) holds R . Seqs is Seqs,D -derivable );

definition
let S be Language;
let D be RuleSet of S;
let Phi be set ;
func(Phi,D) -termEq  ->  set  equals :: FOMODEL4:def 15
 {  [t1,t2] where t1, t2 is termal string of S : (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is Phi,D -provable  }  ;
coherence
  {  [t1,t2] where t1, t2 is termal string of S : (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is Phi,D -provable  }  is set ;
end;

:: deftheorem  defines -termEq FOMODEL4:def_15_:_
 for S being Language
for D being RuleSet of S
for Phi being set holds (Phi,D) -termEq =  {  [t1,t2] where t1, t2 is termal string of S : (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is Phi,D -provable  }  ;

definition
let S be Language;
let D be RuleSet of S;
let Phi be set ;
attrPhi is D -expanded  means :Def16: :: FOMODEL4:def 16
 for x being set st x is Phi,D -provable holds
{x} c= Phi;
end;

:: deftheorem Def16 defines -expanded FOMODEL4:def_16_:_
 for S being Language
for D being RuleSet of S
for Phi being set holds
( Phi is D -expanded iff for x being set st x is Phi,D -provable holds
{x} c= Phi );

definition
let S be Language;
let x be set ;
attrx is S -null  means :Def17: :: FOMODEL4:def 17
verum;
end;

:: deftheorem Def17 defines -null FOMODEL4:def_17_:_
 for S being Language
for x being set holds
( x is S -null iff verum );

Lm5:  for X, Y, x being set
for S being Language
for D being RuleSet of S st X c= Y & x is X,D -provable holds
x is Y,D -provable
proof
let X, Y, x be set ; ::_thesis: for S being Language
for D being RuleSet of S st X c= Y & x is X,D -provable holds
x is Y,D -provable
let S be Language; ::_thesis: for D being RuleSet of S st X c= Y & x is X,D -provable holds
x is Y,D -provable
let D be RuleSet of S; ::_thesis: ( X c= Y & x is X,D -provable implies x is Y,D -provable )
assume A1: X c= Y ; ::_thesis: ( not x is X,D -provable or x is Y,D -provable )
assume  x is X,D -provable ; ::_thesis: x is Y,D -provable
then consider seqt being set  such that
A2: ( seqt `1 c= X & seqt `2 = x & {seqt} is D -derivable ) by Def13;
 ( seqt `1 c= Y & seqt `2 = x & {seqt} is D -derivable ) by A2, A1, XBOOLE_1:1;
hence  x is Y,D -provable by Def13; ::_thesis: verum
end;

definition
let S be Language;
let D be RuleSet of S;
let Phi be set ;
:: original: -termEq
redefine func(Phi,D) -termEq  ->  Relation of (AllTermsOf S);
coherence
 (Phi,D) -termEq is Relation of (AllTermsOf S)
proof
now__::_thesis:_for_x_being_set_st_x_in_(Phi,D)_-termEq_holds_
x_in_[:(AllTermsOf_S),(AllTermsOf_S):]
let x be set ; ::_thesis: ( x in (Phi,D) -termEq implies x in [:(AllTermsOf S),(AllTermsOf S):] )
assume  x in (Phi,D) -termEq ; ::_thesis: x in [:(AllTermsOf S),(AllTermsOf S):]
then consider t1, t2 being termal string of S such that
A1: ( x = [t1,t2] & (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is Phi,D -provable ) ;
reconsider t1b = t1 as Element of AllTermsOf S by FOMODEL1:def_32;
reconsider t2b = t2 as Element of AllTermsOf S by FOMODEL1:def_32;
 x = [t1b,t2b] by A1;
hence  x in [:(AllTermsOf S),(AllTermsOf S):] ; ::_thesis: verum
end;
hence  (Phi,D) -termEq is Relation of (AllTermsOf S) by TARSKI:def_3; ::_thesis: verum
end;
end;

registration
let S be Language;
let phi be wff string of S;
let Phi1, Phi2 be finite Subset of (AllFormulasOf S);
cluster[(Phi1 \/ Phi2),phi] -> S -sequent-like ;
coherence
 [(Phi1 \/ Phi2),phi] is S -sequent-like
proof
set AF = AllFormulasOf S;
reconsider Phi = Phi1 \/ Phi2 as finite Subset of (AllFormulasOf S) ;
 [Phi,phi] in S -sequents ;
hence  [(Phi1 \/ Phi2),phi] is S -sequent-like ; ::_thesis: verum
end;
end;

definition
let S be Language;
let x be empty set ;
let phi be wff string of S;
:: original: [
redefine func[x,phi] ->  Element of S -sequents ;
coherence
 [x,phi] is Element of S -sequents
proof
reconsider premises = x as finite Subset of ((AllSymbolsOf S) *) by XBOOLE_1:2;
 premises c= AllFormulasOf S by XBOOLE_1:2;
then  [premises,phi] in S -sequents ;
hence  [x,phi] is Element of S -sequents ; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster{} /\ S -> S -sequents-like for set ;
coherence
 for b1 being set st b1 = {} /\ S holds
b1 is S -sequents-like
proof
set Q = S -sequents ;
set IT = {} /\ S;
reconsider ITT = {} /\ S as Subset of (S -sequents) by XBOOLE_1:2;
 ITT is S -sequents-like ;
hence  for b1 being set st b1 = {} /\ S holds
b1 is S -sequents-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
clusterS -null for set ;
existence
 ex b1 being set st b1 is S -null by Def17;
end;

registration
let S be Language;
clusterS -sequent-like  -> S -null for set ;
coherence
 for b1 being set st b1 is S -sequent-like holds
b1 is S -null by Def17;
end;

registration
let S be Language;
cluster  -> S -null for Element of S -sequents ;
coherence
 for b1 being Element of S -sequents holds b1 is S -null ;
end;

registration
let m be Nat;
let S be Language;
let D be RuleSet of S;
let X be set ;
cluster((m,D) -derivables) . X -> S -sequents-like ;
coherence
 ((m,D) -derivables) . X is S -sequents-like
proof
set Q = S -sequents ;
reconsider f = (m,D) -derivables as Function of (bool (S -sequents)),(bool (S -sequents)) ;
percases ( not X in bool (S -sequents) or X in bool (S -sequents) ) ;
suppose not X in bool (S -sequents) ; ::_thesis: ((m,D) -derivables) . X is S -sequents-like
then  not X in dom f ;
then  f . X = {} by FUNCT_1:def_2;
then  f . X c= S -sequents by XBOOLE_1:2;
hence  ((m,D) -derivables) . X is S -sequents-like ; ::_thesis: verum
end;
suppose X in bool (S -sequents) ; ::_thesis: ((m,D) -derivables) . X is S -sequents-like
then reconsider XX = X as Element of bool (S -sequents) ;
 f . XX is Element of bool (S -sequents) ;
hence  ((m,D) -derivables) . X is S -sequents-like ; ::_thesis: verum
end;
end;
end;
end;

registration
let S be Language;
let Y be set ;
let X be S -sequents-like set ;
clusterX /\ Y -> S -sequents-like for set ;
coherence
 for b1 being set st b1 = X /\ Y holds
b1 is S -sequents-like
proof
set Q = S -sequents ;
 X c= S -sequents by Def3;
then  X /\ Y c= S -sequents by XBOOLE_1:1;
hence  for b1 being set st b1 = X /\ Y holds
b1 is S -sequents-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
let D be RuleSet of S;
let m be Nat;
let X be set ;
clusterm,X,D -derivable  -> S -sequent-like for set ;
coherence
 for b1 being set st b1 is m,X,D -derivable holds
b1 is S -sequent-like
proof
set O = OneStep D;
set All = union (((OneStep D) [*]) .: {X});
set Q = S -sequents ;
A1: ((m,D) -derivables) . X c= S -sequents by Def3;
let x be set ; ::_thesis: ( x is m,X,D -derivable implies x is S -sequent-like )
assume  x is m,X,D -derivable ; ::_thesis: x is S -sequent-like
then  x in ((m,D) -derivables) . X by Def7;
hence  x is S -sequent-like by A1; ::_thesis: verum
end;
end;

registration
let S be Language;
let D be RuleSet of S;
let Phi1, Phi2 be set ;
clusterPhi1 \ Phi2,D -provable  -> Phi1,D -provable for set ;
coherence
 for b1 being set st b1 is Phi1 \ Phi2,D -provable holds
b1 is Phi1,D -provable by Lm5;
end;

registration
let S be Language;
let D be RuleSet of S;
let Phi1, Phi2 be set ;
clusterPhi1 \ Phi2,D -provable  -> Phi1 \/ Phi2,D -provable for set ;
coherence
 for b1 being set st b1 is Phi1 \ Phi2,D -provable holds
b1 is Phi1 \/ Phi2,D -provable by Lm5, XBOOLE_1:7;
end;

registration
let S be Language;
let D be RuleSet of S;
let Phi1, Phi2 be set ;
clusterPhi1 /\ Phi2,D -provable  -> Phi1,D -provable for set ;
coherence
 for b1 being set st b1 is Phi1 /\ Phi2,D -provable holds
b1 is Phi1,D -provable by Lm5;
end;

registration
let S be Language;
let D be RuleSet of S;
let X be set ;
let x be Subset of X;
clusterx,D -provable  -> X,D -provable for set ;
coherence
 for b1 being set st b1 is x,D -provable holds
b1 is X,D -provable
proof
let y be set ; ::_thesis: ( y is x,D -provable implies y is X,D -provable )
assume  y is x,D -provable ; ::_thesis: y is X,D -provable
then consider seqt being set  such that
A1: ( seqt `1 c= x & seqt `2 = y & {seqt} is D -derivable ) by Def13;
 ( seqt `1 c= X & seqt `2 = y & {seqt} is D -derivable ) by A1, XBOOLE_1:1;
hence  y is X,D -provable by Def13; ::_thesis: verum
end;
end;

registration
let S be Language;
let premises be finite Subset of (AllFormulasOf S);
let phi be wff string of S;
cluster[premises,phi] -> S -sequent-like for set ;
coherence
 for b1 being set st b1 = [premises,phi] holds
b1 is S -sequent-like
proof
set x = [premises,phi];
set Q = S -sequents ;
 [premises,phi] in S -sequents ;
hence  for b1 being set st b1 = [premises,phi] holds
b1 is S -sequent-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
let phi1, phi2 be wff string of S;
cluster[{phi1},phi2] -> S -sequent-like for set ;
coherence
 for b1 being set st b1 = [{phi1},phi2] holds
b1 is S -sequent-like
proof
set AF = AllFormulasOf S;
reconsider phi11 = phi1 as Element of AllFormulasOf S by FOMODEL2:16;
reconsider Phi = {phi11} as finite Subset of (AllFormulasOf S) ;
 [Phi,phi2] is S -sequent-like ;
hence  for b1 being set st b1 = [{phi1},phi2] holds
b1 is S -sequent-like ; ::_thesis: verum
end;
let phi3 be wff string of S;
cluster[{phi1,phi2},phi3] -> S -sequent-like for set ;
coherence
 for b1 being set st b1 = [{phi1,phi2},phi3] holds
b1 is S -sequent-like
proof
set AF = AllFormulasOf S;
reconsider phi11 = phi1, phi22 = phi2 as Element of AllFormulasOf S by FOMODEL2:16;
reconsider Phi = {phi11} \/ {phi22} as finite Subset of (AllFormulasOf S) ;
 ( [Phi,phi2] is S -sequent-like & Phi = {phi11,phi22} ) by ENUMSET1:1;
hence  for b1 being set st b1 = [{phi1,phi2},phi3] holds
b1 is S -sequent-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
let phi1, phi2 be wff string of S;
let Phi be finite Subset of (AllFormulasOf S);
cluster[(Phi \/ {phi1}),phi2] -> S -sequent-like for set ;
coherence
 for b1 being set st b1 = [(Phi \/ {phi1}),phi2] holds
b1 is S -sequent-like
proof
set AF = AllFormulasOf S;
reconsider phi11 = phi1 as Element of AllFormulasOf S by FOMODEL2:16;
reconsider Phi2 = Phi \/ {phi11} as finite Subset of (AllFormulasOf S) ;
 [Phi2,phi2] is S -sequent-like ;
hence  for b1 being set st b1 = [(Phi \/ {phi1}),phi2] holds
b1 is S -sequent-like ; ::_thesis: verum
end;
end;

Lm6:  for X being set
for S being Language
for D being RuleSet of S st X c= S -sequents holds
(OneStep D) . X = union (union  {  (R .: {X}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  )
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S st X c= S -sequents holds
(OneStep D) . X = union (union  {  (R .: {X}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  )
let S be Language; ::_thesis: for D being RuleSet of S st X c= S -sequents holds
(OneStep D) . X = union (union  {  (R .: {X}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  )
let D be RuleSet of S; ::_thesis: ( X c= S -sequents implies (OneStep D) . X = union (union  {  (R .: {X}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  ) )
set Q = S -sequents ;
set F =  {  (R .: {X}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  ;
set O = OneStep D;
assume  X c= S -sequents ; ::_thesis: (OneStep D) . X = union (union  {  (R .: {X}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  )
then reconsider Seqts = X as Element of bool (S -sequents) ;
reconsider DD = D as Subset-Family of [:(bool (S -sequents)),(bool (S -sequents)):] by Lm1;
(OneStep D) . Seqts = union ((union DD) .: {Seqts}) by Def4
.= union (union  {  (R .: {X}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  ) by RELSET_2:34 ;
hence  (OneStep D) . X = union (union  {  (R .: {X}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  ) ; ::_thesis: verum
end;

Lm7:  for S being Language
for R being Rule of S holds R = OneStep {R}
proof
let S be Language; ::_thesis: for R being Rule of S holds R = OneStep {R}
let R be Rule of S; ::_thesis: R = OneStep {R}
set IT = OneStep {R};
A1: dom R = bool (S -sequents) by FUNCT_2:def_1
.= dom (OneStep {R}) by FUNCT_2:def_1 ;
now__::_thesis:_for_x_being_set_st_x_in_dom_R_holds_
R_._x_=_(OneStep_{R})_._x
let x be set ; ::_thesis: ( x in dom R implies R . x = (OneStep {R}) . x )
assume A2: x in dom R ; ::_thesis: R . x = (OneStep {R}) . x
thus R . x = union {(R . x)} by ZFMISC_1:25
.= union (Im (R,x)) by A2, FUNCT_1:59
.= union ((union {R}) .: {x}) by ZFMISC_1:25
.= (OneStep {R}) . x by Def4, A2 ; ::_thesis: verum
end;
hence  R = OneStep {R} by A1, FUNCT_1:2; ::_thesis: verum
end;

Lm8:  for y being set
for S being Language
for D being RuleSet of S
for Seqts being Subset of (S -sequents) st {y} is Seqts,D -derivable holds
ex mm being Element of NAT st y is mm,Seqts,D -derivable
proof
let y be set ; ::_thesis: for S being Language
for D being RuleSet of S
for Seqts being Subset of (S -sequents) st {y} is Seqts,D -derivable holds
ex mm being Element of NAT st y is mm,Seqts,D -derivable
let S be Language; ::_thesis: for D being RuleSet of S
for Seqts being Subset of (S -sequents) st {y} is Seqts,D -derivable holds
ex mm being Element of NAT st y is mm,Seqts,D -derivable
let D be RuleSet of S; ::_thesis: for Seqts being Subset of (S -sequents) st {y} is Seqts,D -derivable holds
ex mm being Element of NAT st y is mm,Seqts,D -derivable
let Seqts be Subset of (S -sequents); ::_thesis: ( {y} is Seqts,D -derivable implies ex mm being Element of NAT st y is mm,Seqts,D -derivable )
set X = Seqts;
set Q = S -sequents ;
set O = OneStep D;
set I = D -iterators ;
assume  {y} is Seqts,D -derivable ; ::_thesis: ex mm being Element of NAT st y is mm,Seqts,D -derivable
then  {y} c= union (((OneStep D) [*]) .: {Seqts}) by Def6;
then  y in union (((OneStep D) [*]) .: {Seqts}) by ZFMISC_1:31;
then consider Y being set  such that
A1: ( y in Y & Y in ((OneStep D) [*]) .: {Seqts} ) by TARSKI:def_4;
 rng (OneStep D) c= dom (OneStep D) by Lm3;
then  (OneStep D) [*] = union (D -iterators) by FOMODEL0:17;
then  ((OneStep D) [*]) .: {Seqts} = union  {  (R .: {Seqts}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D -iterators  }  by RELSET_2:34;
then consider Z being set  such that
A2: ( Y in Z & Z in  {  (R .: {Seqts}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D -iterators  }  ) by A1, TARSKI:def_4;
consider f being Subset of [:(bool (S -sequents)),(bool (S -sequents)):] such that
A3: ( Z = f .: {Seqts} & f in D -iterators ) by A2;
consider mm being Element of NAT  such that
A4: f = iter ((OneStep D),mm) by A3;
take  mm ; ::_thesis: y is mm,Seqts,D -derivable
 iter ((OneStep D),mm) is Function of (bool (S -sequents)),(bool (S -sequents)) by FUNCT_7:83;
then A5: dom (iter ((OneStep D),mm)) = bool (S -sequents) by FUNCT_2:def_1;
 ( y in Y & Y in Im ((iter ((OneStep D),mm)),Seqts) ) by A1, A2, A4, A3;
then  ( y in Y & Y in {((iter ((OneStep D),mm)) . Seqts)} ) by A5, FUNCT_1:59;
then  y in ((mm,D) -derivables) . Seqts by TARSKI:def_1;
hence  y is mm,Seqts,D -derivable by Def7; ::_thesis: verum
end;

Lm9:  for m being Nat
for X being set
for S being Language
for D being RuleSet of S st X c= S -sequents holds
((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X})
proof
let m be Nat; ::_thesis: for X being set
for S being Language
for D being RuleSet of S st X c= S -sequents holds
((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X})
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S st X c= S -sequents holds
((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X})
let S be Language; ::_thesis: for D being RuleSet of S st X c= S -sequents holds
((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X})
let D be RuleSet of S; ::_thesis: ( X c= S -sequents implies ((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X}) )
set O = OneStep D;
set RH = union (((OneStep D) [*]) .: {X});
set Q = S -sequents ;
assume  X c= S -sequents ; ::_thesis: ((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X})
then A1: X in dom (iter ((OneStep D),m)) by Lm3;
reconsider f = union {(iter ((OneStep D),m))} as Function by ZFMISC_1:25;
A2: (iter ((OneStep D),m)) . X = union {((iter ((OneStep D),m)) . X)} by ZFMISC_1:25
.= union (Im ((iter ((OneStep D),m)),X)) by A1, FUNCT_1:59
.= union (f .: {X}) by ZFMISC_1:25 ;
reconsider mm = m as Element of NAT by ORDINAL1:def_12;
 iter ((OneStep D),mm) = iter ((OneStep D),m) ;
then  iter ((OneStep D),m) in D -iterators ;
then  {(iter ((OneStep D),m))} c= D -iterators by ZFMISC_1:31;
then  union {(iter ((OneStep D),m))} c= union (D -iterators) by ZFMISC_1:77;
then  f .: {X} c= (union (D -iterators)) .: {X} by RELAT_1:124;
then A3: ((m,D) -derivables) . X c= union ((union (D -iterators)) .: {X}) by A2, ZFMISC_1:77;
 rng (OneStep D) c= dom (OneStep D) by Lm3;
hence  ((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X}) by A3, FOMODEL0:17; ::_thesis: verum
end;

Lm10:  for X being set
for S being Language
for D being RuleSet of S holds union (((OneStep D) [*]) .: {X}) = union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S holds union (((OneStep D) [*]) .: {X}) = union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }
let S be Language; ::_thesis: for D being RuleSet of S holds union (((OneStep D) [*]) .: {X}) = union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }
let D be RuleSet of S; ::_thesis: union (((OneStep D) [*]) .: {X}) = union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }
set F =  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ;
set LH = union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ;
set Q = S -sequents ;
set O = OneStep D;
set RH = union (((OneStep D) [*]) .: {X});
percases ( not X in bool (S -sequents) or X in bool (S -sequents) ) ;
supposeA1: not X in bool (S -sequents) ; ::_thesis: union (((OneStep D) [*]) .: {X}) = union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }
then  {X} misses bool (S -sequents) by ZFMISC_1:50;
then  {X} misses dom ((OneStep D) [*]) by XBOOLE_1:63;
then  {X} /\ (dom ((OneStep D) [*])) = {} by XBOOLE_0:def_7;
then ((OneStep D) [*]) .: {X} = ((OneStep D) [*]) .: {} by RELAT_1:112
.= {} ;
then reconsider E = ((OneStep D) [*]) .: {X} as empty set ;
now__::_thesis:_for_x_being_set_st_x_in__{__(((mm,D)_-derivables)_._X)_where_mm_is_Element_of_NAT_:_verum__}__holds_
x_in_{{}}
let x be set ; ::_thesis: ( x in  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  implies x in {{}} )
assume  x in  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ; ::_thesis: x in {{}}
then consider mm being Element of NAT  such that
A2: x = ((mm,D) -derivables) . X ;
 not X in dom ((mm,D) -derivables) by A1;
then  x = {} by A2, FUNCT_1:def_2;
hence  x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
then   {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  c= {{}} by TARSKI:def_3;
then  union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  c= union {{}} by ZFMISC_1:77;
then  union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  c= {} by ZFMISC_1:25;
then  ( union E = {} & union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  = {} ) ;
hence  union (((OneStep D) [*]) .: {X}) = union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ; ::_thesis: verum
end;
supposeA3: X in bool (S -sequents) ; ::_thesis: union (((OneStep D) [*]) .: {X}) = union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }
then reconsider Seqts = X as Element of bool (S -sequents) ;
 for Y being set st Y in  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  holds
Y c= union (((OneStep D) [*]) .: {X})
proof
let Y be set ; ::_thesis: ( Y in  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  implies Y c= union (((OneStep D) [*]) .: {X}) )
assume  Y in  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ; ::_thesis: Y c= union (((OneStep D) [*]) .: {X})
then consider mm being Element of NAT  such that
A4: Y = ((mm,D) -derivables) . X ;
thus  Y c= union (((OneStep D) [*]) .: {X}) by A4, A3, Lm9; ::_thesis: verum
end;
then A5: union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  c= union (((OneStep D) [*]) .: {X}) by ZFMISC_1:76;
now__::_thesis:_for_y_being_set_st_y_in_union_(((OneStep_D)_[*])_.:_{X})_holds_
y_in_union__{__(((mm,D)_-derivables)_._X)_where_mm_is_Element_of_NAT_:_verum__}_
let y be set ; ::_thesis: ( y in union (((OneStep D) [*]) .: {X}) implies y in union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  )
assume  y in union (((OneStep D) [*]) .: {X}) ; ::_thesis: y in union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }
then  {y} c= union (((OneStep D) [*]) .: {X}) by ZFMISC_1:31;
then  {y} is Seqts,D -derivable by Def6;
then consider mm being Element of NAT  such that
A6: y is mm,Seqts,D -derivable by Lm8;
set Y = ((mm,D) -derivables) . Seqts;
 ( y in ((mm,D) -derivables) . Seqts & ((mm,D) -derivables) . Seqts in  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ) by Def7, A6;
hence  y in union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  by TARSKI:def_4; ::_thesis: verum
end;
then  union (((OneStep D) [*]) .: {X}) c= union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  by TARSKI:def_3;
hence  union (((OneStep D) [*]) .: {X}) = union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  by A5, XBOOLE_0:def_10; ::_thesis: verum
end;
end;
end;

Lm11:  for m being Nat
for X being set
for S being Language
for D being RuleSet of S holds ((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X})
proof
let m be Nat; ::_thesis: for X being set
for S being Language
for D being RuleSet of S holds ((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X})
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S holds ((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X})
let S be Language; ::_thesis: for D being RuleSet of S holds ((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X})
let D be RuleSet of S; ::_thesis: ((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X})
set F =  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ;
set LH = union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ;
set Q = S -sequents ;
set O = OneStep D;
set RH = union (((OneStep D) [*]) .: {X});
reconsider mm = m as Element of NAT by ORDINAL1:def_12;
 ((mm,D) -derivables) . X in  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ;
then  ((mm,D) -derivables) . X c= union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  by ZFMISC_1:74;
hence  ((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X}) by Lm10; ::_thesis: verum
end;

Lm12:  for m being Nat
for x, X being set
for S being Language
for D being RuleSet of S st x is m,X,D -derivable holds
{x} is X,D -derivable
proof
let m be Nat; ::_thesis: for x, X being set
for S being Language
for D being RuleSet of S st x is m,X,D -derivable holds
{x} is X,D -derivable
let x, X be set ; ::_thesis: for S being Language
for D being RuleSet of S st x is m,X,D -derivable holds
{x} is X,D -derivable
let S be Language; ::_thesis: for D being RuleSet of S st x is m,X,D -derivable holds
{x} is X,D -derivable
let D be RuleSet of S; ::_thesis: ( x is m,X,D -derivable implies {x} is X,D -derivable )
set Q = S -sequents ;
set O = OneStep D;
set RH = union (((OneStep D) [*]) .: {X});
assume  x is m,X,D -derivable ; ::_thesis: {x} is X,D -derivable
then A1: x in ((m,D) -derivables) . X by Def7;
 ((m,D) -derivables) . X c= union (((OneStep D) [*]) .: {X}) by Lm11;
then  {x} c= union (((OneStep D) [*]) .: {X}) by A1, ZFMISC_1:31;
hence  {x} is X,D -derivable by Def6; ::_thesis: verum
end;

Lm13:  for S being Language
for D1, D2 being RuleSet of S
for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
(OneStep D1) . Seqts1 c= (OneStep D2) . Seqts2
proof
let S be Language; ::_thesis: for D1, D2 being RuleSet of S
for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
(OneStep D1) . Seqts1 c= (OneStep D2) . Seqts2
let D1, D2 be RuleSet of S; ::_thesis: for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
(OneStep D1) . Seqts1 c= (OneStep D2) . Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) implies (OneStep D1) . Seqts1 c= (OneStep D2) . Seqts2 )
set Q = S -sequents ;
set U1 = union D1;
set U2 = union D2;
set O1 = OneStep D1;
set O2 = OneStep D2;
set F1 =  {  (R .: {Seqts1}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D1  }  ;
set F2 =  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  ;
set X1 = Seqts1;
set X2 = Seqts2;
A1: ( (OneStep D1) . Seqts1 = union (union  {  (R .: {Seqts1}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D1  }  ) & (OneStep D2) . Seqts2 = union (union  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  ) ) by Lm6;
assume A2: ( Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) ) ; ::_thesis: (OneStep D1) . Seqts1 c= (OneStep D2) . Seqts2
now__::_thesis:_for_z_being_set_st_z_in_union_(union__{__(R_.:_{Seqts1})_where_R_is_Subset_of_[:(bool_(S_-sequents)),(bool_(S_-sequents)):]_:_R_in_D1__}__)_holds_
z_in_union_(union__{__(R_.:_{Seqts2})_where_R_is_Subset_of_[:(bool_(S_-sequents)),(bool_(S_-sequents)):]_:_R_in_D2__}__)
let z be set ; ::_thesis: ( z in union (union  {  (R .: {Seqts1}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D1  }  ) implies z in union (union  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  ) )
assume  z in union (union  {  (R .: {Seqts1}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D1  }  ) ; ::_thesis: z in union (union  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  )
then consider x being set  such that
A3: ( z in x & x in union  {  (R .: {Seqts1}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D1  }  ) by TARSKI:def_4;
consider X being set  such that
A4: ( x in X & X in  {  (R .: {Seqts1}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D1  }  ) by A3, TARSKI:def_4;
consider R being Subset of [:(bool (S -sequents)),(bool (S -sequents)):] such that
A5: ( X = R .: {Seqts1} & R in D1 ) by A4;
reconsider RR = R as Rule of S by A5;
 ( X = Im (RR,Seqts1) & R in D1 & Seqts1 in dom RR ) by A5, Lm3;
then A6: X = {(RR . Seqts1)} by FUNCT_1:59;
now__::_thesis:_ex_g_being_Function_st_
(_g_in_D2_&_RR_._Seqts1_c=_g_._Seqts2_)
percases ( D2 is isotone or not D2 is isotone ) ;
suppose D2 is isotone ; ::_thesis: ex g being Function st
( g in D2 & RR . Seqts1 c= g . Seqts2 )
hence  ex g being Function st
( g in D2 & RR . Seqts1 c= g . Seqts2 ) by A5, A2, Def10; ::_thesis: verum
end;
suppose not D2 is isotone ; ::_thesis: ex g being Function st
( g in D2 & RR . Seqts1 c= g . Seqts2 )
then consider g being Function such that
A7: ( g in D1 & RR . Seqts1 c= g . Seqts2 ) by A5, A2, Def10;
thus  ex g being Function st
( g in D2 & RR . Seqts1 c= g . Seqts2 ) by A7, A2; ::_thesis: verum
end;
end;
end;
then consider g being Function such that
A8: ( g in D2 & RR . Seqts1 c= g . Seqts2 ) ;
reconsider Rg = g as Rule of S by A8;
A9: Seqts2 in dom Rg by Lm3;
A10: x c= g . Seqts2 by A8, A6, A4, TARSKI:def_1;
 Im (Rg,Seqts2) in  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  by A8;
then  {(Rg . Seqts2)} in  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  by A9, FUNCT_1:59;
then  {{(Rg . Seqts2)}} c=  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  by ZFMISC_1:31;
then  union {{(g . Seqts2)}} c= union  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  by ZFMISC_1:77;
then  {(g . Seqts2)} c= union  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  by ZFMISC_1:25;
then  union {(g . Seqts2)} c= union (union  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  ) by ZFMISC_1:77;
then  g . Seqts2 c= union (union  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  ) by ZFMISC_1:25;
then  x c= union (union  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  ) by A10, XBOOLE_1:1;
hence  z in union (union  {  (R .: {Seqts2}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D2  }  ) by A3; ::_thesis: verum
end;
hence  (OneStep D1) . Seqts1 c= (OneStep D2) . Seqts2 by A1, TARSKI:def_3; ::_thesis: verum
end;

Lm14:  for m being Nat
for S being Language
for D1, D2 being RuleSet of S
for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . Seqts1 c= ((m,D2) -derivables) . Seqts2
proof
let m be Nat; ::_thesis: for S being Language
for D1, D2 being RuleSet of S
for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . Seqts1 c= ((m,D2) -derivables) . Seqts2
let S be Language; ::_thesis: for D1, D2 being RuleSet of S
for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . Seqts1 c= ((m,D2) -derivables) . Seqts2
let D1, D2 be RuleSet of S; ::_thesis: for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . Seqts1 c= ((m,D2) -derivables) . Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) implies ((m,D1) -derivables) . Seqts1 c= ((m,D2) -derivables) . Seqts2 )
set O1 = OneStep D1;
set O2 = OneStep D2;
set Q = S -sequents ;
set X1 = Seqts1;
set X2 = Seqts2;
assume A1: ( Seqts1 c= Seqts2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) ) ; ::_thesis: ((m,D1) -derivables) . Seqts1 c= ((m,D2) -derivables) . Seqts2
defpred S1[ Nat] means (($1,D1) -derivables) . Seqts1 c= (($1,D2) -derivables) . Seqts2;
A2: S1[ 0 ]
proof
A3: (iter ((OneStep D1),0)) . Seqts1 = (id (field (OneStep D1))) . Seqts1 by FUNCT_7:68
.= (id (bool (S -sequents))) . Seqts1 by Lm3
.= Seqts1 by FUNCT_1:17 ;
(iter ((OneStep D2),0)) . Seqts2 = (id (field (OneStep D2))) . Seqts2 by FUNCT_7:68
.= (id (bool (S -sequents))) . Seqts2 by Lm3
.= Seqts2 by FUNCT_1:17 ;
hence  S1[ 0 ] by A3, A1; ::_thesis: verum
end;
A4: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
A5: ( Seqts1 in dom (iter ((OneStep D1),n)) & Seqts2 in dom (iter ((OneStep D2),n)) ) by Lm3;
reconsider X11 = ((n,D1) -derivables) . Seqts1, X22 = ((n,D2) -derivables) . Seqts2 as Subset of (S -sequents) ;
assume A6: S1[n] ; ::_thesis: S1[n + 1]
A7: (((n + 1),D1) -derivables) . Seqts1 = ((OneStep D1) * (iter ((OneStep D1),n))) . Seqts1 by FUNCT_7:71
.= (OneStep D1) . X11 by A5, FUNCT_1:13 ;
(((n + 1),D2) -derivables) . Seqts2 = ((OneStep D2) * (iter ((OneStep D2),n))) . Seqts2 by FUNCT_7:71
.= (OneStep D2) . X22 by A5, FUNCT_1:13 ;
hence  S1[n + 1] by A7, A6, Lm13, A1; ::_thesis: verum
end;
 for k being Nat holds S1[k] from NAT_1:sch_2(A2, A4);
hence  ((m,D1) -derivables) . Seqts1 c= ((m,D2) -derivables) . Seqts2 ; ::_thesis: verum
end;

Lm15:  for m being Nat
for S being Language
for D1, D2 being RuleSet of S
for SQ1, SQ2 being b2 -sequents-like set st SQ1 c= SQ2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . SQ1 c= ((m,D2) -derivables) . SQ2
proof
let m be Nat; ::_thesis: for S being Language
for D1, D2 being RuleSet of S
for SQ1, SQ2 being b1 -sequents-like set st SQ1 c= SQ2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . SQ1 c= ((m,D2) -derivables) . SQ2
let S be Language; ::_thesis: for D1, D2 being RuleSet of S
for SQ1, SQ2 being S -sequents-like set st SQ1 c= SQ2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . SQ1 c= ((m,D2) -derivables) . SQ2
let D1, D2 be RuleSet of S; ::_thesis: for SQ1, SQ2 being S -sequents-like set st SQ1 c= SQ2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . SQ1 c= ((m,D2) -derivables) . SQ2
let SQ1, SQ2 be S -sequents-like set ; ::_thesis: ( SQ1 c= SQ2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) implies ((m,D1) -derivables) . SQ1 c= ((m,D2) -derivables) . SQ2 )
reconsider Seqts1 = SQ1, Seqts2 = SQ2 as Subset of (S -sequents) by Def3;
assume  ( SQ1 c= SQ2 & D1 c= D2 & ( D2 is isotone or D1 is isotone ) ) ; ::_thesis: ((m,D1) -derivables) . SQ1 c= ((m,D2) -derivables) . SQ2
then  ((m,D1) -derivables) . Seqts1 c= ((m,D2) -derivables) . Seqts2 by Lm14;
hence  ((m,D1) -derivables) . SQ1 c= ((m,D2) -derivables) . SQ2 ; ::_thesis: verum
end;

Lm16:  for m being Nat
for X being set
for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . X c= ((m,D2) -derivables) . X
proof
let m be Nat; ::_thesis: for X being set
for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . X c= ((m,D2) -derivables) . X
let X be set ; ::_thesis: for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . X c= ((m,D2) -derivables) . X
let S be Language; ::_thesis: for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
((m,D1) -derivables) . X c= ((m,D2) -derivables) . X
let D1, D2 be RuleSet of S; ::_thesis: ( D1 c= D2 & ( D2 is isotone or D1 is isotone ) implies ((m,D1) -derivables) . X c= ((m,D2) -derivables) . X )
set Q = S -sequents ;
set f1 = (m,D1) -derivables ;
set f2 = (m,D2) -derivables ;
assume A1: ( D1 c= D2 & ( D2 is isotone or D1 is isotone ) ) ; ::_thesis: ((m,D1) -derivables) . X c= ((m,D2) -derivables) . X
percases ( X in bool (S -sequents) or not X in bool (S -sequents) ) ;
suppose X in bool (S -sequents) ; ::_thesis: ((m,D1) -derivables) . X c= ((m,D2) -derivables) . X
then reconsider Seqts1 = X as Element of bool (S -sequents) ;
 ((m,D1) -derivables) . Seqts1 c= ((m,D2) -derivables) . Seqts1 by Lm14, A1;
hence  ((m,D1) -derivables) . X c= ((m,D2) -derivables) . X ; ::_thesis: verum
end;
suppose not X in bool (S -sequents) ; ::_thesis: ((m,D1) -derivables) . X c= ((m,D2) -derivables) . X
then  ( not X in dom ((m,D1) -derivables) & not X in dom ((m,D2) -derivables) ) ;
then  ( ((m,D1) -derivables) . X = {} & ((m,D2) -derivables) . X = {} ) by FUNCT_1:def_2;
hence  ((m,D1) -derivables) . X c= ((m,D2) -derivables) . X ; ::_thesis: verum
end;
end;
end;

Lm17:  for X being set
for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
union (((OneStep D1) [*]) .: {X}) c= union (((OneStep D2) [*]) .: {X})
proof
let X be set ; ::_thesis: for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
union (((OneStep D1) [*]) .: {X}) c= union (((OneStep D2) [*]) .: {X})
let S be Language; ::_thesis: for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) holds
union (((OneStep D1) [*]) .: {X}) c= union (((OneStep D2) [*]) .: {X})
let D1, D2 be RuleSet of S; ::_thesis: ( D1 c= D2 & ( D2 is isotone or D1 is isotone ) implies union (((OneStep D1) [*]) .: {X}) c= union (((OneStep D2) [*]) .: {X}) )
set F1 =  {  (((mm,D1) -derivables) . X) where mm is Element of NAT : verum  }  ;
set F2 =  {  (((mm,D2) -derivables) . X) where mm is Element of NAT : verum  }  ;
set O1 = OneStep D1;
set O2 = OneStep D2;
set Q = S -sequents ;
set LH = union (((OneStep D1) [*]) .: {X});
set RH = union (((OneStep D2) [*]) .: {X});
A1: ( union (((OneStep D1) [*]) .: {X}) = union  {  (((mm,D1) -derivables) . X) where mm is Element of NAT : verum  }  & union (((OneStep D2) [*]) .: {X}) = union  {  (((mm,D2) -derivables) . X) where mm is Element of NAT : verum  }  ) by Lm10;
assume A2: ( D1 c= D2 & ( D2 is isotone or D1 is isotone ) ) ; ::_thesis: union (((OneStep D1) [*]) .: {X}) c= union (((OneStep D2) [*]) .: {X})
now__::_thesis:_for_x_being_set_st_x_in__{__(((mm,D1)_-derivables)_._X)_where_mm_is_Element_of_NAT_:_verum__}__holds_
x_c=_union__{__(((mm,D2)_-derivables)_._X)_where_mm_is_Element_of_NAT_:_verum__}_
let x be set ; ::_thesis: ( x in  {  (((mm,D1) -derivables) . X) where mm is Element of NAT : verum  }  implies x c= union  {  (((mm,D2) -derivables) . X) where mm is Element of NAT : verum  }  )
assume  x in  {  (((mm,D1) -derivables) . X) where mm is Element of NAT : verum  }  ; ::_thesis: x c= union  {  (((mm,D2) -derivables) . X) where mm is Element of NAT : verum  }
then consider mm being Element of NAT  such that
A3: x = ((mm,D1) -derivables) . X ;
A4: x c= ((mm,D2) -derivables) . X by A2, Lm16, A3;
 ((mm,D2) -derivables) . X in  {  (((mm,D2) -derivables) . X) where mm is Element of NAT : verum  }  ;
then  ((mm,D2) -derivables) . X c= union  {  (((mm,D2) -derivables) . X) where mm is Element of NAT : verum  }  by ZFMISC_1:74;
hence  x c= union  {  (((mm,D2) -derivables) . X) where mm is Element of NAT : verum  }  by A4, XBOOLE_1:1; ::_thesis: verum
end;
hence  union (((OneStep D1) [*]) .: {X}) c= union (((OneStep D2) [*]) .: {X}) by A1, ZFMISC_1:76; ::_thesis: verum
end;

Lm18:  for Y, X being set
for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) & Y is X,D1 -derivable holds
Y is X,D2 -derivable
proof
let Y, X be set ; ::_thesis: for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) & Y is X,D1 -derivable holds
Y is X,D2 -derivable
let S be Language; ::_thesis: for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) & Y is X,D1 -derivable holds
Y is X,D2 -derivable
let D1, D2 be RuleSet of S; ::_thesis: ( D1 c= D2 & ( D2 is isotone or D1 is isotone ) & Y is X,D1 -derivable implies Y is X,D2 -derivable )
set O1 = OneStep D1;
set O2 = OneStep D2;
set Q = S -sequents ;
set LH = union (((OneStep D1) [*]) .: {X});
set RH = union (((OneStep D2) [*]) .: {X});
assume  ( D1 c= D2 & ( D2 is isotone or D1 is isotone ) & Y is X,D1 -derivable ) ; ::_thesis: Y is X,D2 -derivable
then  ( union (((OneStep D1) [*]) .: {X}) c= union (((OneStep D2) [*]) .: {X}) & Y c= union (((OneStep D1) [*]) .: {X}) ) by Def6, Lm17;
then  Y c= union (((OneStep D2) [*]) .: {X}) by XBOOLE_1:1;
hence  Y is X,D2 -derivable by Def6; ::_thesis: verum
end;

Lm19:  for Y being set
for S being Language
for R being Rule of S
for Seqts being Subset of (S -sequents) st Y c= R . Seqts holds
Y is Seqts,{R} -derivable
proof
let Y be set ; ::_thesis: for S being Language
for R being Rule of S
for Seqts being Subset of (S -sequents) st Y c= R . Seqts holds
Y is Seqts,{R} -derivable
let S be Language; ::_thesis: for R being Rule of S
for Seqts being Subset of (S -sequents) st Y c= R . Seqts holds
Y is Seqts,{R} -derivable
let R be Rule of S; ::_thesis: for Seqts being Subset of (S -sequents) st Y c= R . Seqts holds
Y is Seqts,{R} -derivable
let Seqts be Subset of (S -sequents); ::_thesis: ( Y c= R . Seqts implies Y is Seqts,{R} -derivable )
set D = {R};
set RR = (OneStep {R}) [*] ;
set Q = S -sequents ;
 ( Seqts in bool (S -sequents) & dom R = bool (S -sequents) ) by FUNCT_2:def_1;
then A1: {(R . Seqts)} = Im (R,Seqts) by FUNCT_1:59
.= R .: {Seqts} ;
 OneStep {R} c= (OneStep {R}) [*] by LANG1:18;
then A2: R c= (OneStep {R}) [*] by Lm7;
 {(R . Seqts)} c= ((OneStep {R}) [*]) .: {Seqts} by A1, A2, RELAT_1:124;
then  union {(R . Seqts)} c= union (((OneStep {R}) [*]) .: {Seqts}) by ZFMISC_1:77;
then A3: R . Seqts c= union (((OneStep {R}) [*]) .: {Seqts}) by ZFMISC_1:25;
assume  Y c= R . Seqts ; ::_thesis: Y is Seqts,{R} -derivable
then  Y c= union (((OneStep {R}) [*]) .: {Seqts}) by A3, XBOOLE_1:1;
hence  Y is Seqts,{R} -derivable by Def6; ::_thesis: verum
end;

Lm20:  for m, n being Nat
for Z being set
for S being Language
for D1, D2 being RuleSet of S
for SQ2, SQ1 being b4 -sequents-like set st D1 is isotone & D1 \/ D2 is isotone & SQ2 c= (iter ((OneStep D1),m)) . SQ1 & Z c= (iter ((OneStep D2),n)) . SQ2 holds
Z c= (iter ((OneStep (D1 \/ D2)),(m + n))) . SQ1
proof
let m, n be Nat; ::_thesis: for Z being set
for S being Language
for D1, D2 being RuleSet of S
for SQ2, SQ1 being b2 -sequents-like set st D1 is isotone & D1 \/ D2 is isotone & SQ2 c= (iter ((OneStep D1),m)) . SQ1 & Z c= (iter ((OneStep D2),n)) . SQ2 holds
Z c= (iter ((OneStep (D1 \/ D2)),(m + n))) . SQ1
let Z be set ; ::_thesis: for S being Language
for D1, D2 being RuleSet of S
for SQ2, SQ1 being b1 -sequents-like set st D1 is isotone & D1 \/ D2 is isotone & SQ2 c= (iter ((OneStep D1),m)) . SQ1 & Z c= (iter ((OneStep D2),n)) . SQ2 holds
Z c= (iter ((OneStep (D1 \/ D2)),(m + n))) . SQ1
let S be Language; ::_thesis: for D1, D2 being RuleSet of S
for SQ2, SQ1 being S -sequents-like set st D1 is isotone & D1 \/ D2 is isotone & SQ2 c= (iter ((OneStep D1),m)) . SQ1 & Z c= (iter ((OneStep D2),n)) . SQ2 holds
Z c= (iter ((OneStep (D1 \/ D2)),(m + n))) . SQ1
let D1, D2 be RuleSet of S; ::_thesis: for SQ2, SQ1 being S -sequents-like set st D1 is isotone & D1 \/ D2 is isotone & SQ2 c= (iter ((OneStep D1),m)) . SQ1 & Z c= (iter ((OneStep D2),n)) . SQ2 holds
Z c= (iter ((OneStep (D1 \/ D2)),(m + n))) . SQ1
let SQ2, SQ1 be S -sequents-like set ; ::_thesis: ( D1 is isotone & D1 \/ D2 is isotone & SQ2 c= (iter ((OneStep D1),m)) . SQ1 & Z c= (iter ((OneStep D2),n)) . SQ2 implies Z c= (iter ((OneStep (D1 \/ D2)),(m + n))) . SQ1 )
reconsider mm = m, nn = n as Element of NAT by ORDINAL1:def_12;
set D3 = D1 \/ D2;
set O1 = OneStep D1;
set O2 = OneStep D2;
set O3 = OneStep (D1 \/ D2);
set X = SQ1;
set Y = SQ2;
assume A1: ( D1 is isotone & D1 \/ D2 is isotone ) ; ::_thesis: ( not SQ2 c= (iter ((OneStep D1),m)) . SQ1 or not Z c= (iter ((OneStep D2),n)) . SQ2 or Z c= (iter ((OneStep (D1 \/ D2)),(m + n))) . SQ1 )
assume A2: ( SQ2 c= (iter ((OneStep D1),m)) . SQ1 & Z c= (iter ((OneStep D2),n)) . SQ2 ) ; ::_thesis: Z c= (iter ((OneStep (D1 \/ D2)),(m + n))) . SQ1
A3: ( D1 \/ D2 c= D1 \/ D2 & D1 c= D1 \/ D2 & D2 c= D1 \/ D2 ) by XBOOLE_1:7;
then  ((m,D1) -derivables) . SQ1 c= ((m,(D1 \/ D2)) -derivables) . SQ1 by Lm16, A1;
then A4: SQ2 c= ((m,(D1 \/ D2)) -derivables) . SQ1 by A2, XBOOLE_1:1;
A5: SQ1 in dom (iter ((OneStep (D1 \/ D2)),m)) by Lm3;
 ((n,D2) -derivables) . SQ2 c= ((n,(D1 \/ D2)) -derivables) . SQ2 by A3, A1, Lm16;
then A6: Z c= ((n,(D1 \/ D2)) -derivables) . SQ2 by A2, XBOOLE_1:1;
(((m + n),(D1 \/ D2)) -derivables) . SQ1 = ((iter ((OneStep (D1 \/ D2)),nn)) * (iter ((OneStep (D1 \/ D2)),mm))) . SQ1 by FUNCT_7:77
.= ((n,(D1 \/ D2)) -derivables) . ((iter ((OneStep (D1 \/ D2)),m)) . SQ1) by A5, FUNCT_1:13 ;
then  ((n,(D1 \/ D2)) -derivables) . SQ2 c= (((m + n),(D1 \/ D2)) -derivables) . SQ1 by A4, A1, Lm15;
hence  Z c= (iter ((OneStep (D1 \/ D2)),(m + n))) . SQ1 by A6, XBOOLE_1:1; ::_thesis: verum
end;

Lm21:  for m, n being Nat
for y, X, z being set
for S being Language
for D1, D2 being RuleSet of S st D1 is isotone & D1 \/ D2 is isotone & y is m,X,D1 -derivable & z is n,{y},D2 -derivable holds
z is m + n,X,D1 \/ D2 -derivable
proof
let m, n be Nat; ::_thesis: for y, X, z being set
for S being Language
for D1, D2 being RuleSet of S st D1 is isotone & D1 \/ D2 is isotone & y is m,X,D1 -derivable & z is n,{y},D2 -derivable holds
z is m + n,X,D1 \/ D2 -derivable
let y, X, z be set ; ::_thesis: for S being Language
for D1, D2 being RuleSet of S st D1 is isotone & D1 \/ D2 is isotone & y is m,X,D1 -derivable & z is n,{y},D2 -derivable holds
z is m + n,X,D1 \/ D2 -derivable
let S be Language; ::_thesis: for D1, D2 being RuleSet of S st D1 is isotone & D1 \/ D2 is isotone & y is m,X,D1 -derivable & z is n,{y},D2 -derivable holds
z is m + n,X,D1 \/ D2 -derivable
let D1, D2 be RuleSet of S; ::_thesis: ( D1 is isotone & D1 \/ D2 is isotone & y is m,X,D1 -derivable & z is n,{y},D2 -derivable implies z is m + n,X,D1 \/ D2 -derivable )
set Q = S -sequents ;
set D3 = D1 \/ D2;
set O1 = OneStep D1;
set O2 = OneStep D2;
set O3 = OneStep (D1 \/ D2);
assume A1: ( D1 is isotone & D1 \/ D2 is isotone ) ; ::_thesis: ( not y is m,X,D1 -derivable or not z is n,{y},D2 -derivable or z is m + n,X,D1 \/ D2 -derivable )
assume A2: ( y is m,X,D1 -derivable & z is n,{y},D2 -derivable ) ; ::_thesis: z is m + n,X,D1 \/ D2 -derivable
then A3: ( y in ((m,D1) -derivables) . X & z in ((n,D2) -derivables) . {y} ) by Def7;
 X in bool (S -sequents)
proof
assume  not X in bool (S -sequents) ; ::_thesis: contradiction
then  not X in dom ((m,D1) -derivables) ;
then A4: ((m,D1) -derivables) . X = {} by FUNCT_1:def_2;
thus  contradiction by A4, A2, Def7; ::_thesis: verum
end;
then reconsider SQ = X as Subset of (S -sequents) ;
reconsider yy = y as Element of S -sequents by Def2, A2;
 ( {yy} c= (iter ((OneStep D1),m)) . SQ & {z} c= (iter ((OneStep D2),n)) . {yy} ) by A3, ZFMISC_1:31;
then  {z} c= (iter ((OneStep (D1 \/ D2)),(m + n))) . SQ by Lm20, A1;
then  z in (((m + n),(D1 \/ D2)) -derivables) . X by ZFMISC_1:31;
hence  z is m + n,X,D1 \/ D2 -derivable by Def7; ::_thesis: verum
end;

Lm22:  for X being set
for S being Language
for t1, t2 being termal string of S
for D being RuleSet of S holds
( [t1,t2] in (X,D) -termEq iff (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable )
proof
let X be set ; ::_thesis: for S being Language
for t1, t2 being termal string of S
for D being RuleSet of S holds
( [t1,t2] in (X,D) -termEq iff (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable )
let S be Language; ::_thesis: for t1, t2 being termal string of S
for D being RuleSet of S holds
( [t1,t2] in (X,D) -termEq iff (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable )
let t1, t2 be termal string of S; ::_thesis: for D being RuleSet of S holds
( [t1,t2] in (X,D) -termEq iff (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable )
let D be RuleSet of S; ::_thesis: ( [t1,t2] in (X,D) -termEq iff (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable )
set E = TheEqSymbOf S;
set R = (X,D) -termEq ;
thus  ( [t1,t2] in (X,D) -termEq implies (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable ) ::_thesis: ( (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable implies [t1,t2] in (X,D) -termEq )
proof
assume  [t1,t2] in (X,D) -termEq ; ::_thesis: (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable
then consider t11, t22 being termal string of S such that
A1: ( [t11,t22] = [t1,t2] & (<*(TheEqSymbOf S)*> ^ t11) ^ t22 is X,D -provable ) ;
 ( t11 = t1 & t22 = t2 & (<*(TheEqSymbOf S)*> ^ t11) ^ t22 is X,D -provable ) by A1, XTUPLE_0:1;
hence  (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable ; ::_thesis: verum
end;
assume  (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable ; ::_thesis: [t1,t2] in (X,D) -termEq
hence  [t1,t2] in (X,D) -termEq ; ::_thesis: verum
end;

Lm23:  for S being Language
for Sq being b1 -sequent-like set holds Sq `2 is wff string of S
proof
let S be Language; ::_thesis: for Sq being S -sequent-like set holds Sq `2 is wff string of S
let Sq be S -sequent-like set ; ::_thesis: Sq `2 is wff string of S
set Q = S -sequents ;
reconsider seqt = Sq as Element of S -sequents by Def2;
 seqt in S -sequents ;
then consider premises being Subset of (AllFormulasOf S), conclusion being wff string of S such that
A1: ( seqt = [premises,conclusion] & premises is finite ) ;
thus  Sq `2 is wff string of S by A1, MCART_1:7; ::_thesis: verum
end;

Lm24:  for x, X being set
for S being Language
for D being RuleSet of S st x is X,D -provable holds
x is wff string of S
proof
let x, X be set ; ::_thesis: for S being Language
for D being RuleSet of S st x is X,D -provable holds
x is wff string of S
let S be Language; ::_thesis: for D being RuleSet of S st x is X,D -provable holds
x is wff string of S
let D be RuleSet of S; ::_thesis: ( x is X,D -provable implies x is wff string of S )
set Q = S -sequents ;
assume  x is X,D -provable ; ::_thesis: x is wff string of S
then consider y being set  such that
A1: ( y `1 c= X & y `2 = x & {y} is D -derivable ) by Def13;
reconsider E = {} as Subset of (S -sequents) by XBOOLE_1:2;
 {y} is {} ,D -derivable by A1;
then consider mm being Element of NAT  such that
A2: y is mm,E,D -derivable by Lm8;
reconsider yy = y as Element of S -sequents by Def2, A2;
 yy `2 is wff string of S by Lm23;
hence  x is wff string of S by A1; ::_thesis: verum
end;

Lm25:  for S being Language
for D being RuleSet of S holds AllFormulasOf S is D -expanded
proof
let S be Language; ::_thesis: for D being RuleSet of S holds AllFormulasOf S is D -expanded
let D be RuleSet of S; ::_thesis: AllFormulasOf S is D -expanded
set AF = AllFormulasOf S;
now__::_thesis:_for_x_being_set_st_x_is_AllFormulasOf_S,D_-provable_holds_
{x}_c=_AllFormulasOf_S
let x be set ; ::_thesis: ( x is AllFormulasOf S,D -provable implies {x} c= AllFormulasOf S )
assume  x is AllFormulasOf S,D -provable ; ::_thesis: {x} c= AllFormulasOf S
then reconsider xx = x as wff string of S by Lm24;
consider m being Nat such that
A1: xx is m -wff by FOMODEL2:def_25;
 xx in AllFormulasOf S by A1;
hence  {x} c= AllFormulasOf S by ZFMISC_1:31; ::_thesis: verum
end;
hence  AllFormulasOf S is D -expanded by Def16; ::_thesis: verum
end;

registration
let S be Language;
let D be RuleSet of S;
cluster functional V36() FinSequence-membered with_non-empty_elements D -expanded for Element of bool (AllFormulasOf S);
existence
 ex b1 being Subset of (AllFormulasOf S) st b1 is D -expanded
proof
set AF = AllFormulasOf S;
reconsider AFF = AllFormulasOf S as Subset of (AllFormulasOf S) by XBOOLE_0:def_10;
take  AFF ; ::_thesis: AFF is D -expanded
thus  AFF is D -expanded by Lm25; ::_thesis: verum
end;
end;

registration
let S be Language;
let D be RuleSet of S;
clusterD -expanded for set ;
existence
 ex b1 being set st b1 is D -expanded
proof
set AF = AllFormulasOf S;
take  the D -expanded Subset of (AllFormulasOf S) ; ::_thesis: the D -expanded Subset of (AllFormulasOf S) is D -expanded
thus  the D -expanded Subset of (AllFormulasOf S) is D -expanded ; ::_thesis: verum
end;
end;

definition
let Seqts be set ;
let S be Language;
let seqt be S -null set ;
predseqt Rule0 Seqts means :Def18: :: FOMODEL4:def 18
seqt `2 in seqt `1 ;
predseqt Rule1 Seqts means :Def19: :: FOMODEL4:def 19
 ex y being set st
( y in Seqts & y `1 c= seqt `1 & seqt `2 = y `2 );
predseqt Rule2 Seqts means :Def20: :: FOMODEL4:def 20
( seqt `1 is empty & ex t being termal string of S st seqt `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t );
predseqt Rule3a Seqts means :Def21: :: FOMODEL4:def 21
 ex t, t1, t2 being termal string of S ex x being set st
( x in Seqts & seqt `1 = (x `1) \/ {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} & x `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t1 & seqt `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t2 );
predseqt Rule3b Seqts means :Def22: :: FOMODEL4:def 22
 ex t1, t2 being termal string of S st
( seqt `1 = {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} & seqt `2 = (<*(TheEqSymbOf S)*> ^ t2) ^ t1 );
predseqt Rule3d Seqts means :Def23: :: FOMODEL4:def 23
 ex s being low-compounding Element of S ex T, U being abs (ar b1) -element Element of (AllTermsOf S) * st
( s is operational & seqt `1 =  {  ((<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j)) where j is Element of Seg (abs (ar s)), TT, UU is Function of (Seg (abs (ar s))),(((AllSymbolsOf S) *) \ {{}}) : ( TT = T & UU = U )  }  & seqt `2 = (<*(TheEqSymbOf S)*> ^ (s -compound T)) ^ (s -compound U) );
predseqt Rule3e Seqts means :Def24: :: FOMODEL4:def 24
 ex s being relational Element of S ex T, U being abs (ar b1) -element Element of (AllTermsOf S) * st
( seqt `1 = {(s -compound T)} \/  {  ((<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j)) where j is Element of Seg (abs (ar s)), TT, UU is Function of (Seg (abs (ar s))),(((AllSymbolsOf S) *) \ {{}}) : ( TT = T & UU = U )  }  & seqt `2 = s -compound U );
predseqt Rule4 Seqts means :Def25: :: FOMODEL4:def 25
 ex l being literal Element of S ex phi being wff string of S ex t being termal string of S st
( seqt `1 = {((l,t) SubstIn phi)} & seqt `2 = <*l*> ^ phi );
predseqt Rule5 Seqts means :Def26: :: FOMODEL4:def 26
 ex v1, v2 being literal Element of S ex x being set ex p being FinSequence st
( seqt `1 = x \/ {(<*v1*> ^ p)} & v2 is (x \/ {p}) \/ {(seqt `2)} -absent & [(x \/ {((v1 SubstWith v2) . p)}),(seqt `2)] in Seqts );
predseqt Rule6 Seqts means :Def27: :: FOMODEL4:def 27
 ex y1, y2 being set ex phi1, phi2 being wff string of S st
( y1 in Seqts & y2 in Seqts & y1 `1 = y2 `1 & y2 `1 = seqt `1 & y1 `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi1 & y2 `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi2 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 );
predseqt Rule7 Seqts means :Def28: :: FOMODEL4:def 28
 ex y being set ex phi1, phi2 being wff string of S st
( y in Seqts & y `1 = seqt `1 & y `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1 );
predseqt Rule8 Seqts means :Def29: :: FOMODEL4:def 29
 ex y1, y2 being set ex phi, phi1, phi2 being wff string of S st
( y1 in Seqts & y2 in Seqts & y1 `1 = y2 `1 & y1 `2 = phi1 & y2 `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & {phi} \/ (seqt `1) = y1 `1 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi) ^ phi );
predseqt Rule9 Seqts means :Def30: :: FOMODEL4:def 30
 ex y being set ex phi being wff string of S st
( y in Seqts & seqt `2 = phi & y `1 = seqt `1 & y `2 = xnot (xnot phi) );
end;

:: deftheorem Def18 defines Rule0 FOMODEL4:def_18_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule0 Seqts iff seqt `2 in seqt `1 );

:: deftheorem Def19 defines Rule1 FOMODEL4:def_19_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule1 Seqts iff ex y being set st
( y in Seqts & y `1 c= seqt `1 & seqt `2 = y `2 ) );

:: deftheorem Def20 defines Rule2 FOMODEL4:def_20_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule2 Seqts iff ( seqt `1 is empty & ex t being termal string of S st seqt `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t ) );

:: deftheorem Def21 defines Rule3a FOMODEL4:def_21_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule3a Seqts iff ex t, t1, t2 being termal string of S ex x being set st
( x in Seqts & seqt `1 = (x `1) \/ {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} & x `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t1 & seqt `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t2 ) );

:: deftheorem Def22 defines Rule3b FOMODEL4:def_22_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule3b Seqts iff ex t1, t2 being termal string of S st
( seqt `1 = {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} & seqt `2 = (<*(TheEqSymbOf S)*> ^ t2) ^ t1 ) );

:: deftheorem Def23 defines Rule3d FOMODEL4:def_23_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule3d Seqts iff ex s being low-compounding Element of S ex T, U being abs (ar b4) -element Element of (AllTermsOf S) * st
( s is operational & seqt `1 =  {  ((<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j)) where j is Element of Seg (abs (ar s)), TT, UU is Function of (Seg (abs (ar s))),(((AllSymbolsOf S) *) \ {{}}) : ( TT = T & UU = U )  }  & seqt `2 = (<*(TheEqSymbOf S)*> ^ (s -compound T)) ^ (s -compound U) ) );

:: deftheorem Def24 defines Rule3e FOMODEL4:def_24_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule3e Seqts iff ex s being relational Element of S ex T, U being abs (ar b4) -element Element of (AllTermsOf S) * st
( seqt `1 = {(s -compound T)} \/  {  ((<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j)) where j is Element of Seg (abs (ar s)), TT, UU is Function of (Seg (abs (ar s))),(((AllSymbolsOf S) *) \ {{}}) : ( TT = T & UU = U )  }  & seqt `2 = s -compound U ) );

:: deftheorem Def25 defines Rule4 FOMODEL4:def_25_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule4 Seqts iff ex l being literal Element of S ex phi being wff string of S ex t being termal string of S st
( seqt `1 = {((l,t) SubstIn phi)} & seqt `2 = <*l*> ^ phi ) );

:: deftheorem Def26 defines Rule5 FOMODEL4:def_26_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule5 Seqts iff ex v1, v2 being literal Element of S ex x being set ex p being FinSequence st
( seqt `1 = x \/ {(<*v1*> ^ p)} & v2 is (x \/ {p}) \/ {(seqt `2)} -absent & [(x \/ {((v1 SubstWith v2) . p)}),(seqt `2)] in Seqts ) );

:: deftheorem Def27 defines Rule6 FOMODEL4:def_27_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule6 Seqts iff ex y1, y2 being set ex phi1, phi2 being wff string of S st
( y1 in Seqts & y2 in Seqts & y1 `1 = y2 `1 & y2 `1 = seqt `1 & y1 `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi1 & y2 `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi2 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 ) );

:: deftheorem Def28 defines Rule7 FOMODEL4:def_28_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule7 Seqts iff ex y being set ex phi1, phi2 being wff string of S st
( y in Seqts & y `1 = seqt `1 & y `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1 ) );

:: deftheorem Def29 defines Rule8 FOMODEL4:def_29_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule8 Seqts iff ex y1, y2 being set ex phi, phi1, phi2 being wff string of S st
( y1 in Seqts & y2 in Seqts & y1 `1 = y2 `1 & y1 `2 = phi1 & y2 `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & {phi} \/ (seqt `1) = y1 `1 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi) ^ phi ) );

:: deftheorem Def30 defines Rule9 FOMODEL4:def_30_:_
 for Seqts being set
for S being Language
for seqt being b2 -null set holds
( seqt Rule9 Seqts iff ex y being set ex phi being wff string of S st
( y in Seqts & seqt `2 = phi & y `1 = seqt `1 & y `2 = xnot (xnot phi) ) );

definition
let S be Language;
func P#0 S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def31: :: FOMODEL4:def 31
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule0 Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule0 Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule0 $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A1: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule0 Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule0 Seqts ) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule0 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule0 Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule0 $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule0 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule0 Seqts ) ) implies IT1 = IT2 )
assume A2: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule0 Seqts ) implies ( seqt Rule0 Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A3: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A2, A3); ::_thesis: verum
end;
func P#1 S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def32: :: FOMODEL4:def 32
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule1 Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule1 Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule1 $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A4: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule1 Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule1 Seqts ) by A4; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule1 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule1 Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule1 $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule1 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule1 Seqts ) ) implies IT1 = IT2 )
assume A5: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule1 Seqts ) implies ( seqt Rule1 Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A6: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A5, A6); ::_thesis: verum
end;
func P#2 S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def33: :: FOMODEL4:def 33
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule2 Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule2 Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule2 $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A7: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule2 Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule2 Seqts ) by A7; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule2 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule2 Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule2 $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule2 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule2 Seqts ) ) implies IT1 = IT2 )
assume A8: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule2 Seqts ) implies ( seqt Rule2 Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A9: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A8, A9); ::_thesis: verum
end;
func P#3a S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def34: :: FOMODEL4:def 34
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule3a Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule3a Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule3a $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A10: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule3a Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule3a Seqts ) by A10; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule3a Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule3a Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule3a $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule3a Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule3a Seqts ) ) implies IT1 = IT2 )
assume A11: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule3a Seqts ) implies ( seqt Rule3a Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A12: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A11, A12); ::_thesis: verum
end;
func P#3b S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def35: :: FOMODEL4:def 35
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule3b Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule3b Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule3b $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A13: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule3b Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule3b Seqts ) by A13; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule3b Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule3b Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule3b $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule3b Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule3b Seqts ) ) implies IT1 = IT2 )
assume A14: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule3b Seqts ) implies ( seqt Rule3b Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A15: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A14, A15); ::_thesis: verum
end;
func P#3d S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def36: :: FOMODEL4:def 36
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule3d Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule3d Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule3d $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A16: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule3d Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule3d Seqts ) by A16; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule3d Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule3d Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule3d $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule3d Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule3d Seqts ) ) implies IT1 = IT2 )
assume A17: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule3d Seqts ) implies ( seqt Rule3d Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A18: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A17, A18); ::_thesis: verum
end;
func P#3e S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def37: :: FOMODEL4:def 37
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule3e Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule3e Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule3e $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A19: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule3e Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule3e Seqts ) by A19; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule3e Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule3e Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule3e $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule3e Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule3e Seqts ) ) implies IT1 = IT2 )
assume A20: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule3e Seqts ) implies ( seqt Rule3e Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A21: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A20, A21); ::_thesis: verum
end;
func P#4 S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def38: :: FOMODEL4:def 38
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule4 Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule4 Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule4 $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A22: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule4 Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule4 Seqts ) by A22; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule4 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule4 Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule4 $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule4 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule4 Seqts ) ) implies IT1 = IT2 )
assume A23: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule4 Seqts ) implies ( seqt Rule4 Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A24: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A23, A24); ::_thesis: verum
end;
func P#5 S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def39: :: FOMODEL4:def 39
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule5 Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule5 Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule5 $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A25: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule5 Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule5 Seqts ) by A25; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule5 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule5 Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule5 $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule5 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule5 Seqts ) ) implies IT1 = IT2 )
assume A26: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule5 Seqts ) implies ( seqt Rule5 Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A27: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A26, A27); ::_thesis: verum
end;
func P#6 S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def40: :: FOMODEL4:def 40
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule6 Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule6 Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule6 $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A28: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule6 Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule6 Seqts ) by A28; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule6 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule6 Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule6 $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule6 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule6 Seqts ) ) implies IT1 = IT2 )
assume A29: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule6 Seqts ) implies ( seqt Rule6 Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A30: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A29, A30); ::_thesis: verum
end;
func P#7 S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def41: :: FOMODEL4:def 41
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule7 Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule7 Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule7 $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A31: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule7 Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule7 Seqts ) by A31; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule7 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule7 Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule7 $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule7 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule7 Seqts ) ) implies IT1 = IT2 )
assume A32: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule7 Seqts ) implies ( seqt Rule7 Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A33: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A32, A33); ::_thesis: verum
end;
func P#8 S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def42: :: FOMODEL4:def 42
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule8 Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule8 Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule8 $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A34: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule8 Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule8 Seqts ) by A34; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule8 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule8 Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule8 $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule8 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule8 Seqts ) ) implies IT1 = IT2 )
assume A35: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule8 Seqts ) implies ( seqt Rule8 Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A36: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A35, A36); ::_thesis: verum
end;
func P#9 S ->  Relation of (bool (S -sequents)),(S -sequents) means :Def43: :: FOMODEL4:def 43
 for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in it iff seqt Rule9 Seqts );
existence
 ex b1 being Relation of (bool (S -sequents)),(S -sequents) st
for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule9 Seqts )
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule9 $1;
consider R being Relation of (bool (S -sequents)),(S -sequents) such that
A37: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in R iff S1[x,y] ) from RELSET_1:sch_2();
take  R ; ::_thesis: for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule9 Seqts )
thus  for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in R iff seqt Rule9 Seqts ) by A37; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation of (bool (S -sequents)),(S -sequents) st ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b1 iff seqt Rule9 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule9 Seqts ) ) holds
b1 = b2
proof
defpred S1[ set , Element of S -sequents ] means $2 Rule9 $1;
let IT1, IT2 be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT1 iff seqt Rule9 Seqts ) ) & ( for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in IT2 iff seqt Rule9 Seqts ) ) implies IT1 = IT2 )
assume A38: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT1 iff S1[x,y] ) ; ::_thesis: ( ex Seqts being Element of bool (S -sequents) ex seqt being Element of S -sequents st
( ( [Seqts,seqt] in IT2 implies seqt Rule9 Seqts ) implies ( seqt Rule9 Seqts & not [Seqts,seqt] in IT2 ) ) or IT1 = IT2 )
assume A39: for x being Element of bool (S -sequents)
for y being Element of S -sequents holds
( [x,y] in IT2 iff S1[x,y] ) ; ::_thesis: IT1 = IT2
thus  IT1 = IT2 from RELSET_1:sch_4(A38, A39); ::_thesis: verum
end;
end;

:: deftheorem Def31 defines P#0 FOMODEL4:def_31_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#0 S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule0 Seqts ) );

:: deftheorem Def32 defines P#1 FOMODEL4:def_32_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#1 S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule1 Seqts ) );

:: deftheorem Def33 defines P#2 FOMODEL4:def_33_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#2 S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule2 Seqts ) );

:: deftheorem Def34 defines P#3a FOMODEL4:def_34_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#3a S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule3a Seqts ) );

:: deftheorem Def35 defines P#3b FOMODEL4:def_35_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#3b S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule3b Seqts ) );

:: deftheorem Def36 defines P#3d FOMODEL4:def_36_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#3d S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule3d Seqts ) );

:: deftheorem Def37 defines P#3e FOMODEL4:def_37_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#3e S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule3e Seqts ) );

:: deftheorem Def38 defines P#4 FOMODEL4:def_38_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#4 S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule4 Seqts ) );

:: deftheorem Def39 defines P#5 FOMODEL4:def_39_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#5 S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule5 Seqts ) );

:: deftheorem Def40 defines P#6 FOMODEL4:def_40_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#6 S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule6 Seqts ) );

:: deftheorem Def41 defines P#7 FOMODEL4:def_41_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#7 S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule7 Seqts ) );

:: deftheorem Def42 defines P#8 FOMODEL4:def_42_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#8 S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule8 Seqts ) );

:: deftheorem Def43 defines P#9 FOMODEL4:def_43_:_
 for S being Language
for b2 being Relation of (bool (S -sequents)),(S -sequents) holds
( b2 = P#9 S iff for Seqts being Element of bool (S -sequents)
for seqt being Element of S -sequents holds
( [Seqts,seqt] in b2 iff seqt Rule9 Seqts ) );

definition
let S be Language;
let R be Relation of (bool (S -sequents)),(S -sequents);
func FuncRule R ->  Rule of S means :Def44: :: FOMODEL4:def 44
 for inseqs being set st inseqs in bool (S -sequents) holds
it . inseqs =  {  x where x is Element of S -sequents : [inseqs,x] in R  }  ;
existence
 ex b1 being Rule of S st
for inseqs being set st inseqs in bool (S -sequents) holds
b1 . inseqs =  {  x where x is Element of S -sequents : [inseqs,x] in R  }
proof
deffunc H1( set ) ->  set =  {  x where x is Element of S -sequents : [$1,x] in R  }  ;
A1: for inseqs being set holds H1(inseqs) in bool (S -sequents)
proof
let inseqs be set ; ::_thesis: H1(inseqs) in bool (S -sequents)
now__::_thesis:_for_x_being_set_st_x_in_H1(inseqs)_holds_
x_in_S_-sequents
let x be set ; ::_thesis: ( x in H1(inseqs) implies x in S -sequents )
assume  x in H1(inseqs) ; ::_thesis: x in S -sequents
then consider seq being Element of S -sequents  such that
A2: ( seq = x & [inseqs,seq] in R ) ;
thus  x in S -sequents by A2; ::_thesis: verum
end;
then  H1(inseqs) c= S -sequents by TARSKI:def_3;
hence  H1(inseqs) in bool (S -sequents) ; ::_thesis: verum
end;
A3: for inseqs being set st inseqs in bool (S -sequents) holds
H1(inseqs) in bool (S -sequents) by A1;
consider f being Function of (bool (S -sequents)),(bool (S -sequents)) such that
A4: for x being set st x in bool (S -sequents) holds
f . x = H1(x) from FUNCT_2:sch_2(A3);
take  f ; ::_thesis: ( f is Rule of S & ( for inseqs being set st inseqs in bool (S -sequents) holds
f . inseqs =  {  x where x is Element of S -sequents : [inseqs,x] in R  }  ) )
thus  ( f is Rule of S & ( for inseqs being set st inseqs in bool (S -sequents) holds
f . inseqs =  {  x where x is Element of S -sequents : [inseqs,x] in R  }  ) ) by A4, FUNCT_2:8; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Rule of S st ( for inseqs being set st inseqs in bool (S -sequents) holds
b1 . inseqs =  {  x where x is Element of S -sequents : [inseqs,x] in R  }  ) & ( for inseqs being set st inseqs in bool (S -sequents) holds
b2 . inseqs =  {  x where x is Element of S -sequents : [inseqs,x] in R  }  ) holds
b1 = b2
proof
set Q = S -sequents ;
let IT1, IT2 be Rule of S; ::_thesis: ( ( for inseqs being set st inseqs in bool (S -sequents) holds
IT1 . inseqs =  {  x where x is Element of S -sequents : [inseqs,x] in R  }  ) & ( for inseqs being set st inseqs in bool (S -sequents) holds
IT2 . inseqs =  {  x where x is Element of S -sequents : [inseqs,x] in R  }  ) implies IT1 = IT2 )
deffunc H1( set ) ->  set =  {  x where x is Element of S -sequents : [$1,x] in R  }  ;
assume A5: for inseqs being set st inseqs in bool (S -sequents) holds
IT1 . inseqs = H1(inseqs) ; ::_thesis: ( ex inseqs being set st
( inseqs in bool (S -sequents) & not IT2 . inseqs =  {  x where x is Element of S -sequents : [inseqs,x] in R  }  ) or IT1 = IT2 )
assume A6: for inseqs being set st inseqs in bool (S -sequents) holds
IT2 . inseqs = H1(inseqs) ; ::_thesis: IT1 = IT2
 for x being set st x in bool (S -sequents) holds
IT1 . x = IT2 . x
proof
let x be set ; ::_thesis: ( x in bool (S -sequents) implies IT1 . x = IT2 . x )
assume A7: x in bool (S -sequents) ; ::_thesis: IT1 . x = IT2 . x
hence IT1 . x = H1(x) by A5
.= IT2 . x by A6, A7 ;
::_thesis: verum
end;
hence  IT1 = IT2 by FUNCT_2:12; ::_thesis: verum
end;
end;

:: deftheorem Def44 defines FuncRule FOMODEL4:def_44_:_
 for S being Language
for R being Relation of (bool (S -sequents)),(S -sequents)
for b3 being Rule of S holds
( b3 = FuncRule R iff for inseqs being set st inseqs in bool (S -sequents) holds
b3 . inseqs =  {  x where x is Element of S -sequents : [inseqs,x] in R  }  );

Lm26:  for S being Language
for Seqts being Subset of (S -sequents)
for seqt being Element of S -sequents
for R being Relation of (bool (S -sequents)),(S -sequents) st [Seqts,seqt] in R holds
seqt in (FuncRule R) . Seqts
proof
let S be Language; ::_thesis: for Seqts being Subset of (S -sequents)
for seqt being Element of S -sequents
for R being Relation of (bool (S -sequents)),(S -sequents) st [Seqts,seqt] in R holds
seqt in (FuncRule R) . Seqts
let Seqts be Subset of (S -sequents); ::_thesis: for seqt being Element of S -sequents
for R being Relation of (bool (S -sequents)),(S -sequents) st [Seqts,seqt] in R holds
seqt in (FuncRule R) . Seqts
let seqt be Element of S -sequents ; ::_thesis: for R being Relation of (bool (S -sequents)),(S -sequents) st [Seqts,seqt] in R holds
seqt in (FuncRule R) . Seqts
let R be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( [Seqts,seqt] in R implies seqt in (FuncRule R) . Seqts )
A1: (FuncRule R) . Seqts =  {  x where x is Element of S -sequents : [Seqts,x] in R  }  by Def44;
assume  [Seqts,seqt] in R ; ::_thesis: seqt in (FuncRule R) . Seqts
hence  seqt in (FuncRule R) . Seqts by A1; ::_thesis: verum
end;

Lm27:  for S being Language
for SQ being b1 -sequents-like set
for Sq being b1 -sequent-like set
for R being Relation of (bool (S -sequents)),(S -sequents) st [SQ,Sq] in R holds
Sq is 1,SQ,{(FuncRule R)} -derivable
proof
let S be Language; ::_thesis: for SQ being S -sequents-like set
for Sq being S -sequent-like set
for R being Relation of (bool (S -sequents)),(S -sequents) st [SQ,Sq] in R holds
Sq is 1,SQ,{(FuncRule R)} -derivable
let SQ be S -sequents-like set ; ::_thesis: for Sq being S -sequent-like set
for R being Relation of (bool (S -sequents)),(S -sequents) st [SQ,Sq] in R holds
Sq is 1,SQ,{(FuncRule R)} -derivable
let Sq be S -sequent-like set ; ::_thesis: for R being Relation of (bool (S -sequents)),(S -sequents) st [SQ,Sq] in R holds
Sq is 1,SQ,{(FuncRule R)} -derivable
set Q = S -sequents ;
let R be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( [SQ,Sq] in R implies Sq is 1,SQ,{(FuncRule R)} -derivable )
set F = FuncRule R;
set O = OneStep {(FuncRule R)};
reconsider Seqts = SQ as Subset of (S -sequents) by Def3;
reconsider seqt = Sq as Element of S -sequents by Def2;
A1: FuncRule R = OneStep {(FuncRule R)} by Lm7
.= (1,{(FuncRule R)}) -derivables by FUNCT_7:70 ;
assume  [SQ,Sq] in R ; ::_thesis: Sq is 1,SQ,{(FuncRule R)} -derivable
then  seqt in ((1,{(FuncRule R)}) -derivables) . Seqts by A1, Lm26;
hence  Sq is 1,SQ,{(FuncRule R)} -derivable by Def7; ::_thesis: verum
end;

Lm28:  for y being set
for S being Language
for SQ being b2 -sequents-like set
for R being Relation of (bool (S -sequents)),(S -sequents) holds
( y in (FuncRule R) . SQ iff ( y in S -sequents & [SQ,y] in R ) )
proof
let y be set ; ::_thesis: for S being Language
for SQ being b1 -sequents-like set
for R being Relation of (bool (S -sequents)),(S -sequents) holds
( y in (FuncRule R) . SQ iff ( y in S -sequents & [SQ,y] in R ) )
let S be Language; ::_thesis: for SQ being S -sequents-like set
for R being Relation of (bool (S -sequents)),(S -sequents) holds
( y in (FuncRule R) . SQ iff ( y in S -sequents & [SQ,y] in R ) )
let SQ be S -sequents-like set ; ::_thesis: for R being Relation of (bool (S -sequents)),(S -sequents) holds
( y in (FuncRule R) . SQ iff ( y in S -sequents & [SQ,y] in R ) )
set Q = S -sequents ;
let R be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( y in (FuncRule R) . SQ iff ( y in S -sequents & [SQ,y] in R ) )
reconsider F = FuncRule R as Function of (bool (S -sequents)),(bool (S -sequents)) ;
reconsider Seqts = SQ as Element of bool (S -sequents) by Def3;
set G =  {  x where x is Element of S -sequents : [Seqts,x] in R  }  ;
A1: F . Seqts =  {  x where x is Element of S -sequents : [Seqts,x] in R  }  by Def44;
A2: F . Seqts c= S -sequents ;
thus  ( y in (FuncRule R) . SQ implies ( y in S -sequents & [SQ,y] in R ) ) ::_thesis: ( y in S -sequents & [SQ,y] in R implies y in (FuncRule R) . SQ )
proof
assume A3: y in (FuncRule R) . SQ ; ::_thesis: ( y in S -sequents & [SQ,y] in R )
thus  y in S -sequents by A2, A3; ::_thesis: [SQ,y] in R
consider x being Element of S -sequents  such that
A4: ( y = x & [Seqts,x] in R ) by A3, A1;
thus  [SQ,y] in R by A4; ::_thesis: verum
end;
assume A5: ( y in S -sequents & [SQ,y] in R ) ; ::_thesis: y in (FuncRule R) . SQ
then reconsider seqt = y as Element of S -sequents ;
 seqt in F . Seqts by Lm26, A5;
hence  y in (FuncRule R) . SQ ; ::_thesis: verum
end;

Lm29:  for y, X being set
for S being Language
for R being Relation of (bool (S -sequents)),(S -sequents) holds
( y in (FuncRule R) . X iff ( y in S -sequents & [X,y] in R ) )
proof
let y, X be set ; ::_thesis: for S being Language
for R being Relation of (bool (S -sequents)),(S -sequents) holds
( y in (FuncRule R) . X iff ( y in S -sequents & [X,y] in R ) )
let S be Language; ::_thesis: for R being Relation of (bool (S -sequents)),(S -sequents) holds
( y in (FuncRule R) . X iff ( y in S -sequents & [X,y] in R ) )
set Q = S -sequents ;
let R be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( y in (FuncRule R) . X iff ( y in S -sequents & [X,y] in R ) )
reconsider F = FuncRule R as Function of (bool (S -sequents)),(bool (S -sequents)) ;
percases ( not X in bool (S -sequents) or X in bool (S -sequents) ) ;
supposeA1: not X in bool (S -sequents) ; ::_thesis: ( y in (FuncRule R) . X iff ( y in S -sequents & [X,y] in R ) )
 not X in dom F by A1;
hence  ( y in (FuncRule R) . X iff ( y in S -sequents & [X,y] in R ) ) by A1, FUNCT_1:def_2, ZFMISC_1:87; ::_thesis: verum
end;
suppose X in bool (S -sequents) ; ::_thesis: ( y in (FuncRule R) . X iff ( y in S -sequents & [X,y] in R ) )
then reconsider Seqts = X as Element of bool (S -sequents) ;
set SQ = Seqts;
 ( y in (FuncRule R) . Seqts iff ( y in S -sequents & [Seqts,y] in R ) ) by Lm28;
hence  ( y in (FuncRule R) . X iff ( y in S -sequents & [X,y] in R ) ) ; ::_thesis: verum
end;
end;
end;

definition
let S be Language;
func R#0 S ->  Rule of S equals :: FOMODEL4:def 45
 FuncRule (P#0 S);
coherence
 FuncRule (P#0 S) is Rule of S ;
func R#1 S ->  Rule of S equals :: FOMODEL4:def 46
 FuncRule (P#1 S);
coherence
 FuncRule (P#1 S) is Rule of S ;
func R#2 S ->  Rule of S equals :: FOMODEL4:def 47
 FuncRule (P#2 S);
coherence
 FuncRule (P#2 S) is Rule of S ;
func R#3a S ->  Rule of S equals :: FOMODEL4:def 48
 FuncRule (P#3a S);
coherence
 FuncRule (P#3a S) is Rule of S ;
func R#3b S ->  Rule of S equals :: FOMODEL4:def 49
 FuncRule (P#3b S);
coherence
 FuncRule (P#3b S) is Rule of S ;
func R#3d S ->  Rule of S equals :: FOMODEL4:def 50
 FuncRule (P#3d S);
coherence
 FuncRule (P#3d S) is Rule of S ;
func R#3e S ->  Rule of S equals :: FOMODEL4:def 51
 FuncRule (P#3e S);
coherence
 FuncRule (P#3e S) is Rule of S ;
func R#4 S ->  Rule of S equals :: FOMODEL4:def 52
 FuncRule (P#4 S);
coherence
 FuncRule (P#4 S) is Rule of S ;
func R#5 S ->  Rule of S equals :: FOMODEL4:def 53
 FuncRule (P#5 S);
coherence
 FuncRule (P#5 S) is Rule of S ;
func R#6 S ->  Rule of S equals :: FOMODEL4:def 54
 FuncRule (P#6 S);
coherence
 FuncRule (P#6 S) is Rule of S ;
func R#7 S ->  Rule of S equals :: FOMODEL4:def 55
 FuncRule (P#7 S);
coherence
 FuncRule (P#7 S) is Rule of S ;
func R#8 S ->  Rule of S equals :: FOMODEL4:def 56
 FuncRule (P#8 S);
coherence
 FuncRule (P#8 S) is Rule of S ;
func R#9 S ->  Rule of S equals :: FOMODEL4:def 57
 FuncRule (P#9 S);
coherence
 FuncRule (P#9 S) is Rule of S ;
end;

:: deftheorem  defines R#0 FOMODEL4:def_45_:_
 for S being Language holds R#0 S = FuncRule (P#0 S);

:: deftheorem  defines R#1 FOMODEL4:def_46_:_
 for S being Language holds R#1 S = FuncRule (P#1 S);

:: deftheorem  defines R#2 FOMODEL4:def_47_:_
 for S being Language holds R#2 S = FuncRule (P#2 S);

:: deftheorem  defines R#3a FOMODEL4:def_48_:_
 for S being Language holds R#3a S = FuncRule (P#3a S);

:: deftheorem  defines R#3b FOMODEL4:def_49_:_
 for S being Language holds R#3b S = FuncRule (P#3b S);

:: deftheorem  defines R#3d FOMODEL4:def_50_:_
 for S being Language holds R#3d S = FuncRule (P#3d S);

:: deftheorem  defines R#3e FOMODEL4:def_51_:_
 for S being Language holds R#3e S = FuncRule (P#3e S);

:: deftheorem  defines R#4 FOMODEL4:def_52_:_
 for S being Language holds R#4 S = FuncRule (P#4 S);

:: deftheorem  defines R#5 FOMODEL4:def_53_:_
 for S being Language holds R#5 S = FuncRule (P#5 S);

:: deftheorem  defines R#6 FOMODEL4:def_54_:_
 for S being Language holds R#6 S = FuncRule (P#6 S);

:: deftheorem  defines R#7 FOMODEL4:def_55_:_
 for S being Language holds R#7 S = FuncRule (P#7 S);

:: deftheorem  defines R#8 FOMODEL4:def_56_:_
 for S being Language holds R#8 S = FuncRule (P#8 S);

:: deftheorem  defines R#9 FOMODEL4:def_57_:_
 for S being Language holds R#9 S = FuncRule (P#9 S);

registration
let S be Language;
let t be termal string of S;
cluster{[{},((<*(TheEqSymbOf S)*> ^ t) ^ t)]} -> {(R#2 S)} -derivable for set ;
coherence
 for b1 being set st b1 = {[{},((<*(TheEqSymbOf S)*> ^ t) ^ t)]} holds
b1 is {(R#2 S)} -derivable
proof
set E = TheEqSymbOf S;
set SS = AllSymbolsOf S;
set T = S -termsOfMaxDepth ;
set C = S -multiCat ;
reconsider phi = (<*(TheEqSymbOf S)*> ^ t) ^ t as wff string of S by Lm2;
reconsider Seqts = {} as Element of bool (S -sequents) by XBOOLE_1:2;
reconsider seqt = [{},phi] as Element of S -sequents ;
 ( seqt `1 is empty & seqt `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t ) by MCART_1:7;
then  seqt Rule2 {} by Def20;
then  [Seqts,seqt] in P#2 S by Def33;
then  seqt in (R#2 S) . Seqts by Lm26;
then  {seqt} c= (R#2 S) . Seqts by ZFMISC_1:31;
then  {seqt} is {} ,{(R#2 S)} -derivable by Lm19;
hence  for b1 being set st b1 = {[{},((<*(TheEqSymbOf S)*> ^ t) ^ t)]} holds
b1 is {(R#2 S)} -derivable ; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster R#2 S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#2 S holds
b1 is isotone
proof
now__::_thesis:_for_Seqts1,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts1_c=_Seqts2_holds_
(R#2_S)_._Seqts1_c=_(R#2_S)_._Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts1 c= Seqts2 implies (R#2 S) . Seqts1 c= (R#2 S) . Seqts2 )
set X = Seqts1;
set Y = Seqts2;
assume  Seqts1 c= Seqts2 ; ::_thesis: (R#2 S) . Seqts1 c= (R#2 S) . Seqts2
set R = R#2 S;
set Q = S -sequents ;
now__::_thesis:_for_x_being_set_st_x_in_(R#2_S)_._Seqts1_holds_
x_in_(R#2_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#2 S) . Seqts1 implies x in (R#2 S) . Seqts2 )
assume A1: x in (R#2 S) . Seqts1 ; ::_thesis: x in (R#2 S) . Seqts2
then A2: ( x in S -sequents & [Seqts1,x] in P#2 S ) by Lm29;
reconsider seqt = x as Element of S -sequents by A1;
 seqt Rule2 Seqts1 by Def33, A2;
then  ( seqt `1 is empty & ex t being termal string of S st seqt `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t ) by Def20;
then  seqt Rule2 Seqts2 by Def20;
then  [Seqts2,seqt] in P#2 S by Def33;
hence  x in (R#2 S) . Seqts2 by Lm26; ::_thesis: verum
end;
hence  (R#2 S) . Seqts1 c= (R#2 S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#2 S holds
b1 is isotone by Def9; ::_thesis: verum
end;
end;

Lm30:  for X being set
for S being Language
for D being RuleSet of S st {(R#2 S)} c= D holds
(X,D) -termEq is total
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S st {(R#2 S)} c= D holds
(X,D) -termEq is total
let S be Language; ::_thesis: for D being RuleSet of S st {(R#2 S)} c= D holds
(X,D) -termEq is total
let D be RuleSet of S; ::_thesis: ( {(R#2 S)} c= D implies (X,D) -termEq is total )
assume A1: {(R#2 S)} c= D ; ::_thesis: (X,D) -termEq is total
set AT = AllTermsOf S;
set E = TheEqSymbOf S;
set Phi = X;
set R = (X,D) -termEq ;
now__::_thesis:_for_x_being_set_st_x_in_AllTermsOf_S_holds_
x_in_dom_((X,D)_-termEq)
let x be set ; ::_thesis: ( x in AllTermsOf S implies x in dom ((X,D) -termEq) )
assume  x in AllTermsOf S ; ::_thesis: x in dom ((X,D) -termEq)
then reconsider t = x as termal string of S ;
set phi = (<*(TheEqSymbOf S)*> ^ t) ^ t;
set seqt = [{},((<*(TheEqSymbOf S)*> ^ t) ^ t)];
 {[{},((<*(TheEqSymbOf S)*> ^ t) ^ t)]} is {} ,D -derivable by A1, Lm18;
then  (<*(TheEqSymbOf S)*> ^ t) ^ t is {} \ (X \ {}),D -provable by Def12;
then  (<*(TheEqSymbOf S)*> ^ t) ^ t is X \/ {},D -provable ;
then  [t,t] in (X,D) -termEq ;
hence  x in dom ((X,D) -termEq) by XTUPLE_0:def_12; ::_thesis: verum
end;
then  AllTermsOf S c= dom ((X,D) -termEq) by TARSKI:def_3;
then  AllTermsOf S = dom ((X,D) -termEq) by XBOOLE_0:def_10;
hence  (X,D) -termEq is total by PARTFUN1:def_2; ::_thesis: verum
end;

registration
let S be Language;
cluster R#3b S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#3b S holds
b1 is isotone
proof
now__::_thesis:_for_Seqts1,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts1_c=_Seqts2_holds_
(R#3b_S)_._Seqts1_c=_(R#3b_S)_._Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts1 c= Seqts2 implies (R#3b S) . Seqts1 c= (R#3b S) . Seqts2 )
set X = Seqts1;
set Y = Seqts2;
assume  Seqts1 c= Seqts2 ; ::_thesis: (R#3b S) . Seqts1 c= (R#3b S) . Seqts2
set R = R#3b S;
set Q = S -sequents ;
now__::_thesis:_for_x_being_set_st_x_in_(R#3b_S)_._Seqts1_holds_
x_in_(R#3b_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#3b S) . Seqts1 implies x in (R#3b S) . Seqts2 )
assume A1: x in (R#3b S) . Seqts1 ; ::_thesis: x in (R#3b S) . Seqts2
reconsider seqt = x as Element of S -sequents by A1;
 [Seqts1,seqt] in P#3b S by A1, Lm29;
then  seqt Rule3b Seqts1 by Def35;
then  ex t1, t2 being termal string of S st
( seqt `1 = {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} & seqt `2 = (<*(TheEqSymbOf S)*> ^ t2) ^ t1 ) by Def22;
then  seqt Rule3b Seqts2 by Def22;
then  [Seqts2,seqt] in P#3b S by Def35;
hence  x in (R#3b S) . Seqts2 by Lm26; ::_thesis: verum
end;
hence  (R#3b S) . Seqts1 c= (R#3b S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#3b S holds
b1 is isotone by Def9; ::_thesis: verum
end;
end;

Lm31:  for X being set
for S being Language
for D being RuleSet of S st {(R#3b S)} c= D & X is D -expanded holds
(X,D) -termEq is symmetric
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S st {(R#3b S)} c= D & X is D -expanded holds
(X,D) -termEq is symmetric
let S be Language; ::_thesis: for D being RuleSet of S st {(R#3b S)} c= D & X is D -expanded holds
(X,D) -termEq is symmetric
let D be RuleSet of S; ::_thesis: ( {(R#3b S)} c= D & X is D -expanded implies (X,D) -termEq is symmetric )
set AT = AllTermsOf S;
set E = TheEqSymbOf S;
set Q = S -sequents ;
set AF = AllFormulasOf S;
set Phi = X;
set R = (X,D) -termEq ;
assume A1: {(R#3b S)} c= D ; ::_thesis: ( not X is D -expanded or (X,D) -termEq is symmetric )
assume A2: X is D -expanded ; ::_thesis: (X,D) -termEq is symmetric
A3: field ((X,D) -termEq) c= (AllTermsOf S) \/ (AllTermsOf S) by RELSET_1:8;
now__::_thesis:_for_x,_y_being_set_st_x_in_field_((X,D)_-termEq)_&_y_in_field_((X,D)_-termEq)_&_[x,y]_in_(X,D)_-termEq_holds_
[y,x]_in_(X,D)_-termEq
let x, y be set ; ::_thesis: ( x in field ((X,D) -termEq) & y in field ((X,D) -termEq) & [x,y] in (X,D) -termEq implies [y,x] in (X,D) -termEq )
assume  ( x in field ((X,D) -termEq) & y in field ((X,D) -termEq) ) ; ::_thesis: ( [x,y] in (X,D) -termEq implies [y,x] in (X,D) -termEq )
then reconsider tt1 = x, tt2 = y as Element of AllTermsOf S by A3;
reconsider t1 = tt1, t2 = tt2 as termal string of S ;
reconsider phi1 = (<*(TheEqSymbOf S)*> ^ t1) ^ t2 as wff string of S by Lm2;
reconsider phi2 = (<*(TheEqSymbOf S)*> ^ t2) ^ t1 as wff string of S by Lm2;
reconsider seqt = [{phi1},phi2] as Element of S -sequents by Def2;
A4: ( seqt `1 = {phi1} & seqt `2 = phi2 ) by MCART_1:7;
reconsider Seqts = {} as Element of bool (S -sequents) by XBOOLE_1:2;
A5: seqt Rule3b {} by A4, Def22;
 [Seqts,seqt] in P#3b S by A5, Def35;
then  seqt in (R#3b S) . Seqts by Lm26;
then  {seqt} c= (R#3b S) . Seqts by ZFMISC_1:31;
then  {seqt} is {} ,{(R#3b S)} -derivable by Lm19;
then  {seqt} is {} ,D -derivable by A1, Lm18;
then A6: phi2 is {phi1},D -provable by Def12;
assume  [x,y] in (X,D) -termEq ; ::_thesis: [y,x] in (X,D) -termEq
then consider t11, t22 being termal string of S such that
A7: ( [x,y] = [t11,t22] & (<*(TheEqSymbOf S)*> ^ t11) ^ t22 is X,D -provable ) ;
 ( t1 = t11 & t2 = t22 ) by A7, XTUPLE_0:1;
then  {phi1} c= X by A2, Def16, A7;
hence  [y,x] in (X,D) -termEq by A6; ::_thesis: verum
end;
then  (X,D) -termEq is_symmetric_in field ((X,D) -termEq) by RELAT_2:def_3;
hence  (X,D) -termEq is symmetric by RELAT_2:def_11; ::_thesis: verum
end;

registration
let S be Language;
let t, t1, t2 be termal string of S;
let premises be finite Subset of (AllFormulasOf S);
cluster[(premises \/ {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)}),((<*(TheEqSymbOf S)*> ^ t) ^ t2)] -> 1,{[premises,((<*(TheEqSymbOf S)*> ^ t) ^ t1)]},{(R#3a S)} -derivable for set ;
coherence
 for b1 being set st b1 = [(premises \/ {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)}),((<*(TheEqSymbOf S)*> ^ t) ^ t2)] holds
b1 is 1,{[premises,((<*(TheEqSymbOf S)*> ^ t) ^ t1)]},{(R#3a S)} -derivable
proof
set E = TheEqSymbOf S;
set AF = AllFormulasOf S;
reconsider phi0 = (<*(TheEqSymbOf S)*> ^ t1) ^ t2, phi1 = (<*(TheEqSymbOf S)*> ^ t) ^ t1, phi2 = (<*(TheEqSymbOf S)*> ^ t) ^ t2 as 0wff string of S by Lm2;
 phi0 in AllFormulasOf S ;
then reconsider Phi0 = {phi0} as finite Subset of (AllFormulasOf S) by ZFMISC_1:31;
reconsider prem2 = premises \/ Phi0 as finite Subset of (AllFormulasOf S) ;
reconsider seqt2 = [prem2,phi2], seqt1 = [premises,phi1] as Element of S -sequents by Def2;
reconsider Seqts = {seqt1} as Subset of (S -sequents) ;
A1: seqt1 in Seqts by TARSKI:def_1;
A2: seqt2 `1 = premises \/ {phi0} by MCART_1:7
.= (seqt1 `1) \/ {phi0} by MCART_1:7 ;
A3: ( seqt1 `2 = phi1 & seqt2 `2 = phi2 ) by MCART_1:7;
 seqt2 Rule3a Seqts by A1, A2, A3, Def21;
then  [Seqts,seqt2] in P#3a S by Def34;
hence  for b1 being set st b1 = [(premises \/ {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)}),((<*(TheEqSymbOf S)*> ^ t) ^ t2)] holds
b1 is 1,{[premises,((<*(TheEqSymbOf S)*> ^ t) ^ t1)]},{(R#3a S)} -derivable by Lm27; ::_thesis: verum
end;
end;

registration
let S be Language;
let t, t1, t2 be termal string of S;
let phi be wff string of S;
cluster[{phi,((<*(TheEqSymbOf S)*> ^ t1) ^ t2)},((<*(TheEqSymbOf S)*> ^ t) ^ t2)] -> 1,{[{phi},((<*(TheEqSymbOf S)*> ^ t) ^ t1)]},{(R#3a S)} -derivable for set ;
coherence
 for b1 being set st b1 = [{phi,((<*(TheEqSymbOf S)*> ^ t1) ^ t2)},((<*(TheEqSymbOf S)*> ^ t) ^ t2)] holds
b1 is 1,{[{phi},((<*(TheEqSymbOf S)*> ^ t) ^ t1)]},{(R#3a S)} -derivable
proof
set AF = AllFormulasOf S;
set E = TheEqSymbOf S;
set allpremises = {phi,((<*(TheEqSymbOf S)*> ^ t1) ^ t2)};
set IT = [{phi,((<*(TheEqSymbOf S)*> ^ t1) ^ t2)},((<*(TheEqSymbOf S)*> ^ t) ^ t2)];
reconsider phii = phi as Element of AllFormulasOf S by FOMODEL2:16;
reconsider premises = {phii} as finite Subset of (AllFormulasOf S) ;
 [(premises \/ {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)}),((<*(TheEqSymbOf S)*> ^ t) ^ t2)] is 1,{[premises,((<*(TheEqSymbOf S)*> ^ t) ^ t1)]},{(R#3a S)} -derivable ;
hence  for b1 being set st b1 = [{phi,((<*(TheEqSymbOf S)*> ^ t1) ^ t2)},((<*(TheEqSymbOf S)*> ^ t) ^ t2)] holds
b1 is 1,{[{phi},((<*(TheEqSymbOf S)*> ^ t) ^ t1)]},{(R#3a S)} -derivable by ENUMSET1:1; ::_thesis: verum
end;
end;

registration
let S be Language;
let phi be wff string of S;
let Phi be finite Subset of (AllFormulasOf S);
cluster[(Phi \/ {phi}),phi] -> 1, {} ,{(R#0 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [(Phi \/ {phi}),phi] holds
b1 is 1, {} ,{(R#0 S)} -derivable
proof
set Q = S -sequents ;
reconsider Sq = [(Phi \/ {phi}),phi] as Element of S -sequents by Def2;
reconsider E = {} as Subset of (S -sequents) by XBOOLE_1:2;
A1: ( phi in {phi} & {phi} c= Phi \/ {phi} ) by TARSKI:def_1, XBOOLE_1:7;
 ( Sq `2 = phi & Sq `1 = Phi \/ {phi} ) by MCART_1:7;
then  Sq Rule0 E by Def18, A1;
then  [E,Sq] in P#0 S by Def31;
hence  for b1 being set st b1 = [(Phi \/ {phi}),phi] holds
b1 is 1, {} ,{(R#0 S)} -derivable by Lm27; ::_thesis: verum
end;
end;

registration
let S be Language;
let phi1, phi2 be wff string of S;
cluster[{phi1,phi2},phi1] -> 1, {} ,{(R#0 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [{phi1,phi2},phi1] holds
b1 is 1, {} ,{(R#0 S)} -derivable
proof
set AF = AllFormulasOf S;
reconsider phi11 = phi1, phi22 = phi2 as Element of AllFormulasOf S by FOMODEL2:16;
reconsider Phi = {phi22} as finite Subset of (AllFormulasOf S) ;
 [(Phi \/ {phi1}),phi1] is 1, {} ,{(R#0 S)} -derivable ;
hence  for b1 being set st b1 = [{phi1,phi2},phi1] holds
b1 is 1, {} ,{(R#0 S)} -derivable by ENUMSET1:1; ::_thesis: verum
end;
end;

registration
let S be Language;
let phi be wff string of S;
cluster[{phi},phi] -> 1, {} ,{(R#0 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [{phi},phi] holds
b1 is 1, {} ,{(R#0 S)} -derivable
proof
set AF = AllFormulasOf S;
reconsider Phi = {} as finite Subset of (AllFormulasOf S) by XBOOLE_1:2;
 [(Phi \/ {phi}),phi] is 1, {} ,{(R#0 S)} -derivable ;
hence  for b1 being set st b1 = [{phi},phi] holds
b1 is 1, {} ,{(R#0 S)} -derivable ; ::_thesis: verum
end;
end;

registration
let S be Language;
let phi be wff string of S;
cluster{[{phi},phi]} ->  {} ,{(R#0 S)} -derivable for set ;
coherence
 for b1 being set st b1 = {[{phi},phi]} holds
b1 is {} ,{(R#0 S)} -derivable by Lm12;
end;

registration
let S be Language;
cluster R#0 S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#0 S holds
b1 is isotone
proof
now__::_thesis:_for_Seqts1,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts1_c=_Seqts2_holds_
(R#0_S)_._Seqts1_c=_(R#0_S)_._Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts1 c= Seqts2 implies (R#0 S) . Seqts1 c= (R#0 S) . Seqts2 )
set X = Seqts1;
set Y = Seqts2;
assume  Seqts1 c= Seqts2 ; ::_thesis: (R#0 S) . Seqts1 c= (R#0 S) . Seqts2
set R = R#0 S;
set Q = S -sequents ;
now__::_thesis:_for_x_being_set_st_x_in_(R#0_S)_._Seqts1_holds_
x_in_(R#0_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#0 S) . Seqts1 implies x in (R#0 S) . Seqts2 )
assume A1: x in (R#0 S) . Seqts1 ; ::_thesis: x in (R#0 S) . Seqts2
reconsider seqt = x as Element of S -sequents by A1;
 [Seqts1,seqt] in P#0 S by A1, Lm29;
then  seqt Rule0 Seqts1 by Def31;
then  seqt `2 in seqt `1 by Def18;
then  seqt Rule0 Seqts2 by Def18;
then  [Seqts2,seqt] in P#0 S by Def31;
hence  x in (R#0 S) . Seqts2 by Lm26; ::_thesis: verum
end;
hence  (R#0 S) . Seqts1 c= (R#0 S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#0 S holds
b1 is isotone by Def9; ::_thesis: verum
end;
cluster R#3a S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#3a S holds
b1 is isotone
proof
now__::_thesis:_for_Seqts1,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts1_c=_Seqts2_holds_
(R#3a_S)_._Seqts1_c=_(R#3a_S)_._Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts1 c= Seqts2 implies (R#3a S) . Seqts1 c= (R#3a S) . Seqts2 )
set X = Seqts1;
set Y = Seqts2;
assume A2: Seqts1 c= Seqts2 ; ::_thesis: (R#3a S) . Seqts1 c= (R#3a S) . Seqts2
set R = R#3a S;
set Q = S -sequents ;
now__::_thesis:_for_x_being_set_st_x_in_(R#3a_S)_._Seqts1_holds_
x_in_(R#3a_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#3a S) . Seqts1 implies x in (R#3a S) . Seqts2 )
assume A3: x in (R#3a S) . Seqts1 ; ::_thesis: x in (R#3a S) . Seqts2
reconsider seqt = x as Element of S -sequents by A3;
 [Seqts1,seqt] in P#3a S by A3, Lm29;
then  seqt Rule3a Seqts1 by Def34;
then consider t, t1, t2 being termal string of S, xx being set  such that
A4: ( xx in Seqts1 & seqt `1 = (xx `1) \/ {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} & xx `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t1 & seqt `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t2 ) by Def21;
 seqt Rule3a Seqts2 by Def21, A4, A2;
then  [Seqts2,seqt] in P#3a S by Def34;
hence  x in (R#3a S) . Seqts2 by Lm26; ::_thesis: verum
end;
hence  (R#3a S) . Seqts1 c= (R#3a S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#3a S holds
b1 is isotone by Def9; ::_thesis: verum
end;
cluster R#3d S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#3d S holds
b1 is isotone
proof
now__::_thesis:_for_Seqts1,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts1_c=_Seqts2_holds_
(R#3d_S)_._Seqts1_c=_(R#3d_S)_._Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts1 c= Seqts2 implies (R#3d S) . Seqts1 c= (R#3d S) . Seqts2 )
set X = Seqts1;
set Y = Seqts2;
assume  Seqts1 c= Seqts2 ; ::_thesis: (R#3d S) . Seqts1 c= (R#3d S) . Seqts2
set R = R#3d S;
set Q = S -sequents ;
now__::_thesis:_for_x_being_set_st_x_in_(R#3d_S)_._Seqts1_holds_
x_in_(R#3d_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#3d S) . Seqts1 implies x in (R#3d S) . Seqts2 )
assume A5: x in (R#3d S) . Seqts1 ; ::_thesis: x in (R#3d S) . Seqts2
reconsider seqt = x as Element of S -sequents by A5;
 [Seqts1,seqt] in P#3d S by A5, Lm29;
then  seqt Rule3d Seqts1 by Def36;
then  ex s being low-compounding Element of S ex T, U being abs (ar b1) -element Element of (AllTermsOf S) * st
( s is operational & seqt `1 =  {  ((<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j)) where j is Element of Seg (abs (ar s)), TT, UU is Function of (Seg (abs (ar s))),(((AllSymbolsOf S) *) \ {{}}) : ( TT = T & UU = U )  }  & seqt `2 = (<*(TheEqSymbOf S)*> ^ (s -compound T)) ^ (s -compound U) ) by Def23;
then  seqt Rule3d Seqts2 by Def23;
then  [Seqts2,seqt] in P#3d S by Def36;
hence  x in (R#3d S) . Seqts2 by Lm26; ::_thesis: verum
end;
hence  (R#3d S) . Seqts1 c= (R#3d S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#3d S holds
b1 is isotone by Def9; ::_thesis: verum
end;
cluster R#3e S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#3e S holds
b1 is isotone
proof
now__::_thesis:_for_Seqts1,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts1_c=_Seqts2_holds_
(R#3e_S)_._Seqts1_c=_(R#3e_S)_._Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts1 c= Seqts2 implies (R#3e S) . Seqts1 c= (R#3e S) . Seqts2 )
set X = Seqts1;
set Y = Seqts2;
assume  Seqts1 c= Seqts2 ; ::_thesis: (R#3e S) . Seqts1 c= (R#3e S) . Seqts2
set R = R#3e S;
set Q = S -sequents ;
now__::_thesis:_for_x_being_set_st_x_in_(R#3e_S)_._Seqts1_holds_
x_in_(R#3e_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#3e S) . Seqts1 implies x in (R#3e S) . Seqts2 )
assume A6: x in (R#3e S) . Seqts1 ; ::_thesis: x in (R#3e S) . Seqts2
reconsider seqt = x as Element of S -sequents by A6;
 [Seqts1,seqt] in P#3e S by A6, Lm29;
then  seqt Rule3e Seqts1 by Def37;
then  ex s being relational Element of S ex T, U being abs (ar b1) -element Element of (AllTermsOf S) * st
( seqt `1 = {(s -compound T)} \/  {  ((<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j)) where j is Element of Seg (abs (ar s)), TT, UU is Function of (Seg (abs (ar s))),(((AllSymbolsOf S) *) \ {{}}) : ( TT = T & UU = U )  }  & seqt `2 = s -compound U ) by Def24;
then  seqt Rule3e Seqts2 by Def24;
then  [Seqts2,seqt] in P#3e S by Def37;
hence  x in (R#3e S) . Seqts2 by Lm26; ::_thesis: verum
end;
hence  (R#3e S) . Seqts1 c= (R#3e S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#3e S holds
b1 is isotone by Def9; ::_thesis: verum
end;
let K1, K2 be isotone RuleSet of S;
clusterK1 \/ K2 ->  isotone for RuleSet of S;
coherence
 for b1 being RuleSet of S st b1 = K1 \/ K2 holds
b1 is isotone
proof
set D = K1 \/ K2;
A7: ( K1 c= K1 \/ K2 & K2 c= K1 \/ K2 ) by XBOOLE_1:7;
 for Seqts1, Seqts2 being Subset of (S -sequents)
for f being Function st Seqts1 c= Seqts2 & f in K1 \/ K2 holds
ex g being Function st
( g in K1 \/ K2 & f . Seqts1 c= g . Seqts2 )
proof
let Seqts1, Seqts2 be Subset of (S -sequents); ::_thesis: for f being Function st Seqts1 c= Seqts2 & f in K1 \/ K2 holds
ex g being Function st
( g in K1 \/ K2 & f . Seqts1 c= g . Seqts2 )
let f be Function; ::_thesis: ( Seqts1 c= Seqts2 & f in K1 \/ K2 implies ex g being Function st
( g in K1 \/ K2 & f . Seqts1 c= g . Seqts2 ) )
set X = Seqts1;
set Y = Seqts2;
assume A8: ( Seqts1 c= Seqts2 & f in K1 \/ K2 ) ; ::_thesis: ex g being Function st
( g in K1 \/ K2 & f . Seqts1 c= g . Seqts2 )
percases ( f in K1 or not f in K1 ) ;
suppose f in K1 ; ::_thesis: ex g being Function st
( g in K1 \/ K2 & f . Seqts1 c= g . Seqts2 )
then consider g being Function such that
A9: ( g in K1 & f . Seqts1 c= g . Seqts2 ) by A8, Def10;
take  g ; ::_thesis: ( g in K1 \/ K2 & f . Seqts1 c= g . Seqts2 )
thus  ( g in K1 \/ K2 & f . Seqts1 c= g . Seqts2 ) by A9, A7; ::_thesis: verum
end;
suppose not f in K1 ; ::_thesis: ex g being Function st
( g in K1 \/ K2 & f . Seqts1 c= g . Seqts2 )
then  f in K2 by A8, XBOOLE_0:def_3;
then consider g being Function such that
A10: ( g in K2 & f . Seqts1 c= g . Seqts2 ) by A8, Def10;
take  g ; ::_thesis: ( g in K1 \/ K2 & f . Seqts1 c= g . Seqts2 )
thus  ( g in K1 \/ K2 & f . Seqts1 c= g . Seqts2 ) by A10, A7; ::_thesis: verum
end;
end;
end;
hence  for b1 being RuleSet of S st b1 = K1 \/ K2 holds
b1 is isotone by Def10; ::_thesis: verum
end;
end;

Lm32:  for X being set
for S being Language
for D being RuleSet of S st {(R#0 S)} \/ {(R#3a S)} c= D & X is D -expanded holds
(X,D) -termEq is transitive
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S st {(R#0 S)} \/ {(R#3a S)} c= D & X is D -expanded holds
(X,D) -termEq is transitive
let S be Language; ::_thesis: for D being RuleSet of S st {(R#0 S)} \/ {(R#3a S)} c= D & X is D -expanded holds
(X,D) -termEq is transitive
let D be RuleSet of S; ::_thesis: ( {(R#0 S)} \/ {(R#3a S)} c= D & X is D -expanded implies (X,D) -termEq is transitive )
set AT = AllTermsOf S;
set E = TheEqSymbOf S;
set Q = S -sequents ;
set AF = AllFormulasOf S;
set Phi = X;
set R = (X,D) -termEq ;
reconsider Seqts = {} as Element of bool (S -sequents) by XBOOLE_1:2;
assume A1: {(R#0 S)} \/ {(R#3a S)} c= D ; ::_thesis: ( not X is D -expanded or (X,D) -termEq is transitive )
assume A2: X is D -expanded ; ::_thesis: (X,D) -termEq is transitive
A3: field ((X,D) -termEq) c= (AllTermsOf S) \/ (AllTermsOf S) by RELSET_1:8;
now__::_thesis:_for_x,_y,_z_being_set_st_x_in_field_((X,D)_-termEq)_&_y_in_field_((X,D)_-termEq)_&_z_in_field_((X,D)_-termEq)_&_[x,y]_in_(X,D)_-termEq_&_[y,z]_in_(X,D)_-termEq_holds_
[x,z]_in_(X,D)_-termEq
let x, y, z be set ; ::_thesis: ( x in field ((X,D) -termEq) & y in field ((X,D) -termEq) & z in field ((X,D) -termEq) & [x,y] in (X,D) -termEq & [y,z] in (X,D) -termEq implies [x,z] in (X,D) -termEq )
assume  ( x in field ((X,D) -termEq) & y in field ((X,D) -termEq) & z in field ((X,D) -termEq) ) ; ::_thesis: ( [x,y] in (X,D) -termEq & [y,z] in (X,D) -termEq implies [x,z] in (X,D) -termEq )
then reconsider tt1 = x, tt2 = y, tt3 = z as Element of AllTermsOf S by A3;
reconsider t1 = tt1, t2 = tt2, t3 = tt3 as termal string of S ;
reconsider phi1 = (<*(TheEqSymbOf S)*> ^ t1) ^ t2 as wff string of S by Lm2;
reconsider phi2 = (<*(TheEqSymbOf S)*> ^ t2) ^ t3 as wff string of S by Lm2;
reconsider phi3 = (<*(TheEqSymbOf S)*> ^ t1) ^ t3 as wff string of S by Lm2;
 [{phi1,((<*(TheEqSymbOf S)*> ^ t2) ^ t3)},((<*(TheEqSymbOf S)*> ^ t1) ^ t3)] is 1,{[{phi1},((<*(TheEqSymbOf S)*> ^ t1) ^ t2)]},{(R#3a S)} -derivable ;
then A4: [{phi1,phi2},phi3] is 1,{[{phi1},phi1]},{(R#3a S)} -derivable ;
 [{phi1,phi2},phi3] is 1 + 1, {} ,{(R#0 S)} \/ {(R#3a S)} -derivable by A4, Lm21;
then  {[{phi1,phi2},phi3]} is {} ,{(R#0 S)} \/ {(R#3a S)} -derivable by Lm12;
then  {[{phi1,phi2},phi3]} is {} ,D -derivable by Lm18, A1;
then A5: ( {phi1,phi2} = {phi1} \/ {phi2} & phi3 is {phi1,phi2},D -provable ) by Def12, ENUMSET1:1;
assume  ( [x,y] in (X,D) -termEq & [y,z] in (X,D) -termEq ) ; ::_thesis: [x,z] in (X,D) -termEq
then  ( phi1 is X,D -provable & phi2 is X,D -provable ) by Lm22;
then reconsider Phi1 = {phi1}, Phi2 = {phi2} as Subset of X by A2, Def16;
reconsider Phi3 = Phi1 \/ Phi2 as Subset of X ;
 phi3 is Phi3,D -provable by A5;
hence  [x,z] in (X,D) -termEq ; ::_thesis: verum
end;
then  (X,D) -termEq is_transitive_in field ((X,D) -termEq) by RELAT_2:def_8;
hence  (X,D) -termEq is transitive by RELAT_2:def_16; ::_thesis: verum
end;

Lm33:  for X being set
for S being Language
for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D & X is D -expanded holds
(X,D) -termEq is Equivalence_Relation of (AllTermsOf S)
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D & X is D -expanded holds
(X,D) -termEq is Equivalence_Relation of (AllTermsOf S)
let S be Language; ::_thesis: for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D & X is D -expanded holds
(X,D) -termEq is Equivalence_Relation of (AllTermsOf S)
let D be RuleSet of S; ::_thesis: ( {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D & X is D -expanded implies (X,D) -termEq is Equivalence_Relation of (AllTermsOf S) )
A1: ( {(R#2 S)} c= {(R#2 S),(R#3b S)} & {(R#3b S)} c= {(R#2 S),(R#3b S)} ) by ZFMISC_1:7;
assume  {(R#0 S),(R#3a S)} c= D ; ::_thesis: ( not {(R#2 S),(R#3b S)} c= D or not X is D -expanded or (X,D) -termEq is Equivalence_Relation of (AllTermsOf S) )
then A2: {(R#0 S)} \/ {(R#3a S)} c= D by ENUMSET1:1;
assume  {(R#2 S),(R#3b S)} c= D ; ::_thesis: ( not X is D -expanded or (X,D) -termEq is Equivalence_Relation of (AllTermsOf S) )
then A3: ( {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} ) by A1, XBOOLE_1:1, XBOOLE_1:28;
assume A4: X is D -expanded ; ::_thesis: (X,D) -termEq is Equivalence_Relation of (AllTermsOf S)
set R = (X,D) -termEq ;
thus  (X,D) -termEq is Equivalence_Relation of (AllTermsOf S) by Lm30, Lm32, A2, A4, Lm31, A3; ::_thesis: verum
end;

registration
let S be Language;
let t1, t2 be termal string of S;
cluster(<*(TheEqSymbOf S)*> ^ t1) ^ t2 ->  0 -wff for string of S;
coherence
 for b1 being string of S st b1 = (<*(TheEqSymbOf S)*> ^ t1) ^ t2 holds
b1 is 0 -wff by Lm2;
end;

definition
let S be Language;
let m be non zero Nat;
let T, U be m -element Element of (AllTermsOf S) * ;
func PairWiseEq (T,U) ->  set  equals :: FOMODEL4:def 58
 {  ((<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j)) where j is Element of Seg m, TT, UU is Function of (Seg m),(((AllSymbolsOf S) *) \ {{}}) : ( TT = T & UU = U )  }  ;
coherence
  {  ((<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j)) where j is Element of Seg m, TT, UU is Function of (Seg m),(((AllSymbolsOf S) *) \ {{}}) : ( TT = T & UU = U )  }  is set ;
end;

:: deftheorem  defines PairWiseEq FOMODEL4:def_58_:_
 for S being Language
for m being non zero Nat
for T, U being b2 -element Element of (AllTermsOf S) * holds PairWiseEq (T,U) =  {  ((<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j)) where j is Element of Seg m, TT, UU is Function of (Seg m),(((AllSymbolsOf S) *) \ {{}}) : ( TT = T & UU = U )  }  ;

definition
let S be Language;
let m be non zero Nat;
let T1, T2 be m -element Element of (AllTermsOf S) * ;
:: original: PairWiseEq
redefine func PairWiseEq (T1,T2) ->  Subset of (AllFormulasOf S);
coherence
 PairWiseEq (T1,T2) is Subset of (AllFormulasOf S)
proof
set P = PairWiseEq (T1,T2);
set SS = AllSymbolsOf S;
set E = TheEqSymbOf S;
set AT = AllTermsOf S;
set AF = AllFormulasOf S;
now__::_thesis:_for_x_being_set_st_x_in_PairWiseEq_(T1,T2)_holds_
x_in_AllFormulasOf_S
let x be set ; ::_thesis: ( x in PairWiseEq (T1,T2) implies x in AllFormulasOf S )
assume  x in PairWiseEq (T1,T2) ; ::_thesis: x in AllFormulasOf S
then consider j being Element of Seg m, T11, T22 being Function of (Seg m),(((AllSymbolsOf S) *) \ {{}}) such that
A1: ( x = (<*(TheEqSymbOf S)*> ^ (T11 . j)) ^ (T22 . j) & T11 = T1 & T22 = T2 ) ;
 m -tuples_on (AllTermsOf S) = Funcs ((Seg m),(AllTermsOf S)) by FOMODEL0:11;
then  ( T1 is Element of Funcs ((Seg m),(AllTermsOf S)) & T2 is Element of Funcs ((Seg m),(AllTermsOf S)) ) by FOMODEL0:16;
then reconsider T111 = T1, T222 = T2 as Function of (Seg m),(AllTermsOf S) ;
 ( T111 . j is Element of AllTermsOf S & T222 . j is Element of AllTermsOf S ) ;
then reconsider t1 = T111 . j, t2 = T222 . j as string of S ;
reconsider w = (<*(TheEqSymbOf S)*> ^ t1) ^ t2 as 0 -wff string of S ;
 w in AllFormulasOf S ;
hence  x in AllFormulasOf S by A1; ::_thesis: verum
end;
hence  PairWiseEq (T1,T2) is Subset of (AllFormulasOf S) by TARSKI:def_3; ::_thesis: verum
end;
end;

registration
let S be Language;
let m be non zero Nat;
let T, U be m -element Element of (AllTermsOf S) * ;
cluster PairWiseEq (T,U) ->  finite ;
coherence
 PairWiseEq (T,U) is finite
proof
set AT = AllTermsOf S;
set E = TheEqSymbOf S;
set SS = AllSymbolsOf S;
 ( T in m -tuples_on (AllTermsOf S) & U in m -tuples_on (AllTermsOf S) ) by FOMODEL0:16;
then  ( T is Element of Funcs ((Seg m),(AllTermsOf S)) & U is Element of Funcs ((Seg m),(AllTermsOf S)) ) by FOMODEL0:11;
then reconsider TT = T, UU = U as Function of (Seg m),(AllTermsOf S) ;
deffunc H1( Element of Seg m) ->  set = (<*(TheEqSymbOf S)*> ^ (TT . S)) ^ (UU . S);
set IT =  {  H1(j) where j is Element of Seg m : j in Seg m  }  ;
A1: Seg m is finite ;
  {  H1(j) where j is Element of Seg m : j in Seg m  }  is finite from FRAENKEL:sch_21(A1);
then reconsider ITT =  {  H1(j) where j is Element of Seg m : j in Seg m  }  as finite set ;
now__::_thesis:_for_x_being_set_st_x_in_PairWiseEq_(T,U)_holds_
x_in__{__H1(j)_where_j_is_Element_of_Seg_m_:_j_in_Seg_m__}_
let x be set ; ::_thesis: ( x in PairWiseEq (T,U) implies x in  {  H1(j) where j is Element of Seg m : j in Seg m  }  )
assume  x in PairWiseEq (T,U) ; ::_thesis: x in  {  H1(j) where j is Element of Seg m : j in Seg m  }
then consider j being Element of Seg m, TTT, UUU being Function of (Seg m),(((AllSymbolsOf S) *) \ {{}}) such that
A2: ( x = (<*(TheEqSymbOf S)*> ^ (TTT . j)) ^ (UUU . j) & TTT = T & UUU = U ) ;
thus  x in  {  H1(j) where j is Element of Seg m : j in Seg m  }  by A2; ::_thesis: verum
end;
then reconsider Y = PairWiseEq (T,U) as Subset of ITT by TARSKI:def_3;
 Y is finite ;
hence  PairWiseEq (T,U) is finite ; ::_thesis: verum
end;
end;

Lm34:  for S being Language
for s being low-compounding Element of S
for T, U being abs (ar b2) -element Element of (AllTermsOf S) * st s is termal holds
{[(PairWiseEq (T,U)),((<*(TheEqSymbOf S)*> ^ (s -compound T)) ^ (s -compound U))]} is {} ,{(R#3d S)} -derivable
proof
let S be Language; ::_thesis: for s being low-compounding Element of S
for T, U being abs (ar b1) -element Element of (AllTermsOf S) * st s is termal holds
{[(PairWiseEq (T,U)),((<*(TheEqSymbOf S)*> ^ (s -compound T)) ^ (s -compound U))]} is {} ,{(R#3d S)} -derivable
let s be low-compounding Element of S; ::_thesis: for T, U being abs (ar s) -element Element of (AllTermsOf S) * st s is termal holds
{[(PairWiseEq (T,U)),((<*(TheEqSymbOf S)*> ^ (s -compound T)) ^ (s -compound U))]} is {} ,{(R#3d S)} -derivable
set m = abs (ar s);
set AT = AllTermsOf S;
set E = TheEqSymbOf S;
let T, U be abs (ar s) -element Element of (AllTermsOf S) * ; ::_thesis: ( s is termal implies {[(PairWiseEq (T,U)),((<*(TheEqSymbOf S)*> ^ (s -compound T)) ^ (s -compound U))]} is {} ,{(R#3d S)} -derivable )
assume  s is termal ; ::_thesis: {[(PairWiseEq (T,U)),((<*(TheEqSymbOf S)*> ^ (s -compound T)) ^ (s -compound U))]} is {} ,{(R#3d S)} -derivable
then reconsider ss = s as termal Element of S ;
reconsider t1 = ss -compound T, t2 = ss -compound U as termal string of S ;
reconsider conclusion = (<*(TheEqSymbOf S)*> ^ t1) ^ t2 as wff string of S ;
reconsider seqt = [(PairWiseEq (T,U)),conclusion] as Element of S -sequents by Def2;
reconsider Seqts = {} as Subset of (S -sequents) by XBOOLE_1:2;
 ss is termal ;
then A1: ( s is operational & seqt `1 = PairWiseEq (T,U) & seqt `2 = (<*(TheEqSymbOf S)*> ^ (s -compound T)) ^ (s -compound U) ) by MCART_1:7;
 seqt Rule3d Seqts by A1, Def23;
then  [Seqts,seqt] in P#3d S by Def36;
then  seqt in (R#3d S) . Seqts by Lm26;
then  {seqt} c= (R#3d S) . Seqts by ZFMISC_1:31;
hence  {[(PairWiseEq (T,U)),((<*(TheEqSymbOf S)*> ^ (s -compound T)) ^ (s -compound U))]} is {} ,{(R#3d S)} -derivable by Lm19; ::_thesis: verum
end;

Lm35:  for S being Language
for s being relational Element of S
for T1, T2 being abs (ar b2) -element Element of (AllTermsOf S) * holds {[((PairWiseEq (T1,T2)) \/ {(s -compound T1)}),(s -compound T2)]} is {} ,{(R#3e S)} -derivable
proof
let S be Language; ::_thesis: for s being relational Element of S
for T1, T2 being abs (ar b1) -element Element of (AllTermsOf S) * holds {[((PairWiseEq (T1,T2)) \/ {(s -compound T1)}),(s -compound T2)]} is {} ,{(R#3e S)} -derivable
let s be relational Element of S; ::_thesis: for T1, T2 being abs (ar s) -element Element of (AllTermsOf S) * holds {[((PairWiseEq (T1,T2)) \/ {(s -compound T1)}),(s -compound T2)]} is {} ,{(R#3e S)} -derivable
set m = abs (ar s);
set AT = AllTermsOf S;
set E = TheEqSymbOf S;
set AF = AllFormulasOf S;
let T1, T2 be abs (ar s) -element Element of (AllTermsOf S) * ; ::_thesis: {[((PairWiseEq (T1,T2)) \/ {(s -compound T1)}),(s -compound T2)]} is {} ,{(R#3e S)} -derivable
reconsider w1 = s -compound T1, conclusion = s -compound T2 as 0 -wff string of S ;
 w1 in AllFormulasOf S ;
then reconsider w11 = w1 as Element of AllFormulasOf S ;
reconsider premises = (PairWiseEq (T1,T2)) \/ {w11} as Subset of (AllFormulasOf S) ;
reconsider seqt = [premises,conclusion] as Element of S -sequents by Def2;
reconsider Seqts = {} as Subset of (S -sequents) by XBOOLE_1:2;
 ( seqt `1 = {(s -compound T1)} \/ (PairWiseEq (T1,T2)) & seqt `2 = s -compound T2 ) by MCART_1:7;
then  seqt Rule3e Seqts by Def24;
then  [Seqts,seqt] in P#3e S by Def37;
then  seqt in (R#3e S) . Seqts by Lm26;
then  {seqt} c= (R#3e S) . Seqts by ZFMISC_1:31;
hence  {[((PairWiseEq (T1,T2)) \/ {(s -compound T1)}),(s -compound T2)]} is {} ,{(R#3e S)} -derivable by Lm19; ::_thesis: verum
end;

registration
let S be Language;
let s be relational Element of S;
let T1, T2 be abs (ar s) -element Element of (AllTermsOf S) * ;
cluster{[((PairWiseEq (T1,T2)) \/ {(s -compound T1)}),(s -compound T2)]} ->  {} ,{(R#3e S)} -derivable ;
coherence
 {[((PairWiseEq (T1,T2)) \/ {(s -compound T1)}),(s -compound T2)]} is {} ,{(R#3e S)} -derivable by Lm35;
end;

Lm36:  for X, x1, x2 being set
for S being Language
for D being RuleSet of S
for s being low-compounding Element of S st X is D -expanded & [x1,x2] in (abs (ar s)) -placesOf ((X,D) -termEq) holds
ex T, U being abs (ar b6) -element Element of (AllTermsOf S) * st
( T = x1 & U = x2 & PairWiseEq (T,U) c= X )
proof
let X, x1, x2 be set ; ::_thesis: for S being Language
for D being RuleSet of S
for s being low-compounding Element of S st X is D -expanded & [x1,x2] in (abs (ar s)) -placesOf ((X,D) -termEq) holds
ex T, U being abs (ar b3) -element Element of (AllTermsOf S) * st
( T = x1 & U = x2 & PairWiseEq (T,U) c= X )
let S be Language; ::_thesis: for D being RuleSet of S
for s being low-compounding Element of S st X is D -expanded & [x1,x2] in (abs (ar s)) -placesOf ((X,D) -termEq) holds
ex T, U being abs (ar b2) -element Element of (AllTermsOf S) * st
( T = x1 & U = x2 & PairWiseEq (T,U) c= X )
let D be RuleSet of S; ::_thesis: for s being low-compounding Element of S st X is D -expanded & [x1,x2] in (abs (ar s)) -placesOf ((X,D) -termEq) holds
ex T, U being abs (ar b1) -element Element of (AllTermsOf S) * st
( T = x1 & U = x2 & PairWiseEq (T,U) c= X )
let s be low-compounding Element of S; ::_thesis: ( X is D -expanded & [x1,x2] in (abs (ar s)) -placesOf ((X,D) -termEq) implies ex T, U being abs (ar s) -element Element of (AllTermsOf S) * st
( T = x1 & U = x2 & PairWiseEq (T,U) c= X ) )
set n = abs (ar s);
set AT = AllTermsOf S;
set E = TheEqSymbOf S;
set Phi = X;
set f = S -cons ;
set SS = AllSymbolsOf S;
set R = (X,D) -termEq ;
set SS = AllSymbolsOf S;
assume A1: X is D -expanded ; ::_thesis: ( not [x1,x2] in (abs (ar s)) -placesOf ((X,D) -termEq) or ex T, U being abs (ar s) -element Element of (AllTermsOf S) * st
( T = x1 & U = x2 & PairWiseEq (T,U) c= X ) )
assume  [x1,x2] in (abs (ar s)) -placesOf ((X,D) -termEq) ; ::_thesis: ex T, U being abs (ar s) -element Element of (AllTermsOf S) * st
( T = x1 & U = x2 & PairWiseEq (T,U) c= X )
then consider p, q being Element of (abs (ar s)) -tuples_on (AllTermsOf S) such that
A2: ( [x1,x2] = [p,q] & ( for i being set st i in Seg (abs (ar s)) holds
[(p . i),(q . i)] in (X,D) -termEq ) ) ;
A3: ( p = x1 & q = x2 ) by A2, XTUPLE_0:1;
reconsider T1 = x1, T2 = x2 as Element of (abs (ar s)) -tuples_on (AllTermsOf S) by A2, XTUPLE_0:1;
reconsider T11 = T1, T22 = T2 as abs (ar s) -element Element of (AllTermsOf S) * by FINSEQ_1:def_11;
take T = T11; ::_thesis: ex U being abs (ar s) -element Element of (AllTermsOf S) * st
( T = x1 & U = x2 & PairWiseEq (T,U) c= X )
take U = T22; ::_thesis: ( T = x1 & U = x2 & PairWiseEq (T,U) c= X )
thus  ( T = x1 & U = x2 ) ; ::_thesis: PairWiseEq (T,U) c= X
set Z = PairWiseEq (T,U);
 ( T1 is Element of Funcs ((Seg (abs (ar s))),(AllTermsOf S)) & T2 is Element of Funcs ((Seg (abs (ar s))),(AllTermsOf S)) ) by FOMODEL0:11;
then reconsider T111 = T1, T222 = T2 as Function of (Seg (abs (ar s))),(AllTermsOf S) ;
now__::_thesis:_for_z_being_set_st_z_in_PairWiseEq_(T,U)_holds_
z_in_X
let z be set ; ::_thesis: ( z in PairWiseEq (T,U) implies z in X )
assume  z in PairWiseEq (T,U) ; ::_thesis: z in X
then consider j being Element of Seg (abs (ar s)), TT, UU being Function of (Seg (abs (ar s))),(((AllSymbolsOf S) *) \ {{}}) such that
A4: ( z = (<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j) & TT = T11 & UU = T22 ) ;
reconsider t11 = T111 . j, t22 = T222 . j as Element of AllTermsOf S ;
reconsider t1 = t11, t2 = t22 as termal string of S ;
 [(T111 . j),(T222 . j)] in (X,D) -termEq by A2, A3;
then  (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is X,D -provable by Lm22;
then  {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} c= X by A1, Def16;
hence  z in X by A4, ZFMISC_1:31; ::_thesis: verum
end;
hence  PairWiseEq (T,U) c= X by TARSKI:def_3; ::_thesis: verum
end;

Lm37:  for X being set
for S being Language
for D being RuleSet of S
for s being low-compounding Element of S st {(R#3d S)} c= D & X is D -expanded & s is termal holds
X -freeInterpreter s is (X,D) -termEq -respecting
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S
for s being low-compounding Element of S st {(R#3d S)} c= D & X is D -expanded & s is termal holds
X -freeInterpreter s is (X,D) -termEq -respecting
let S be Language; ::_thesis: for D being RuleSet of S
for s being low-compounding Element of S st {(R#3d S)} c= D & X is D -expanded & s is termal holds
X -freeInterpreter s is (X,D) -termEq -respecting
let D be RuleSet of S; ::_thesis: for s being low-compounding Element of S st {(R#3d S)} c= D & X is D -expanded & s is termal holds
X -freeInterpreter s is (X,D) -termEq -respecting
let s be low-compounding Element of S; ::_thesis: ( {(R#3d S)} c= D & X is D -expanded & s is termal implies X -freeInterpreter s is (X,D) -termEq -respecting )
set n = abs (ar s);
set AT = AllTermsOf S;
set E = TheEqSymbOf S;
set Phi = X;
set R = (X,D) -termEq ;
set I = X -freeInterpreter s;
assume A1: {(R#3d S)} c= D ; ::_thesis: ( not X is D -expanded or not s is termal or X -freeInterpreter s is (X,D) -termEq -respecting )
assume A2: X is D -expanded ; ::_thesis: ( not s is termal or X -freeInterpreter s is (X,D) -termEq -respecting )
assume  s is termal ; ::_thesis: X -freeInterpreter s is (X,D) -termEq -respecting
then reconsider ss = s as termal Element of S ;
A3: not ss is relational ;
now__::_thesis:_for_x1,_x2_being_set_st_[x1,x2]_in_(abs_(ar_s))_-placesOf_((X,D)_-termEq)_holds_
[((X_-freeInterpreter_s)_._x1),((X_-freeInterpreter_s)_._x2)]_in_(X,D)_-termEq
let x1, x2 be set ; ::_thesis: ( [x1,x2] in (abs (ar s)) -placesOf ((X,D) -termEq) implies [((X -freeInterpreter s) . x1),((X -freeInterpreter s) . x2)] in (X,D) -termEq )
assume  [x1,x2] in (abs (ar s)) -placesOf ((X,D) -termEq) ; ::_thesis: [((X -freeInterpreter s) . x1),((X -freeInterpreter s) . x2)] in (X,D) -termEq
then consider T1, T2 being abs (ar s) -element Element of (AllTermsOf S) *  such that
A4: ( T1 = x1 & T2 = x2 & PairWiseEq (T1,T2) c= X ) by Lm36, A2;
set Z = PairWiseEq (T1,T2);
reconsider t1 = ss -compound T1, t2 = ss -compound T2 as termal string of S ;
reconsider ZZ = PairWiseEq (T1,T2) as Subset of X by A4;
 {[(PairWiseEq (T1,T2)),((<*(TheEqSymbOf S)*> ^ t1) ^ t2)]} is {} ,{(R#3d S)} -derivable by Lm34;
then A5: {[(PairWiseEq (T1,T2)),((<*(TheEqSymbOf S)*> ^ t1) ^ t2)]} is {} ,D -derivable by A1, Lm18;
A6: (<*(TheEqSymbOf S)*> ^ t1) ^ t2 is ZZ,D -provable by A5, Def12;
 ( (X -freeInterpreter s) . T1 = t1 & (X -freeInterpreter s) . T2 = t2 ) by FOMODEL3:6;
hence  [((X -freeInterpreter s) . x1),((X -freeInterpreter s) . x2)] in (X,D) -termEq by A6, A4; ::_thesis: verum
end;
then  X -freeInterpreter s is (abs (ar s)) -placesOf ((X,D) -termEq),(X,D) -termEq -respecting by FOMODEL3:def_9;
hence  X -freeInterpreter s is (X,D) -termEq -respecting by A3, FOMODEL3:def_10; ::_thesis: verum
end;

Lm38:  for X, x1, x2 being set
for S being Language
for r being relational Element of S
for D being RuleSet of S st {(R#3e S)} c= D & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) & (r -compound) . x1 in X holds
(r -compound) . x2 in X
proof
let X, x1, x2 be set ; ::_thesis: for S being Language
for r being relational Element of S
for D being RuleSet of S st {(R#3e S)} c= D & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) & (r -compound) . x1 in X holds
(r -compound) . x2 in X
let S be Language; ::_thesis: for r being relational Element of S
for D being RuleSet of S st {(R#3e S)} c= D & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) & (r -compound) . x1 in X holds
(r -compound) . x2 in X
let r be relational Element of S; ::_thesis: for D being RuleSet of S st {(R#3e S)} c= D & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) & (r -compound) . x1 in X holds
(r -compound) . x2 in X
let D be RuleSet of S; ::_thesis: ( {(R#3e S)} c= D & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) & (r -compound) . x1 in X implies (r -compound) . x2 in X )
set s = r;
set n = abs (ar r);
set AT = AllTermsOf S;
set E = TheEqSymbOf S;
set Phi = X;
set f = r -compound ;
set R = (X,D) -termEq ;
assume A1: {(R#3e S)} c= D ; ::_thesis: ( not X is D -expanded or not [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) or not (r -compound) . x1 in X or (r -compound) . x2 in X )
assume A2: X is D -expanded ; ::_thesis: ( not [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) or not (r -compound) . x1 in X or (r -compound) . x2 in X )
assume  [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) ; ::_thesis: ( not (r -compound) . x1 in X or (r -compound) . x2 in X )
then consider T1, T2 being abs (ar r) -element Element of (AllTermsOf S) *  such that
A3: ( T1 = x1 & T2 = x2 & PairWiseEq (T1,T2) c= X ) by Lm36, A2;
set Z = PairWiseEq (T1,T2);
reconsider w1 = r -compound T1, w2 = r -compound T2 as 0 -wff string of S ;
A4: ( (r -compound) . x1 = w1 & (r -compound) . x2 = w2 ) by A3, FOMODEL3:def_2;
assume  (r -compound) . x1 in X ; ::_thesis: (r -compound) . x2 in X
then reconsider X1 = {w1} as Subset of X by A4, ZFMISC_1:31;
reconsider ZZ = PairWiseEq (T1,T2) as Subset of X by A3;
reconsider ZZZ = ZZ \/ X1 as Subset of X ;
 {[((PairWiseEq (T1,T2)) \/ {(r -compound T1)}),(r -compound T2)]} is {} ,{(R#3e S)} -derivable ;
then  {[ZZZ,w2]} is {} ,D -derivable by Lm18, A1;
then  w2 is ZZZ,D -provable by Def12;
then  {w2} c= X by A2, Def16;
hence  (r -compound) . x2 in X by A4, ZFMISC_1:31; ::_thesis: verum
end;

Lm39:  for X, x1, x2 being set
for S being Language
for r being relational Element of S
for D being RuleSet of S st {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) holds
( (r -compound) . x1 in X iff (r -compound) . x2 in X )
proof
let X, x1, x2 be set ; ::_thesis: for S being Language
for r being relational Element of S
for D being RuleSet of S st {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) holds
( (r -compound) . x1 in X iff (r -compound) . x2 in X )
let S be Language; ::_thesis: for r being relational Element of S
for D being RuleSet of S st {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) holds
( (r -compound) . x1 in X iff (r -compound) . x2 in X )
let r be relational Element of S; ::_thesis: for D being RuleSet of S st {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) holds
( (r -compound) . x1 in X iff (r -compound) . x2 in X )
let D be RuleSet of S; ::_thesis: ( {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) implies ( (r -compound) . x1 in X iff (r -compound) . x2 in X ) )
set s = r;
set n = abs (ar r);
set AT = AllTermsOf S;
set E = TheEqSymbOf S;
set Phi = X;
set f = r -compound ;
set R = (X,D) -termEq ;
assume A1: ( {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded & [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) ) ; ::_thesis: ( (r -compound) . x1 in X iff (r -compound) . x2 in X )
then reconsider Rr = (X,D) -termEq as total symmetric Relation of (AllTermsOf S) by Lm30, Lm31;
thus  ( (r -compound) . x1 in X implies (r -compound) . x2 in X ) by A1, Lm38; ::_thesis: ( (r -compound) . x2 in X implies (r -compound) . x1 in X )
reconsider RR = (abs (ar r)) -placesOf Rr as total symmetric Relation of ((abs (ar r)) -tuples_on (AllTermsOf S)) ;
 [x2,x1] in RR by A1, EQREL_1:6;
hence  ( (r -compound) . x2 in X implies (r -compound) . x1 in X ) by A1, Lm38; ::_thesis: verum
end;

Lm40:  for U being non empty set
for Y being set
for x, y being Element of U holds
not ( ( x in Y implies y in Y ) & ( y in Y implies x in Y ) & not [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN )
proof
let U be non empty set ; ::_thesis: for Y being set
for x, y being Element of U holds
not ( ( x in Y implies y in Y ) & ( y in Y implies x in Y ) & not [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN )
let Y be set ; ::_thesis: for x, y being Element of U holds
not ( ( x in Y implies y in Y ) & ( y in Y implies x in Y ) & not [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN )
let x, y be Element of U; ::_thesis: not ( ( x in Y implies y in Y ) & ( y in Y implies x in Y ) & not [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN )
set f = chi (Y,U);
assume A1: ( x in Y iff y in Y ) ; ::_thesis: [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN
percases ( x in Y or not x in Y ) ;
suppose x in Y ; ::_thesis: [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN
then  ( (chi (Y,U)) . x = 1 & y in Y ) by A1, RFUNCT_1:63;
then  ( (chi (Y,U)) . x = 1 & (chi (Y,U)) . y = 1 & 1 in BOOLEAN ) by RFUNCT_1:63;
hence  [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN by RELAT_1:def_10; ::_thesis: verum
end;
suppose not x in Y ; ::_thesis: [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN
then  ( (chi (Y,U)) . x = 0 & not y in Y ) by A1, RFUNCT_1:64;
then  ( (chi (Y,U)) . x = 0 & (chi (Y,U)) . y = 0 & 0 in BOOLEAN ) by RFUNCT_1:64;
hence  [((chi (Y,U)) . x),((chi (Y,U)) . y)] in id BOOLEAN by RELAT_1:def_10; ::_thesis: verum
end;
end;
end;

Lm41:  for X being set
for S being Language
for r being relational Element of S
for D being RuleSet of S st {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded holds
X -freeInterpreter r is (X,D) -termEq -respecting
proof
let X be set ; ::_thesis: for S being Language
for r being relational Element of S
for D being RuleSet of S st {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded holds
X -freeInterpreter r is (X,D) -termEq -respecting
let S be Language; ::_thesis: for r being relational Element of S
for D being RuleSet of S st {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded holds
X -freeInterpreter r is (X,D) -termEq -respecting
let r be relational Element of S; ::_thesis: for D being RuleSet of S st {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded holds
X -freeInterpreter r is (X,D) -termEq -respecting
let D be RuleSet of S; ::_thesis: ( {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded implies X -freeInterpreter r is (X,D) -termEq -respecting )
assume A1: ( {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded ) ; ::_thesis: X -freeInterpreter r is (X,D) -termEq -respecting
set AT = AllTermsOf S;
set R = (X,D) -termEq ;
set I = X -freeInterpreter r;
set AF = AtomicFormulasOf S;
set ch = chi (X,(AtomicFormulasOf S));
set SS = AllSymbolsOf S;
set g = r -compound ;
set m = abs (ar r);
now__::_thesis:_for_x1,_x2_being_set_st_[x1,x2]_in_(abs_(ar_r))_-placesOf_((X,D)_-termEq)_holds_
[((X_-freeInterpreter_r)_._x1),((X_-freeInterpreter_r)_._x2)]_in_id_BOOLEAN
let x1, x2 be set ; ::_thesis: ( [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) implies [((X -freeInterpreter r) . x1),((X -freeInterpreter r) . x2)] in id BOOLEAN )
assume A2: [x1,x2] in (abs (ar r)) -placesOf ((X,D) -termEq) ; ::_thesis: [((X -freeInterpreter r) . x1),((X -freeInterpreter r) . x2)] in id BOOLEAN
then consider T1, T2 being abs (ar r) -element Element of (AllTermsOf S) *  such that
A3: ( T1 = x1 & T2 = x2 & PairWiseEq (T1,T2) c= X ) by Lm36, A1;
reconsider w1 = r -compound T1, w2 = r -compound T2 as 0 -wff string of S ;
 ( w1 in AtomicFormulasOf S & w2 in AtomicFormulasOf S ) ;
then reconsider w11 = w1, w22 = w2 as Element of AtomicFormulasOf S ;
A4: ( (r -compound) . x1 = w11 & (r -compound) . x2 = w22 ) by A3, FOMODEL3:def_2;
 ( w11 in X iff w22 in X ) by A4, A1, A2, Lm39;
then  ( [((chi (X,(AtomicFormulasOf S))) . w1),((chi (X,(AtomicFormulasOf S))) . w2)] in id BOOLEAN & (X -freeInterpreter r) . T1 = (chi (X,(AtomicFormulasOf S))) . w1 & (X -freeInterpreter r) . T2 = (chi (X,(AtomicFormulasOf S))) . w2 ) by Lm40, FOMODEL3:6;
hence  [((X -freeInterpreter r) . x1),((X -freeInterpreter r) . x2)] in id BOOLEAN by A3; ::_thesis: verum
end;
then  X -freeInterpreter r is (abs (ar r)) -placesOf ((X,D) -termEq), id BOOLEAN -respecting by FOMODEL3:def_9;
hence  X -freeInterpreter r is (X,D) -termEq -respecting by FOMODEL3:def_10; ::_thesis: verum
end;

Lm42:  for U being non empty set
for X, x being set
for R being total reflexive Relation of U
for f being Function of X,U st x in X holds
f is {[x,x]},R -respecting
proof
let U be non empty set ; ::_thesis: for X, x being set
for R being total reflexive Relation of U
for f being Function of X,U st x in X holds
f is {[x,x]},R -respecting
let X, x be set ; ::_thesis: for R being total reflexive Relation of U
for f being Function of X,U st x in X holds
f is {[x,x]},R -respecting
let R be total reflexive Relation of U; ::_thesis: for f being Function of X,U st x in X holds
f is {[x,x]},R -respecting
let f be Function of X,U; ::_thesis: ( x in X implies f is {[x,x]},R -respecting )
reconsider E = {[x,x]} as Relation ;
assume A1: x in X ; ::_thesis: f is {[x,x]},R -respecting
then reconsider XX = X as non empty set ;
reconsider ff = f as Function of XX,U ;
now__::_thesis:_for_x1,_x2_being_set_st_[x1,x2]_in_E_holds_
[(f_._x1),(f_._x2)]_in_R
let x1, x2 be set ; ::_thesis: ( [x1,x2] in E implies [(f . x1),(f . x2)] in R )
assume  [x1,x2] in E ; ::_thesis: [(f . x1),(f . x2)] in R
then A2: [x1,x2] = [x,x] by TARSKI:def_1;
then A3: ( x1 = x & x2 = x ) by XTUPLE_0:1;
reconsider x11 = x1, x22 = x2 as Element of XX by A1, A2, XTUPLE_0:1;
 ( ff . x11 in U & ff . x22 in U & ff . x11 = ff . x22 ) by A3;
hence  [(f . x1),(f . x2)] in R by EQREL_1:5; ::_thesis: verum
end;
hence  f is {[x,x]},R -respecting by FOMODEL3:def_9; ::_thesis: verum
end;

Lm43:  for U being non empty set
for S being Language
for l being literal Element of S
for E being total reflexive Relation of U
for f being Interpreter of l,U holds f is E -respecting
proof
let U be non empty set ; ::_thesis: for S being Language
for l being literal Element of S
for E being total reflexive Relation of U
for f being Interpreter of l,U holds f is E -respecting
let S be Language; ::_thesis: for l being literal Element of S
for E being total reflexive Relation of U
for f being Interpreter of l,U holds f is E -respecting
let l be literal Element of S; ::_thesis: for E being total reflexive Relation of U
for f being Interpreter of l,U holds f is E -respecting
let E be total reflexive Relation of U; ::_thesis: for f being Interpreter of l,U holds f is E -respecting
reconsider m = abs (ar l) as zero Nat ;
let f be Interpreter of l,U; ::_thesis: f is E -respecting
 m -tuples_on U = {{}} by FOMODEL0:10;
then reconsider ff = f as Function of {{}},U by FOMODEL2:def_2;
 {} in {{}} by TARSKI:def_1;
then  ff is {[{},{}]},E -respecting by Lm42;
then  f is m -placesOf E,E -respecting ;
hence  f is E -respecting by FOMODEL3:def_10; ::_thesis: verum
end;

Lm44:  for X being set
for S being Language
for l being literal Element of S
for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D & X is D -expanded holds
X -freeInterpreter l is (X,D) -termEq -respecting
proof
let X be set ; ::_thesis: for S being Language
for l being literal Element of S
for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D & X is D -expanded holds
X -freeInterpreter l is (X,D) -termEq -respecting
let S be Language; ::_thesis: for l being literal Element of S
for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D & X is D -expanded holds
X -freeInterpreter l is (X,D) -termEq -respecting
let l be literal Element of S; ::_thesis: for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D & X is D -expanded holds
X -freeInterpreter l is (X,D) -termEq -respecting
let D be RuleSet of S; ::_thesis: ( {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D & X is D -expanded implies X -freeInterpreter l is (X,D) -termEq -respecting )
set AT = AllTermsOf S;
set I = X -freeInterpreter l;
assume  ( {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D & X is D -expanded ) ; ::_thesis: X -freeInterpreter l is (X,D) -termEq -respecting
then reconsider R = (X,D) -termEq as Equivalence_Relation of (AllTermsOf S) by Lm33;
 X -freeInterpreter l is R -respecting by Lm43;
hence  X -freeInterpreter l is (X,D) -termEq -respecting ; ::_thesis: verum
end;

Lm45:  for X being set
for S being Language
for a being ofAtomicFormula Element of S
for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & D /\ {(R#3d S)} = {(R#3d S)} & {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded holds
X -freeInterpreter a is (X,D) -termEq -respecting
proof
let X be set ; ::_thesis: for S being Language
for a being ofAtomicFormula Element of S
for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & D /\ {(R#3d S)} = {(R#3d S)} & {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded holds
X -freeInterpreter a is (X,D) -termEq -respecting
let S be Language; ::_thesis: for a being ofAtomicFormula Element of S
for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & D /\ {(R#3d S)} = {(R#3d S)} & {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded holds
X -freeInterpreter a is (X,D) -termEq -respecting
let a be ofAtomicFormula Element of S; ::_thesis: for D being RuleSet of S st {(R#0 S),(R#3a S)} c= D & D /\ {(R#3d S)} = {(R#3d S)} & {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded holds
X -freeInterpreter a is (X,D) -termEq -respecting
let D be RuleSet of S; ::_thesis: ( {(R#0 S),(R#3a S)} c= D & D /\ {(R#3d S)} = {(R#3d S)} & {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded implies X -freeInterpreter a is (X,D) -termEq -respecting )
set s = a;
set AT = AllTermsOf S;
set I = X -freeInterpreter a;
set AF = AtomicFormulasOf S;
set ch = chi (X,(AtomicFormulasOf S));
set SS = AllSymbolsOf S;
set n = abs (ar a);
set f = a -compound ;
set R = (X,D) -termEq ;
assume A1: ( {(R#0 S),(R#3a S)} c= D & D /\ {(R#3d S)} = {(R#3d S)} & {(R#2 S)} /\ D = {(R#2 S)} & {(R#3b S)} /\ D = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} & X is D -expanded ) ; ::_thesis: X -freeInterpreter a is (X,D) -termEq -respecting
then reconsider S2 = {(R#2 S)}, S3 = {(R#3b S)} as Subset of D ;
A2: S2 \/ S3 c= D ;
percases ( not a is relational or a is relational ) ;
suppose not a is relational ; ::_thesis: X -freeInterpreter a is (X,D) -termEq -respecting
then reconsider ss = a as termal Element of S ;
percases ( ss is literal or not ss is literal ) ;
suppose ss is literal ; ::_thesis: X -freeInterpreter a is (X,D) -termEq -respecting
then reconsider l = ss as literal Element of S ;
 {(R#2 S),(R#3b S)} c= D by A2, ENUMSET1:1;
then  X -freeInterpreter l is (X,D) -termEq -respecting by Lm44, A1;
hence  X -freeInterpreter a is (X,D) -termEq -respecting ; ::_thesis: verum
end;
suppose not ss is literal ; ::_thesis: X -freeInterpreter a is (X,D) -termEq -respecting
then reconsider sss = ss as low-compounding Element of S ;
 X -freeInterpreter sss is (X,D) -termEq -respecting by Lm37, A1;
hence  X -freeInterpreter a is (X,D) -termEq -respecting ; ::_thesis: verum
end;
end;
end;
suppose a is relational ; ::_thesis: X -freeInterpreter a is (X,D) -termEq -respecting
then reconsider r = a as relational Element of S ;
 X -freeInterpreter r is (X,D) -termEq -respecting by Lm41, A1;
hence  X -freeInterpreter a is (X,D) -termEq -respecting ; ::_thesis: verum
end;
end;
end;

definition
let m be Nat;
let S be Language;
let D be RuleSet of S;
attrD is m -ranked  means :Def59: :: FOMODEL4:def 59
( R#0 S in D & R#2 S in D & R#3a S in D & R#3b S in D ) if m = 0
( R#0 S in D & R#2 S in D & R#3a S in D & R#3b S in D & R#3d S in D & R#3e S in D ) if m = 1
( R#0 S in D & R#1 S in D & R#2 S in D & R#3a S in D & R#3b S in D & R#3d S in D & R#3e S in D & R#4 S in D & R#5 S in D & R#6 S in D & R#7 S in D & R#8 S in D ) if m = 2
otherwise D = {} ;
consistency
 ( ( m = 0 & m = 1 implies ( R#0 S in D & R#2 S in D & R#3a S in D & R#3b S in D iff ( R#0 S in D & R#2 S in D & R#3a S in D & R#3b S in D & R#3d S in D & R#3e S in D ) ) ) & ( m = 0 & m = 2 implies ( R#0 S in D & R#2 S in D & R#3a S in D & R#3b S in D iff ( R#0 S in D & R#1 S in D & R#2 S in D & R#3a S in D & R#3b S in D & R#3d S in D & R#3e S in D & R#4 S in D & R#5 S in D & R#6 S in D & R#7 S in D & R#8 S in D ) ) ) & ( m = 1 & m = 2 implies ( R#0 S in D & R#2 S in D & R#3a S in D & R#3b S in D & R#3d S in D & R#3e S in D iff ( R#0 S in D & R#1 S in D & R#2 S in D & R#3a S in D & R#3b S in D & R#3d S in D & R#3e S in D & R#4 S in D & R#5 S in D & R#6 S in D & R#7 S in D & R#8 S in D ) ) ) ) ;
end;

:: deftheorem Def59 defines -ranked FOMODEL4:def_59_:_
 for m being Nat
for S being Language
for D being RuleSet of S holds
( ( m = 0 implies ( D is m -ranked iff ( R#0 S in D & R#2 S in D & R#3a S in D & R#3b S in D ) ) ) & ( m = 1 implies ( D is m -ranked iff ( R#0 S in D & R#2 S in D & R#3a S in D & R#3b S in D & R#3d S in D & R#3e S in D ) ) ) & ( m = 2 implies ( D is m -ranked iff ( R#0 S in D & R#1 S in D & R#2 S in D & R#3a S in D & R#3b S in D & R#3d S in D & R#3e S in D & R#4 S in D & R#5 S in D & R#6 S in D & R#7 S in D & R#8 S in D ) ) ) & ( not m = 0 & not m = 1 & not m = 2 implies ( D is m -ranked iff D = {} ) ) );

registration
let S be Language;
cluster1 -ranked  ->  0 -ranked for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
coherence
 for b1 being RuleSet of S st b1 is 1 -ranked holds
b1 is 0 -ranked
proof
let D be RuleSet of S; ::_thesis: ( D is 1 -ranked implies D is 0 -ranked )
assume  D is 1 -ranked ; ::_thesis: D is 0 -ranked
then  ( R#0 S in D & R#2 S in D & R#3a S in D & R#3b S in D ) by Def59;
hence  D is 0 -ranked by Def59; ::_thesis: verum
end;
cluster2 -ranked  -> 1 -ranked for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
coherence
 for b1 being RuleSet of S st b1 is 2 -ranked holds
b1 is 1 -ranked
proof
let D be RuleSet of S; ::_thesis: ( D is 2 -ranked implies D is 1 -ranked )
assume  D is 2 -ranked ; ::_thesis: D is 1 -ranked
then  ( R#0 S in D & R#2 S in D & R#3a S in D & R#3b S in D & R#3d S in D & R#3e S in D ) by Def59;
hence  D is 1 -ranked by Def59; ::_thesis: verum
end;
end;

definition
let S be Language;
funcS -rules  ->  RuleSet of S equals :: FOMODEL4:def 60
{(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)};
coherence
 {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} is RuleSet of S ;
end;

:: deftheorem  defines -rules FOMODEL4:def_60_:_
 for S being Language holds S -rules = {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)};

registration
let S be Language;
clusterS -rules  -> 2 -ranked for RuleSet of S;
coherence
 for b1 being RuleSet of S st b1 = S -rules holds
b1 is 2 -ranked
proof
set A = {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)};
set B = {(R#5 S),(R#6 S),(R#7 S),(R#8 S)};
set C = {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)};
 ( R#0 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} & R#1 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} & R#2 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} & R#3a S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} & R#3b S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} & R#3d S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} & R#3e S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} & R#4 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} & R#5 S in {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#6 S in {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#7 S in {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#8 S in {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} ) by ENUMSET1:def_2, ENUMSET1:def_6;
then  ( R#0 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#1 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#2 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#3a S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#3b S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#3d S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#3e S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#4 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#5 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#6 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#7 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} & R#8 S in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} ) by XBOOLE_0:def_3;
hence  for b1 being RuleSet of S st b1 = S -rules holds
b1 is 2 -ranked by Def59; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster functional 2 -ranked for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
existence
 ex b1 being RuleSet of S st b1 is 2 -ranked
proof
take  S -rules ; ::_thesis: S -rules is 2 -ranked
thus  S -rules is 2 -ranked ; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster functional 1 -ranked for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
existence
 ex b1 being RuleSet of S st b1 is 1 -ranked
proof
take  the 2 -ranked RuleSet of S ; ::_thesis: the 2 -ranked RuleSet of S is 1 -ranked
thus  the 2 -ranked RuleSet of S is 1 -ranked ; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster functional 0 -ranked for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
existence
 ex b1 being RuleSet of S st b1 is 0 -ranked
proof
take  the 1 -ranked RuleSet of S ; ::_thesis: the 1 -ranked RuleSet of S is 0 -ranked
thus  the 1 -ranked RuleSet of S is 0 -ranked ; ::_thesis: verum
end;
end;

Lm46:  for X being set
for S being Language
for a being ofAtomicFormula Element of S
for D being RuleSet of S st D is 1 -ranked & X is D -expanded holds
X -freeInterpreter a is (X,D) -termEq -respecting
proof
let X be set ; ::_thesis: for S being Language
for a being ofAtomicFormula Element of S
for D being RuleSet of S st D is 1 -ranked & X is D -expanded holds
X -freeInterpreter a is (X,D) -termEq -respecting
let S be Language; ::_thesis: for a being ofAtomicFormula Element of S
for D being RuleSet of S st D is 1 -ranked & X is D -expanded holds
X -freeInterpreter a is (X,D) -termEq -respecting
let a be ofAtomicFormula Element of S; ::_thesis: for D being RuleSet of S st D is 1 -ranked & X is D -expanded holds
X -freeInterpreter a is (X,D) -termEq -respecting
let D be RuleSet of S; ::_thesis: ( D is 1 -ranked & X is D -expanded implies X -freeInterpreter a is (X,D) -termEq -respecting )
assume A1: D is 1 -ranked ; ::_thesis: ( not X is D -expanded or X -freeInterpreter a is (X,D) -termEq -respecting )
then  ( R#0 S in D & R#3a S in D ) by Def59;
then  ( {(R#0 S)} c= D & {(R#3a S)} c= D ) by ZFMISC_1:31;
then  {(R#0 S)} \/ {(R#3a S)} c= D by XBOOLE_1:8;
then A2: {(R#0 S),(R#3a S)} c= D by ENUMSET1:1;
 ( R#3d S in D & R#2 S in D & R#3b S in D & R#3e S in D ) by A1, Def59;
then  ( {(R#3d S)} c= D & {(R#2 S)} c= D & {(R#3b S)} c= D & {(R#3e S)} c= D ) by ZFMISC_1:31;
then A3: ( D /\ {(R#3d S)} = {(R#3d S)} & D /\ {(R#2 S)} = {(R#2 S)} & D /\ {(R#3b S)} = {(R#3b S)} & D /\ {(R#3e S)} = {(R#3e S)} ) by XBOOLE_1:28;
assume  X is D -expanded ; ::_thesis: X -freeInterpreter a is (X,D) -termEq -respecting
hence  X -freeInterpreter a is (X,D) -termEq -respecting by Lm45, A2, A3; ::_thesis: verum
end;

registration
let S be Language;
let D be 1 -ranked RuleSet of S;
let X be D -expanded set ;
let a be ofAtomicFormula Element of S;
clusterX -freeInterpreter a -> (X,D) -termEq -respecting for Interpreter of a, AllTermsOf S;
coherence
 for b1 being Interpreter of a, AllTermsOf S st b1 = X -freeInterpreter a holds
b1 is (X,D) -termEq -respecting by Lm46;
end;

Lm47:  for X being set
for S being Language
for D being RuleSet of S st D is 0 -ranked & X is D -expanded holds
(X,D) -termEq is Equivalence_Relation of (AllTermsOf S)
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S st D is 0 -ranked & X is D -expanded holds
(X,D) -termEq is Equivalence_Relation of (AllTermsOf S)
let S be Language; ::_thesis: for D being RuleSet of S st D is 0 -ranked & X is D -expanded holds
(X,D) -termEq is Equivalence_Relation of (AllTermsOf S)
let D be RuleSet of S; ::_thesis: ( D is 0 -ranked & X is D -expanded implies (X,D) -termEq is Equivalence_Relation of (AllTermsOf S) )
assume  D is 0 -ranked ; ::_thesis: ( not X is D -expanded or (X,D) -termEq is Equivalence_Relation of (AllTermsOf S) )
then  ( R#0 S in D & R#3a S in D & R#2 S in D & R#3b S in D ) by Def59;
then  ( {(R#0 S)} c= D & {(R#3a S)} c= D & {(R#2 S)} c= D & {(R#3b S)} c= D ) by ZFMISC_1:31;
then  ( {(R#0 S)} \/ {(R#3a S)} c= D & {(R#2 S)} \/ {(R#3b S)} c= D ) by XBOOLE_1:8;
then A1: ( {(R#0 S),(R#3a S)} c= D & {(R#2 S),(R#3b S)} c= D ) by ENUMSET1:1;
assume  X is D -expanded ; ::_thesis: (X,D) -termEq is Equivalence_Relation of (AllTermsOf S)
hence  (X,D) -termEq is Equivalence_Relation of (AllTermsOf S) by A1, Lm33; ::_thesis: verum
end;

registration
let S be Language;
let D be 0 -ranked RuleSet of S;
let X be D -expanded set ;
cluster(X,D) -termEq  ->  total symmetric transitive for Relation of (AllTermsOf S);
coherence
 for b1 being Relation of (AllTermsOf S) st b1 = (X,D) -termEq holds
( b1 is total & b1 is symmetric & b1 is transitive ) by Lm47;
end;

registration
let S be Language;
cluster functional 0 -ranked 1 -ranked for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
existence
 ex b1 being 0 -ranked RuleSet of S st b1 is 1 -ranked
proof
set D = the 1 -ranked RuleSet of S;
reconsider DD = the 1 -ranked RuleSet of S as 0 -ranked RuleSet of S ;
take  DD ; ::_thesis: DD is 1 -ranked
thus  DD is 1 -ranked ; ::_thesis: verum
end;
end;

theorem :: FOMODEL4:1
for Y, X being set
for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) & Y is X,D1 -derivable holds
Y is X,D2 -derivable by Lm18;

registration
let S be Language;
let Sq be S -sequent-like set ;
cluster{Sq} -> S -sequents-like ;
coherence
 {Sq} is S -sequents-like
proof
set Q = S -sequents ;
 Sq in S -sequents by Def2;
then  {Sq} c= S -sequents by ZFMISC_1:31;
hence  {Sq} is S -sequents-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
let SQ1, SQ2 be S -sequents-like set ;
clusterSQ1 \/ SQ2 -> S -sequents-like for set ;
coherence
 for b1 being set st b1 = SQ1 \/ SQ2 holds
b1 is S -sequents-like
proof
set Q = S -sequents ;
reconsider X = SQ1, Y = SQ2 as Subset of (S -sequents) by Def3;
 X \/ Y c= S -sequents ;
hence  for b1 being set st b1 = SQ1 \/ SQ2 holds
b1 is S -sequents-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
let x, y be S -sequent-like set ;
cluster{x,y} -> S -sequents-like for set ;
coherence
 for b1 being set st b1 = {x,y} holds
b1 is S -sequents-like
proof
 {x,y} = {x} \/ {y} by ENUMSET1:1;
hence  for b1 being set st b1 = {x,y} holds
b1 is S -sequents-like ; ::_thesis: verum
end;
end;

Lm48:  for m, n being Nat
for x, y, z being set
for S being Language
for D1, D2, D3 being RuleSet of S
for SQ1, SQ2 being b6 -sequents-like set st D1 \/ D2 is isotone & (D1 \/ D2) \/ D3 is isotone & x is m,SQ1,D1 -derivable & y is m,SQ2,D2 -derivable & z is n,{x,y},D3 -derivable holds
z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable
proof
let m, n be Nat; ::_thesis: for x, y, z being set
for S being Language
for D1, D2, D3 being RuleSet of S
for SQ1, SQ2 being b4 -sequents-like set st D1 \/ D2 is isotone & (D1 \/ D2) \/ D3 is isotone & x is m,SQ1,D1 -derivable & y is m,SQ2,D2 -derivable & z is n,{x,y},D3 -derivable holds
z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable
let x, y, z be set ; ::_thesis: for S being Language
for D1, D2, D3 being RuleSet of S
for SQ1, SQ2 being b1 -sequents-like set st D1 \/ D2 is isotone & (D1 \/ D2) \/ D3 is isotone & x is m,SQ1,D1 -derivable & y is m,SQ2,D2 -derivable & z is n,{x,y},D3 -derivable holds
z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable
let S be Language; ::_thesis: for D1, D2, D3 being RuleSet of S
for SQ1, SQ2 being S -sequents-like set st D1 \/ D2 is isotone & (D1 \/ D2) \/ D3 is isotone & x is m,SQ1,D1 -derivable & y is m,SQ2,D2 -derivable & z is n,{x,y},D3 -derivable holds
z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable
let D1, D2, D3 be RuleSet of S; ::_thesis: for SQ1, SQ2 being S -sequents-like set st D1 \/ D2 is isotone & (D1 \/ D2) \/ D3 is isotone & x is m,SQ1,D1 -derivable & y is m,SQ2,D2 -derivable & z is n,{x,y},D3 -derivable holds
z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable
let SQ1, SQ2 be S -sequents-like set ; ::_thesis: ( D1 \/ D2 is isotone & (D1 \/ D2) \/ D3 is isotone & x is m,SQ1,D1 -derivable & y is m,SQ2,D2 -derivable & z is n,{x,y},D3 -derivable implies z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable )
set Q = S -sequents ;
set D = D1 \/ D2;
set O1 = OneStep D1;
set O2 = OneStep D2;
set O3 = OneStep D3;
set O = OneStep (D1 \/ D2);
set OO = OneStep ((D1 \/ D2) \/ D3);
reconsider X = SQ1, Y = SQ2 as Subset of (S -sequents) by Def3;
set Z = X \/ Y;
assume A1: ( D1 \/ D2 is isotone & (D1 \/ D2) \/ D3 is isotone ) ; ::_thesis: ( not x is m,SQ1,D1 -derivable or not y is m,SQ2,D2 -derivable or not z is n,{x,y},D3 -derivable or z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable )
assume A2: ( x is m,SQ1,D1 -derivable & y is m,SQ2,D2 -derivable ) ; ::_thesis: ( not z is n,{x,y},D3 -derivable or z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable )
then A3: ( x in ((m,D1) -derivables) . X & y in ((m,D2) -derivables) . Y ) by Def7;
reconsider sq1 = x, sq2 = y as S -sequent-like set by A2;
 ( X null Y c= X \/ Y & Y null X c= X \/ Y & D1 null D2 c= D1 \/ D2 & D2 null D1 c= D1 \/ D2 ) ;
then  ( ((m,D1) -derivables) . X c= ((m,(D1 \/ D2)) -derivables) . (X \/ Y) & ((m,D2) -derivables) . Y c= ((m,(D1 \/ D2)) -derivables) . (X \/ Y) ) by Lm14, A1;
then A4: {sq1,sq2} c= (iter ((OneStep (D1 \/ D2)),m)) . (X \/ Y) by A3, ZFMISC_1:32;
assume  z is n,{x,y},D3 -derivable ; ::_thesis: z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable
then  z in ((n,D3) -derivables) . {x,y} by Def7;
then  {z} c= (iter ((OneStep D3),n)) . {x,y} by ZFMISC_1:31;
then  {z} c= (iter ((OneStep ((D1 \/ D2) \/ D3)),(m + n))) . (X \/ Y) by A4, A1, Lm20;
then  z in (((m + n),((D1 \/ D2) \/ D3)) -derivables) . (X \/ Y) by ZFMISC_1:31;
hence  z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable by Def7; ::_thesis: verum
end;

Lm49:  for x, X being set
for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D1 is isotone or D2 is isotone ) & x is X,D1 -provable holds
x is X,D2 -provable
proof
let x, X be set ; ::_thesis: for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D1 is isotone or D2 is isotone ) & x is X,D1 -provable holds
x is X,D2 -provable
let S be Language; ::_thesis: for D1, D2 being RuleSet of S st D1 c= D2 & ( D1 is isotone or D2 is isotone ) & x is X,D1 -provable holds
x is X,D2 -provable
let D1, D2 be RuleSet of S; ::_thesis: ( D1 c= D2 & ( D1 is isotone or D2 is isotone ) & x is X,D1 -provable implies x is X,D2 -provable )
assume A1: ( D1 c= D2 & ( D1 is isotone or D2 is isotone ) ) ; ::_thesis: ( not x is X,D1 -provable or x is X,D2 -provable )
assume  x is X,D1 -provable ; ::_thesis: x is X,D2 -provable
then consider seqt being set  such that
A2: ( seqt `1 c= X & seqt `2 = x & {seqt} is D1 -derivable ) by Def13;
 {seqt} is {} ,D2 -derivable by A2, A1, Lm18;
hence  x is X,D2 -provable by A2, Def13; ::_thesis: verum
end;

Lm50:  for x, X being set
for S being Language
for R being Rule of S st x in R . X holds
x is 1,X,{R} -derivable
proof
let x, X be set ; ::_thesis: for S being Language
for R being Rule of S st x in R . X holds
x is 1,X,{R} -derivable
let S be Language; ::_thesis: for R being Rule of S st x in R . X holds
x is 1,X,{R} -derivable
let R be Rule of S; ::_thesis: ( x in R . X implies x is 1,X,{R} -derivable )
set Q = S -sequents ;
set D = {R};
set O = OneStep {R};
set f = iter ((OneStep {R}),1);
assume A1: x in R . X ; ::_thesis: x is 1,X,{R} -derivable
then  X in dom R by FUNCT_1:def_2;
then reconsider Seqts = X as Element of bool (S -sequents) ;
iter ((OneStep {R}),1) = OneStep {R} by FUNCT_7:70
.= R by Lm7 ;
then  x in ((1,{R}) -derivables) . Seqts by A1;
hence  x is 1,X,{R} -derivable by Def7; ::_thesis: verum
end;

registration
let S be Language;
let phi1, phi2 be wff string of S;
cluster[{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] -> 1,{[{(xnot phi1),(xnot phi2)},(xnot phi1)],[{(xnot phi1),(xnot phi2)},(xnot phi2)]},{(R#6 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] holds
b1 is 1,{[{(xnot phi1),(xnot phi2)},(xnot phi1)],[{(xnot phi1),(xnot phi2)},(xnot phi2)]},{(R#6 S)} -derivable
proof
set Q = S -sequents ;
set x1 = xnot phi1;
set x2 = xnot phi2;
set N = TheNorSymbOf S;
set prem = {(xnot phi1),(xnot phi2)};
set sq = [{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)];
set sq1 = [{(xnot phi1),(xnot phi2)},(xnot phi1)];
set sq2 = [{(xnot phi1),(xnot phi2)},(xnot phi2)];
set SQ = {[{(xnot phi1),(xnot phi2)},(xnot phi1)],[{(xnot phi1),(xnot phi2)},(xnot phi2)]};
reconsider seqt = [{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] as Element of S -sequents by Def2;
reconsider Seqts = {[{(xnot phi1),(xnot phi2)},(xnot phi1)],[{(xnot phi1),(xnot phi2)},(xnot phi2)]} as Element of bool (S -sequents) by Def3;
 ( {[{(xnot phi1),(xnot phi2)},(xnot phi1)]} \ Seqts = {} & {[{(xnot phi1),(xnot phi2)},(xnot phi2)]} \ Seqts = {} & ([{(xnot phi1),(xnot phi2)},(xnot phi1)] `1) \+\ {(xnot phi1),(xnot phi2)} = {} & ([{(xnot phi1),(xnot phi2)},(xnot phi2)] `1) \+\ {(xnot phi1),(xnot phi2)} = {} & ([{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `1) \+\ {(xnot phi1),(xnot phi2)} = {} & ([{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `1) \+\ {(xnot phi1),(xnot phi2)} = {} & ([{(xnot phi1),(xnot phi2)},(xnot phi1)] `2) \+\ (xnot phi1) = {} & ([{(xnot phi1),(xnot phi2)},(xnot phi2)] `2) \+\ (xnot phi2) = {} & ([{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `2) \+\ ((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2) = {} ) ;
then  ( [{(xnot phi1),(xnot phi2)},(xnot phi1)] in Seqts & [{(xnot phi1),(xnot phi2)},(xnot phi2)] in Seqts & [{(xnot phi1),(xnot phi2)},(xnot phi1)] `1 = {(xnot phi1),(xnot phi2)} & [{(xnot phi1),(xnot phi2)},(xnot phi2)] `1 = {(xnot phi1),(xnot phi2)} & [{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `1 = {(xnot phi1),(xnot phi2)} & [{(xnot phi1),(xnot phi2)},(xnot phi1)] `2 = xnot phi1 & [{(xnot phi1),(xnot phi2)},(xnot phi2)] `2 = xnot phi2 & [{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 ) by ZFMISC_1:60;
then  seqt Rule6 Seqts by Def27;
then  [Seqts,seqt] in P#6 S by Def40;
then  seqt in (R#6 S) . Seqts by Lm26;
hence  for b1 being set st b1 = [{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] holds
b1 is 1,{[{(xnot phi1),(xnot phi2)},(xnot phi1)],[{(xnot phi1),(xnot phi2)},(xnot phi2)]},{(R#6 S)} -derivable by Lm50; ::_thesis: verum
end;
end;

registration
let S be Language;
let phi1, phi2 be wff string of S;
cluster[{phi1,phi2},phi2] -> 1, {} ,{(R#0 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [{phi1,phi2},phi2] holds
b1 is 1, {} ,{(R#0 S)} -derivable
proof
 [{phi1,phi2},phi2] = [{phi2,phi1},phi2] ;
hence  for b1 being set st b1 = [{phi1,phi2},phi2] holds
b1 is 1, {} ,{(R#0 S)} -derivable ; ::_thesis: verum
end;
end;

theorem Th2: :: FOMODEL4:2
for S being Language
for SQ being b1 -sequents-like set
for Sq being b1 -sequent-like set
for R being Relation of (bool (S -sequents)),(S -sequents) st [SQ,Sq] in R holds
Sq in (FuncRule R) . SQ
proof
let S be Language; ::_thesis: for SQ being S -sequents-like set
for Sq being S -sequent-like set
for R being Relation of (bool (S -sequents)),(S -sequents) st [SQ,Sq] in R holds
Sq in (FuncRule R) . SQ
let SQ be S -sequents-like set ; ::_thesis: for Sq being S -sequent-like set
for R being Relation of (bool (S -sequents)),(S -sequents) st [SQ,Sq] in R holds
Sq in (FuncRule R) . SQ
let Sq be S -sequent-like set ; ::_thesis: for R being Relation of (bool (S -sequents)),(S -sequents) st [SQ,Sq] in R holds
Sq in (FuncRule R) . SQ
set Q = S -sequents ;
reconsider seqt = Sq as Element of S -sequents by Def2;
reconsider Seqts = SQ as Element of bool (S -sequents) by Def3;
let R be Relation of (bool (S -sequents)),(S -sequents); ::_thesis: ( [SQ,Sq] in R implies Sq in (FuncRule R) . SQ )
 ( [Seqts,seqt] in R implies seqt in (FuncRule R) . Seqts ) by Lm26;
hence  ( [SQ,Sq] in R implies Sq in (FuncRule R) . SQ ) ; ::_thesis: verum
end;

theorem :: FOMODEL4:3
for x, X being set
for S being Language
for R being Rule of S st x in R . X holds
x is 1,X,{R} -derivable by Lm50;

definition
let S be Language;
let D be RuleSet of S;
let X be set ;
redefine attr X is D -expanded  means :Def61: :: FOMODEL4:def 61
 for x being set st x is X,D -provable holds
x in X;
compatibility
 ( X is D -expanded iff for x being set st x is X,D -provable holds
x in X )
proof
defpred S1[] means  for x being set st x is X,D -provable holds
x in X;
thus  ( X is D -expanded implies S1[] ) ::_thesis: ( ( for x being set st x is X,D -provable holds
x in X ) implies X is D -expanded )
proof
assume A1: X is D -expanded ; ::_thesis: S1[]
hereby  ::_thesis: verum
let x be set ; ::_thesis: ( x is X,D -provable implies x in X )
assume  x is X,D -provable ; ::_thesis: x in X
then  {x} c= X by A1, Def16;
hence  x in X by ZFMISC_1:31; ::_thesis: verum
end;
end;
assume A2: S1[] ; ::_thesis: X is D -expanded
hereby  :: according to FOMODEL4:def_16 ::_thesis: verum
let x be set ; ::_thesis: ( x is X,D -provable implies {x} c= X )
assume  x is X,D -provable ; ::_thesis: {x} c= X
then  x in X by A2;
hence  {x} c= X by ZFMISC_1:31; ::_thesis: verum
end;
end;
end;

:: deftheorem Def61 defines -expanded FOMODEL4:def_61_:_
 for S being Language
for D being RuleSet of S
for X being set holds
( X is D -expanded iff for x being set st x is X,D -provable holds
x in X );

theorem Th4: :: FOMODEL4:4
for X being set
for S being Language
for phi being wff string of S st phi in X holds
phi is X,{(R#0 S)} -provable
proof
let X be set ; ::_thesis: for S being Language
for phi being wff string of S st phi in X holds
phi is X,{(R#0 S)} -provable
let S be Language; ::_thesis: for phi being wff string of S st phi in X holds
phi is X,{(R#0 S)} -provable
let phi be wff string of S; ::_thesis: ( phi in X implies phi is X,{(R#0 S)} -provable )
assume  phi in X ; ::_thesis: phi is X,{(R#0 S)} -provable
then reconsider Xphi = {phi} as Subset of X by ZFMISC_1:31;
 {[{phi},phi]} is {} ,{(R#0 S)} -derivable ;
then  phi is Xphi,{(R#0 S)} -provable by Def12;
hence  phi is X,{(R#0 S)} -provable ; ::_thesis: verum
end;

theorem :: FOMODEL4:5
for m, n being Nat
for x, y, z being set
for S being Language
for D1, D2, D3 being RuleSet of S
for SQ1, SQ2 being b6 -sequents-like set st D1 \/ D2 is isotone & (D1 \/ D2) \/ D3 is isotone & x is m,SQ1,D1 -derivable & y is m,SQ2,D2 -derivable & z is n,{x,y},D3 -derivable holds
z is m + n,SQ1 \/ SQ2,(D1 \/ D2) \/ D3 -derivable by Lm48;

theorem :: FOMODEL4:6
for m, n being Nat
for y, X, z being set
for S being Language
for D1, D2 being RuleSet of S st D1 is isotone & D1 \/ D2 is isotone & y is m,X,D1 -derivable & z is n,{y},D2 -derivable holds
z is m + n,X,D1 \/ D2 -derivable by Lm21;

theorem :: FOMODEL4:7
for m being Nat
for x, X being set
for S being Language
for D being RuleSet of S st x is m,X,D -derivable holds
{x} is X,D -derivable by Lm12;

registration
let S be Language;
cluster R#6 S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#6 S holds
b1 is isotone
proof
set R = R#6 S;
set Q = S -sequents ;
now__::_thesis:_for_Seqts,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts_c=_Seqts2_holds_
(R#6_S)_._Seqts_c=_(R#6_S)_._Seqts2
let Seqts, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts c= Seqts2 implies (R#6 S) . Seqts c= (R#6 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume A1: Seqts c= Seqts2 ; ::_thesis: (R#6 S) . Seqts c= (R#6 S) . Seqts2
now__::_thesis:_for_x_being_set_st_x_in_(R#6_S)_._Seqts_holds_
x_in_(R#6_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#6 S) . Seqts implies x in (R#6 S) . Seqts2 )
assume A2: x in (R#6 S) . Seqts ; ::_thesis: x in (R#6 S) . Seqts2
reconsider seqt = x as Element of S -sequents by A2;
 [Seqts,seqt] in P#6 S by A2, Lm29;
then  seqt Rule6 Seqts by Def40;
then consider y1, y2 being set , phi1, phi2 being wff string of S such that
A3: ( y1 in Seqts & y2 in Seqts & y1 `1 = y2 `1 & y2 `1 = seqt `1 & y1 `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi1 & y2 `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi2 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 ) by Def27;
 seqt Rule6 Seqts2 by Def27, A3, A1;
then  [Seqts2,seqt] in P#6 S by Def40;
hence  x in (R#6 S) . Seqts2 by Th2; ::_thesis: verum
end;
hence  (R#6 S) . Seqts c= (R#6 S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#6 S holds
b1 is isotone by Def9; ::_thesis: verum
end;
end;

theorem :: FOMODEL4:8
for x, X being set
for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D1 is isotone or D2 is isotone ) & x is X,D1 -provable holds
x is X,D2 -provable by Lm49;

theorem :: FOMODEL4:9
for X, Y, x being set
for S being Language
for D being RuleSet of S st X c= Y & x is X,D -provable holds
x is Y,D -provable ;

registration
let S be Language;
cluster R#8 S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#8 S holds
b1 is isotone
proof
set R = R#8 S;
set Q = S -sequents ;
now__::_thesis:_for_Seqts,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts_c=_Seqts2_holds_
(R#8_S)_._Seqts_c=_(R#8_S)_._Seqts2
let Seqts, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts c= Seqts2 implies (R#8 S) . Seqts c= (R#8 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume A1: Seqts c= Seqts2 ; ::_thesis: (R#8 S) . Seqts c= (R#8 S) . Seqts2
now__::_thesis:_for_x_being_set_st_x_in_(R#8_S)_._Seqts_holds_
x_in_(R#8_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#8 S) . Seqts implies x in (R#8 S) . Seqts2 )
assume A2: x in (R#8 S) . Seqts ; ::_thesis: x in (R#8 S) . Seqts2
reconsider seqt = x as Element of S -sequents by A2;
 [Seqts,seqt] in P#8 S by A2, Lm29;
then  seqt Rule8 Seqts by Def42;
then consider y1, y2 being set , phi, phi1, phi2 being wff string of S such that
A3: ( y1 in Seqts & y2 in Seqts & y1 `1 = y2 `1 & y1 `2 = phi1 & y2 `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & {phi} \/ (seqt `1) = y1 `1 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi) ^ phi ) by Def29;
 seqt Rule8 Seqts2 by Def29, A3, A1;
then  [Seqts2,seqt] in P#8 S by Def42;
hence  x in (R#8 S) . Seqts2 by Th2; ::_thesis: verum
end;
hence  (R#8 S) . Seqts c= (R#8 S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#8 S holds
b1 is isotone by Def9; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster R#1 S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#1 S holds
b1 is isotone
proof
set R = R#1 S;
set Q = S -sequents ;
now__::_thesis:_for_Seqts,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts_c=_Seqts2_holds_
(R#1_S)_._Seqts_c=_(R#1_S)_._Seqts2
let Seqts, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts c= Seqts2 implies (R#1 S) . Seqts c= (R#1 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume A1: Seqts c= Seqts2 ; ::_thesis: (R#1 S) . Seqts c= (R#1 S) . Seqts2
now__::_thesis:_for_x_being_set_st_x_in_(R#1_S)_._Seqts_holds_
x_in_(R#1_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#1 S) . Seqts implies x in (R#1 S) . Seqts2 )
assume A2: x in (R#1 S) . Seqts ; ::_thesis: x in (R#1 S) . Seqts2
reconsider seqt = x as Element of S -sequents by A2;
 [Seqts,seqt] in P#1 S by A2, Lm29;
then  seqt Rule1 Seqts by Def32;
then consider y being set  such that
A3: ( y in Seqts & y `1 c= seqt `1 & seqt `2 = y `2 ) by Def19;
 seqt Rule1 Seqts2 by Def19, A3, A1;
then  [Seqts2,seqt] in P#1 S by Def32;
hence  x in (R#1 S) . Seqts2 by Th2; ::_thesis: verum
end;
hence  (R#1 S) . Seqts c= (R#1 S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#1 S holds
b1 is isotone by Def9; ::_thesis: verum
end;
end;

theorem :: FOMODEL4:10
for y being set
for S being Language
for D being RuleSet of S
for SQ being b2 -sequents-like set st {y} is SQ,D -derivable holds
ex mm being Element of NAT st y is mm,SQ,D -derivable
proof
let y be set ; ::_thesis: for S being Language
for D being RuleSet of S
for SQ being b1 -sequents-like set st {y} is SQ,D -derivable holds
ex mm being Element of NAT st y is mm,SQ,D -derivable
let S be Language; ::_thesis: for D being RuleSet of S
for SQ being S -sequents-like set st {y} is SQ,D -derivable holds
ex mm being Element of NAT st y is mm,SQ,D -derivable
let D be RuleSet of S; ::_thesis: for SQ being S -sequents-like set st {y} is SQ,D -derivable holds
ex mm being Element of NAT st y is mm,SQ,D -derivable
let SQ be S -sequents-like set ; ::_thesis: ( {y} is SQ,D -derivable implies ex mm being Element of NAT st y is mm,SQ,D -derivable )
set Q = S -sequents ;
reconsider Seqts = SQ as Element of bool (S -sequents) by Def3;
 ( {y} is Seqts,D -derivable implies ex mm being Element of NAT st y is mm,Seqts,D -derivable ) by Lm8;
hence  ( {y} is SQ,D -derivable implies ex mm being Element of NAT st y is mm,SQ,D -derivable ) ; ::_thesis: verum
end;

registration
let S be Language;
let D be RuleSet of S;
let X be set ;
clusterX,D -derivable  -> S -sequents-like for set ;
coherence
 for b1 being set st b1 is X,D -derivable holds
b1 is S -sequents-like
proof
set Q = S -sequents ;
set O = OneStep D;
set F =  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ;
let IT be set ; ::_thesis: ( IT is X,D -derivable implies IT is S -sequents-like )
assume  IT is X,D -derivable ; ::_thesis: IT is S -sequents-like
then  IT c= union (((OneStep D) [*]) .: {X}) by Def6;
then A1: IT c= union  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  by Lm10;
now__::_thesis:_for_x_being_set_st_x_in_IT_holds_
x_in_S_-sequents
let x be set ; ::_thesis: ( x in IT implies x in S -sequents )
assume  x in IT ; ::_thesis: x in S -sequents
then consider Y being set  such that
A2: ( x in Y & Y in  {  (((mm,D) -derivables) . X) where mm is Element of NAT : verum  }  ) by A1, TARSKI:def_4;
consider mm being Element of NAT  such that
A3: Y = ((mm,D) -derivables) . X by A2;
 x is mm,X,D -derivable by Def7, A2, A3;
hence  x in S -sequents by Def2; ::_thesis: verum
end;
then  IT c= S -sequents by TARSKI:def_3;
hence  IT is S -sequents-like ; ::_thesis: verum
end;
end;

definition
let S be Language;
let D be RuleSet of S;
let X, x be set ;
redefine attr x is X,D -provable  means :Def62: :: FOMODEL4:def 62
 ex H being set ex m being Nat st
( H c= X & [H,x] is m, {} ,D -derivable );
compatibility
 ( x is X,D -provable iff ex H being set ex m being Nat st
( H c= X & [H,x] is m, {} ,D -derivable ) )
proof
set FF = AllFormulasOf S;
set Q = S -sequents ;
defpred S1[] means  ex H being set ex m being Nat st
( H c= X & [H,x] is m, {} ,D -derivable );
 {} /\ S is S -sequents-like ;
then reconsider e = {} as Element of bool (S -sequents) by Def3;
thus  ( x is X,D -provable implies S1[] ) ::_thesis: ( ex H being set ex m being Nat st
( H c= X & [H,x] is m, {} ,D -derivable ) implies x is X,D -provable )
proof
assume  x is X,D -provable ; ::_thesis: S1[]
then consider seqt being set  such that
A1: ( seqt `1 c= X & seqt `2 = x & {seqt} is D -derivable ) by Def13;
A2: ( seqt `1 c= X & seqt `2 = x & {seqt} is {} ,D -derivable ) by A1;
 ( {seqt} c= S -sequents & seqt in {seqt} ) by A2, Def3, TARSKI:def_1;
then  seqt in S -sequents ;
then consider premises being Subset of (AllFormulasOf S), conclusion being wff string of S such that
A3: ( seqt = [premises,conclusion] & premises is finite ) ;
consider mm being Element of NAT  such that
A4: seqt is mm,e,D -derivable by A2, Lm8;
take H = seqt `1 ; ::_thesis: ex m being Nat st
( H c= X & [H,x] is m, {} ,D -derivable )
take m = mm; ::_thesis: ( H c= X & [H,x] is m, {} ,D -derivable )
 ( H = premises & seqt `2 = conclusion ) by A3, MCART_1:7;
hence  ( H c= X & [H,x] is m, {} ,D -derivable ) by A1, A4, A3; ::_thesis: verum
end;
assume  S1[] ; ::_thesis: x is X,D -provable
then consider H being set , m being Nat such that
A5: ( H c= X & [H,x] is m, {} ,D -derivable ) ;
now__::_thesis:_ex_seqt_being_set_st_
(_seqt_`1_c=_X_&_seqt_`2_=_x_&_{seqt}_is_D_-derivable_)
take seqt = [H,x]; ::_thesis: ( seqt `1 c= X & seqt `2 = x & {seqt} is D -derivable )
 ( seqt `1 c= X & seqt `2 = x & {seqt} is {} ,D -derivable ) by A5, Lm12, MCART_1:7;
hence  ( seqt `1 c= X & seqt `2 = x & {seqt} is D -derivable ) ; ::_thesis: verum
end;
hence  x is X,D -provable by Def12; ::_thesis: verum
end;
end;

:: deftheorem Def62 defines -provable FOMODEL4:def_62_:_
 for S being Language
for D being RuleSet of S
for X, x being set holds
( x is X,D -provable iff ex H being set ex m being Nat st
( H c= X & [H,x] is m, {} ,D -derivable ) );

theorem Th11: :: FOMODEL4:11
for m being Nat
for x, X being set
for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) & x is m,X,D1 -derivable holds
x is m,X,D2 -derivable
proof
let m be Nat; ::_thesis: for x, X being set
for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) & x is m,X,D1 -derivable holds
x is m,X,D2 -derivable
let x, X be set ; ::_thesis: for S being Language
for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) & x is m,X,D1 -derivable holds
x is m,X,D2 -derivable
let S be Language; ::_thesis: for D1, D2 being RuleSet of S st D1 c= D2 & ( D2 is isotone or D1 is isotone ) & x is m,X,D1 -derivable holds
x is m,X,D2 -derivable
let D1, D2 be RuleSet of S; ::_thesis: ( D1 c= D2 & ( D2 is isotone or D1 is isotone ) & x is m,X,D1 -derivable implies x is m,X,D2 -derivable )
set f1 = (m,D1) -derivables ;
set f2 = (m,D2) -derivables ;
assume  ( D1 c= D2 & ( D2 is isotone or D1 is isotone ) ) ; ::_thesis: ( not x is m,X,D1 -derivable or x is m,X,D2 -derivable )
then A1: ((m,D1) -derivables) . X c= ((m,D2) -derivables) . X by Lm16;
assume  x is m,X,D1 -derivable ; ::_thesis: x is m,X,D2 -derivable
then  x in ((m,D1) -derivables) . X by Def7;
hence  x is m,X,D2 -derivable by Def7, A1; ::_thesis: verum
end;

registration
let S be Language;
cluster R#7 S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#7 S holds
b1 is isotone
proof
set R = R#7 S;
set Q = S -sequents ;
now__::_thesis:_for_Seqts,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts_c=_Seqts2_holds_
(R#7_S)_._Seqts_c=_(R#7_S)_._Seqts2
let Seqts, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts c= Seqts2 implies (R#7 S) . Seqts c= (R#7 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume A1: Seqts c= Seqts2 ; ::_thesis: (R#7 S) . Seqts c= (R#7 S) . Seqts2
now__::_thesis:_for_x_being_set_st_x_in_(R#7_S)_._Seqts_holds_
x_in_(R#7_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#7 S) . Seqts implies x in (R#7 S) . Seqts2 )
assume A2: x in (R#7 S) . Seqts ; ::_thesis: x in (R#7 S) . Seqts2
reconsider seqt = x as Element of S -sequents by A2;
 [Seqts,seqt] in P#7 S by A2, Lm29;
then  seqt Rule7 Seqts by Def41;
then consider y being set , phi1, phi2 being wff string of S such that
A3: ( y in Seqts & y `1 = seqt `1 & y `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1 ) by Def28;
 seqt Rule7 Seqts2 by Def28, A3, A1;
then  [Seqts2,seqt] in P#7 S by Def41;
hence  x in (R#7 S) . Seqts2 by Th2; ::_thesis: verum
end;
hence  (R#7 S) . Seqts c= (R#7 S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#7 S holds
b1 is isotone by Def9; ::_thesis: verum
end;
end;

theorem :: FOMODEL4:12
for x, X being set
for S being Language
for D being RuleSet of S st x is X,D -provable holds
x is wff string of S by Lm24;


registration
let S be Language;
let D1 be 0 -ranked 1 -ranked RuleSet of S;
let X be D1 -expanded set ;
cluster(S,X) -freeInterpreter  -> S, AllTermsOf S -interpreter-like (X,D1) -termEq -respecting for S, AllTermsOf S -interpreter-like Function;
coherence
 for b1 being S, AllTermsOf S -interpreter-like Function st b1 = (S,X) -freeInterpreter holds
b1 is (X,D1) -termEq -respecting
proof
set TT = AllTermsOf S;
set E = (X,D1) -termEq ;
set I = (S,X) -freeInterpreter ;
now__::_thesis:_for_o_being_own_Element_of_S_holds_((S,X)_-freeInterpreter)_._o_is_(X,D1)_-termEq_-respecting
let o be own Element of S; ::_thesis: ((S,X) -freeInterpreter) . o is (X,D1) -termEq -respecting
 ((S,X) -freeInterpreter) . o = X -freeInterpreter o by FOMODEL3:def_4;
hence  ((S,X) -freeInterpreter) . o is (X,D1) -termEq -respecting ; ::_thesis: verum
end;
hence  for b1 being S, AllTermsOf S -interpreter-like Function st b1 = (S,X) -freeInterpreter holds
b1 is (X,D1) -termEq -respecting by FOMODEL3:def_16; ::_thesis: verum
end;
end;

definition
let S be Language;
let D be 0 -ranked RuleSet of S;
let X be D -expanded set ;
funcD Henkin X ->  Function equals :: FOMODEL4:def 63
((S,X) -freeInterpreter) quotient ((X,D) -termEq);
coherence
 ((S,X) -freeInterpreter) quotient ((X,D) -termEq) is Function ;
end;

:: deftheorem  defines Henkin FOMODEL4:def_63_:_
 for S being Language
for D being 0 -ranked RuleSet of S
for X being b2 -expanded set holds D Henkin X = ((S,X) -freeInterpreter) quotient ((X,D) -termEq);

registration
let S be Language;
let D be 0 -ranked RuleSet of S;
let X be D -expanded set ;
clusterD Henkin X ->  OwnSymbolsOf S -defined ;
coherence
 D Henkin X is OwnSymbolsOf S -defined ;
end;

registration
let S be Language;
let D1 be 0 -ranked 1 -ranked RuleSet of S;
let X be D1 -expanded set ;
clusterD1 Henkin X -> S, Class ((X,D1) -termEq) -interpreter-like for Function;
coherence
 for b1 being Function st b1 = D1 Henkin X holds
b1 is S, Class ((X,D1) -termEq) -interpreter-like ;
end;

definition
let S be Language;
let D1 be 0 -ranked 1 -ranked RuleSet of S;
let X be D1 -expanded set ;
:: original: Henkin
redefine funcD1 Henkin X ->  Element of (Class ((X,D1) -termEq)) -InterpretersOf S;
coherence
 D1 Henkin X is Element of (Class ((X,D1) -termEq)) -InterpretersOf S
proof
set TT = AllTermsOf S;
set R = (X,D1) -termEq ;
set I = (S,X) -freeInterpreter ;
 ((S,X) -freeInterpreter) quotient ((X,D1) -termEq) is Element of (Class ((X,D1) -termEq)) -InterpretersOf S ;
hence  D1 Henkin X is Element of (Class ((X,D1) -termEq)) -InterpretersOf S ; ::_thesis: verum
end;
end;

registration
let S be Language;
let phi1, phi2 be wff string of S;
cluster(<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 -> {(xnot phi1),(xnot phi2)},{(R#0 S)} \/ {(R#6 S)} -provable for set ;
coherence
 for b1 being set st b1 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 holds
b1 is {(xnot phi1),(xnot phi2)},{(R#0 S)} \/ {(R#6 S)} -provable
proof
set N = TheNorSymbOf S;
set phi = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2;
set x1 = xnot phi1;
set x2 = xnot phi2;
set prem = {(xnot phi1),(xnot phi2)};
set sq = [{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)];
set sq1 = [{(xnot phi1),(xnot phi2)},(xnot phi1)];
set sq2 = [{(xnot phi1),(xnot phi2)},(xnot phi2)];
 {} /\ S is S -sequents-like ;
then reconsider SQe = {} as S -sequents-like set ;
 [{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] is 1 + 1,SQe \/ SQe,({(R#0 S)} \/ {(R#0 S)}) \/ {(R#6 S)} -derivable by Lm48;
then  {[{(xnot phi1),(xnot phi2)},((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} is {} ,{(R#0 S)} \/ {(R#6 S)} -derivable by Lm12;
hence  for b1 being set st b1 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 holds
b1 is {(xnot phi1),(xnot phi2)},{(R#0 S)} \/ {(R#6 S)} -provable by Def12; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster 0 -ranked  ->  non empty 0 -ranked for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
coherence
 for b1 being 0 -ranked RuleSet of S holds not b1 is empty by Def59;
end;

definition
let S be Language;
let x be set ;
attrx is S -premises-like  means :Def64: :: FOMODEL4:def 64
( x c= AllFormulasOf S & x is finite );
end;

:: deftheorem Def64 defines -premises-like FOMODEL4:def_64_:_
 for S being Language
for x being set holds
( x is S -premises-like iff ( x c= AllFormulasOf S & x is finite ) );

registration
let S be Language;
clusterS -premises-like  ->  finite for set ;
coherence
 for b1 being set st b1 is S -premises-like holds
b1 is finite by Def64;
end;

registration
let S be Language;
let phi be wff string of S;
cluster{phi} -> S -premises-like for set ;
coherence
 for b1 being set st b1 = {phi} holds
b1 is S -premises-like
proof
set FF = AllFormulasOf S;
 phi in AllFormulasOf S by FOMODEL2:16;
then  ( {phi} c= AllFormulasOf S & {phi} is finite ) by ZFMISC_1:31;
hence  for b1 being set st b1 = {phi} holds
b1 is S -premises-like by Def64; ::_thesis: verum
end;
end;

registration
let S be Language;
let e be empty set ;
clustere null S -> S -premises-like for set ;
coherence
 for b1 being set st b1 = e null S holds
b1 is S -premises-like
proof
set FF = AllFormulasOf S;
 e /\ (AllFormulasOf S) = e ;
hence  for b1 being set st b1 = e null S holds
b1 is S -premises-like by Def64; ::_thesis: verum
end;
end;

registration
let X be set ;
let S be Language;
clusterS -premises-like for Element of bool X;
existence
 ex b1 being Subset of X st b1 is S -premises-like
proof
 {} /\ X = {} ;
then reconsider e = {} null S as Subset of X ;
take  e ; ::_thesis: e is S -premises-like
thus  e is S -premises-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
clusterS -premises-like for set ;
existence
 ex b1 being set st b1 is S -premises-like
proof
take  the S -premises-like Subset of S ; ::_thesis: the S -premises-like Subset of S is S -premises-like
thus  the S -premises-like Subset of S is S -premises-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
let X be S -premises-like set ;
cluster  -> S -premises-like for Element of bool X;
coherence
 for b1 being Subset of X holds b1 is S -premises-like
proof
set FF = AllFormulasOf S;
reconsider XX = X as finite Subset of (AllFormulasOf S) by Def64;
let Y be Subset of X; ::_thesis: Y is S -premises-like
 Y is Subset of XX ;
then  Y is finite Subset of (AllFormulasOf S) by XBOOLE_1:1;
hence  Y is S -premises-like by Def64; ::_thesis: verum
end;
end;


definition
let S be Language;
let H2, H1 be S -premises-like set ;
:: original: null
redefine funcH1 null H2 ->  Subset of (AllFormulasOf S);
coherence
 H1 null H2 is Subset of (AllFormulasOf S) by Def64;
end;

registration
let S be Language;
let H be S -premises-like set ;
let x be set ;
clusterH null x -> S -premises-like ;
coherence
 H null x is S -premises-like ;
end;

registration
let S be Language;
let H1, H2 be S -premises-like set ;
clusterH1 \/ H2 -> S -premises-like for set ;
coherence
 for b1 being set st b1 = H1 \/ H2 holds
b1 is S -premises-like
proof
set FF = AllFormulasOf S;
 (H1 null H1) \/ (H2 null H2) c= AllFormulasOf S ;
hence  for b1 being set st b1 = H1 \/ H2 holds
b1 is S -premises-like by Def64; ::_thesis: verum
end;
end;

registration
let S be Language;
let H be S -premises-like set ;
let phi be wff string of S;
cluster[H,phi] -> S -sequent-like ;
coherence
 [H,phi] is S -sequent-like
proof
set FF = AllFormulasOf S;
reconsider HH = H as finite Subset of (AllFormulasOf S) by Def64;
 [HH,phi] is S -sequent-like ;
hence  [H,phi] is S -sequent-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
let H1, H2 be S -premises-like set ;
let phi be wff string of S;
cluster[(H1 \/ H2),phi] -> 1,{[H1,phi]},{(R#1 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [(H1 \/ H2),phi] holds
b1 is 1,{[H1,phi]},{(R#1 S)} -derivable
proof
set y = [H1,phi];
set SQ = {[H1,phi]};
set H = H1 \/ H2;
set Sq = [(H1 \/ H2),phi];
set Q = S -sequents ;
reconsider seqt = [(H1 \/ H2),phi] as Element of S -sequents by Def2;
reconsider Seqts = {[H1,phi]} as Element of bool (S -sequents) by Def3;
 ( H1 null H2 c= H1 \/ H2 & {[H1,phi]} \ {[H1,phi]} = {} & ([H1,phi] `1) \+\ H1 = {} & ([(H1 \/ H2),phi] `1) \+\ (H1 \/ H2) = {} & ([(H1 \/ H2),phi] `2) \+\ phi = {} & ([H1,phi] `2) \+\ phi = {} ) ;
then  ( H1 c= H1 \/ H2 & [H1,phi] in {[H1,phi]} & [H1,phi] `1 = H1 & [(H1 \/ H2),phi] `1 = H1 \/ H2 & [(H1 \/ H2),phi] `2 = phi & [H1,phi] `2 = phi ) by ZFMISC_1:60;
then  seqt Rule1 Seqts by Def19;
then  [Seqts,seqt] in P#1 S by Def32;
then  [(H1 \/ H2),phi] in (R#1 S) . {[H1,phi]} by Th2;
hence  for b1 being set st b1 = [(H1 \/ H2),phi] holds
b1 is 1,{[H1,phi]},{(R#1 S)} -derivable by Lm50; ::_thesis: verum
end;
end;

registration
let S be Language;
let H be S -premises-like set ;
let phi, phi1, phi2 be wff string of S;
cluster[(H null (phi1 ^ phi2)),(xnot phi)] -> 1,{[(H \/ {phi}),phi1],[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},{(R#8 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [(H null (phi1 ^ phi2)),(xnot phi)] holds
b1 is 1,{[(H \/ {phi}),phi1],[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},{(R#8 S)} -derivable
proof
set N = TheNorSymbOf S;
set H1 = H \/ {phi};
set psi = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2;
set y1 = [(H \/ {phi}),phi1];
set y2 = [(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)];
set SQ = {[(H \/ {phi}),phi1],[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]};
set Sq = [H,(xnot phi)];
set Q = S -sequents ;
reconsider seqt = [H,(xnot phi)] as Element of S -sequents by Def2;
reconsider Seqts = {[(H \/ {phi}),phi1],[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} as Element of bool (S -sequents) by Def3;
 ( {[(H \/ {phi}),phi1]} \ {[(H \/ {phi}),phi1],[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} = {} & {[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} \ {[(H \/ {phi}),phi1],[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} = {} & ([(H \/ {phi}),phi1] `1) \+\ (H \/ {phi}) = {} & ([(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `1) \+\ (H \/ {phi}) = {} & ([(H \/ {phi}),phi1] `2) \+\ phi1 = {} & ([(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `2) \+\ ((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2) = {} & ([H,(xnot phi)] `1) \+\ H = {} & ([H,(xnot phi)] `2) \+\ (xnot phi) = {} ) ;
then  ( [(H \/ {phi}),phi1] in {[(H \/ {phi}),phi1],[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} & [(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] in {[(H \/ {phi}),phi1],[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} & [(H \/ {phi}),phi1] `1 = H \/ {phi} & [(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `1 = H \/ {phi} & [(H \/ {phi}),phi1] `2 = phi1 & [(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & [H,(xnot phi)] `1 = H & [H,(xnot phi)] `2 = xnot phi ) by ZFMISC_1:60;
then  seqt Rule8 Seqts by Def29;
then  [Seqts,seqt] in P#8 S by Def42;
then  [H,(xnot phi)] in (R#8 S) . {[(H \/ {phi}),phi1],[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} by Th2;
hence  for b1 being set st b1 = [(H null (phi1 ^ phi2)),(xnot phi)] holds
b1 is 1,{[(H \/ {phi}),phi1],[(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},{(R#8 S)} -derivable by Lm50; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster{} null S -> S -sequents-like for set ;
coherence
 for b1 being set st b1 = {} null S holds
b1 is S -sequents-like
proof
 {} null S = {} /\ S ;
hence  for b1 being set st b1 = {} null S holds
b1 is S -sequents-like ; ::_thesis: verum
end;
end;

registration
let S be Language;
let H be S -premises-like set ;
let phi be wff string of S;
cluster[(H \/ {phi}),phi] -> 1, {} ,{(R#0 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [(H \/ {phi}),phi] holds
b1 is 1, {} ,{(R#0 S)} -derivable
proof
set H1 = H \/ {phi};
set Sq = [(H \/ {phi}),phi];
set SQ = {} null S;
set Q = S -sequents ;
reconsider seqt = [(H \/ {phi}),phi] as Element of S -sequents by Def2;
reconsider Seqts = {} null S as Element of bool (S -sequents) by Def3;
 ( ([(H \/ {phi}),phi] `2) \+\ phi = {} & ([(H \/ {phi}),phi] `1) \+\ (H \/ {phi}) = {} & ({phi} null H) \ (H \/ {phi}) = {} ) ;
then  ( [(H \/ {phi}),phi] `2 = phi & [(H \/ {phi}),phi] `1 = H \/ {phi} & phi in H \/ {phi} ) by ZFMISC_1:60;
then  seqt Rule0 Seqts by Def18;
then  [Seqts,seqt] in P#0 S by Def31;
then  [(H \/ {phi}),phi] in (R#0 S) . ({} null S) by Th2;
hence  for b1 being set st b1 = [(H \/ {phi}),phi] holds
b1 is 1, {} ,{(R#0 S)} -derivable by Lm50; ::_thesis: verum
end;
end;

registration
let S be Language;
let H be S -premises-like set ;
let phi1, phi2 be wff string of S;
cluster[(H null phi2),(xnot phi1)] -> 2,{[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},({(R#0 S)} \/ {(R#1 S)}) \/ {(R#8 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [(H null phi2),(xnot phi1)] holds
b1 is 2,{[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},({(R#0 S)} \/ {(R#1 S)}) \/ {(R#8 S)} -derivable
proof
set N = TheNorSymbOf S;
set psi1 = xnot phi1;
set psi2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2;
set Sq = [H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)];
set Sq1 = [(H \/ {phi1}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)];
set Sq2 = [(H \/ {phi1}),phi1];
set SQ = {} null S;
set goal = [(H null (phi1 ^ phi2)),(xnot phi1)];
 [(H null (phi1 ^ phi2)),(xnot phi1)] is 1 + 1,({} null S) \/ {[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},({(R#0 S)} \/ {(R#1 S)}) \/ {(R#8 S)} -derivable by Lm48;
hence  for b1 being set st b1 = [(H null phi2),(xnot phi1)] holds
b1 is 2,{[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},({(R#0 S)} \/ {(R#1 S)}) \/ {(R#8 S)} -derivable ; ::_thesis: verum
end;
end;

registration
let S be Language;
let H be S -premises-like set ;
let phi1, phi2 be wff string of S;
cluster[H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)] -> 1,{[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},{(R#7 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)] holds
b1 is 1,{[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},{(R#7 S)} -derivable
proof
set N = TheNorSymbOf S;
set psi1 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2;
set psi2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1;
set Sq = [H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)];
set y = [H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)];
set SQ = {[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]};
set Q = S -sequents ;
reconsider seqt = [H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)] as Element of S -sequents by Def2;
reconsider Seqts = {[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} as Element of bool (S -sequents) by Def3;
 ( {[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} \ {[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} = {} & ([H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `1) \+\ H = {} & ([H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)] `1) \+\ H = {} & ([H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `2) \+\ ((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2) = {} & ([H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)] `2) \+\ ((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1) = {} ) ;
then  ( [H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] in {[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} & [H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `1 = H & [H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)] `1 = H & [H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & [H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)] `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1 ) by ZFMISC_1:60;
then  seqt Rule7 Seqts by Def28;
then  [Seqts,seqt] in P#7 S by Def41;
then  [H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)] in (R#7 S) . {[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} by Th2;
hence  for b1 being set st b1 = [H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)] holds
b1 is 1,{[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},{(R#7 S)} -derivable by Lm50; ::_thesis: verum
end;
end;

registration
let S be Language;
let H be S -premises-like set ;
let phi1, phi2 be wff string of S;
cluster[(H null phi1),(xnot phi2)] -> 3,{[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},(({(R#0 S)} \/ {(R#1 S)}) \/ {(R#8 S)}) \/ {(R#7 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [(H null phi1),(xnot phi2)] holds
b1 is 3,{[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},(({(R#0 S)} \/ {(R#1 S)}) \/ {(R#8 S)}) \/ {(R#7 S)} -derivable
proof
set N = TheNorSymbOf S;
set psi2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1;
set Sq1 = [H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)];
set D1 = {(R#7 S)};
set D2 = {(R#7 S)};
set D3 = ({(R#0 S)} \/ {(R#1 S)}) \/ {(R#8 S)};
set goal = [(H null phi1),(xnot phi2)];
set SQ1 = {[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]};
set SQ2 = {[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]};
A1: ( {(R#7 S)} \/ {(R#7 S)} is isotone & ({(R#7 S)} \/ {(R#7 S)}) \/ (({(R#0 S)} \/ {(R#1 S)}) \/ {(R#8 S)}) is isotone & {[H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)],[H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)]} = {[H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)]} \/ {[H,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi1)]} ) by ENUMSET1:1;
 [(H null phi1),(xnot phi2)] is 1 + 2,{[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]} \/ {[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},({(R#7 S)} \/ {(R#7 S)}) \/ (({(R#0 S)} \/ {(R#1 S)}) \/ {(R#8 S)}) -derivable by A1, Lm48;
hence  for b1 being set st b1 = [(H null phi1),(xnot phi2)] holds
b1 is 3,{[H,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},(({(R#0 S)} \/ {(R#1 S)}) \/ {(R#8 S)}) \/ {(R#7 S)} -derivable ; ::_thesis: verum
end;
end;

registration
let S be Language;
let Sq be S -sequent-like set ;
clusterSq `1  -> S -premises-like for set ;
coherence
 for b1 being set st b1 = Sq `1 holds
b1 is S -premises-like
proof
set FF = AllFormulasOf S;
set Q = S -sequents ;
 Sq in S -sequents by Def2;
then consider premises being Subset of (AllFormulasOf S), conclusion being wff string of S such that
A1: ( Sq = [premises,conclusion] & premises is finite ) ;
 Sq `1 = premises by A1, MCART_1:7;
hence  for b1 being set st b1 = Sq `1 holds
b1 is S -premises-like by A1, Def64; ::_thesis: verum
end;
end;

definition
let S be Language;
let X be set ;
let D be RuleSet of S;
:: original: null
redefine funcD null X ->  RuleSet of S;
coherence
 D null X is RuleSet of S ;
end;

registration
let S be Language;
let phi1, phi2 be wff string of S;
let l1 be literal Element of S;
let H be S -premises-like set ;
let l2 be literal (H \/ {phi1}) \/ {phi2} -absent Element of S;
cluster[((H \/ {(<*l1*> ^ phi1)}) null l2),phi2] -> 1,{[(H \/ {((l1,l2) -SymbolSubstIn phi1)}),phi2]},{(R#5 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [((H \/ {(<*l1*> ^ phi1)}) null l2),phi2] holds
b1 is 1,{[(H \/ {((l1,l2) -SymbolSubstIn phi1)}),phi2]},{(R#5 S)} -derivable
proof
reconsider phi11 = (l1,l2) -SymbolSubstIn phi1 as wff string of S ;
set H1 = H \/ {phi11};
set Sq1 = [(H \/ {phi11}),phi2];
set H2 = H \/ {(<*l1*> ^ phi1)};
set Sq2 = [(H \/ {(<*l1*> ^ phi1)}),phi2];
set R = R#5 S;
set Q = S -sequents ;
set x = H;
set SS = AllSymbolsOf S;
set SQ = {[(H \/ {phi11}),phi2]};
set s = l1 SubstWith l2;
reconsider p = phi1 as AllSymbolsOf S -valued FinSequence ;
reconsider seqt = [(H \/ {(<*l1*> ^ phi1)}),phi2] as Element of S -sequents by Def2;
reconsider Seqts = {[(H \/ {phi11}),phi2]} as Element of bool (S -sequents) by Def3;
 ( (seqt `1) \+\ (H \/ {(<*l1*> ^ phi1)}) = {} & (seqt `2) \+\ phi2 = {} ) ;
then A1: ( seqt `1 = H \/ {(<*l1*> ^ phi1)} & seqt `2 = phi2 ) by FOMODEL0:29;
 H \/ {((l1 SubstWith l2) . p)} = H \/ {phi11} by FOMODEL0:def_23;
then  [(H \/ {((l1 SubstWith l2) . p)}),(seqt `2)] in Seqts by A1, TARSKI:def_1;
then  seqt Rule5 Seqts by A1, Def26;
then  [Seqts,seqt] in P#5 S by Def39;
then  seqt in (R#5 S) . Seqts by Th2;
hence  for b1 being set st b1 = [((H \/ {(<*l1*> ^ phi1)}) null l2),phi2] holds
b1 is 1,{[(H \/ {((l1,l2) -SymbolSubstIn phi1)}),phi2]},{(R#5 S)} -derivable by Lm50; ::_thesis: verum
end;
end;

definition
let S be Language;
let D be RuleSet of S;
let X be set ;
attrX is D -inconsistent  means :Def65: :: FOMODEL4:def 65
 ex phi1, phi2 being wff string of S st
( phi1 is X,D -provable & (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 is X,D -provable );
end;

:: deftheorem Def65 defines -inconsistent FOMODEL4:def_65_:_
 for S being Language
for D being RuleSet of S
for X being set holds
( X is D -inconsistent iff ex phi1, phi2 being wff string of S st
( phi1 is X,D -provable & (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 is X,D -provable ) );

registration
let m1 be non zero Nat;
let S be Language;
let H1, H2 be S -premises-like set ;
let phi be wff string of S;
cluster[((H1 \/ H2) null m1),phi] -> m1,{[H1,phi]},{(R#1 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [((H1 \/ H2) null m1),phi] holds
b1 is m1,{[H1,phi]},{(R#1 S)} -derivable
proof
set H = H1 \/ H2;
set sq1 = [H1,phi];
set sq = [(H1 \/ H2),phi];
consider m being Nat such that
A1: m1 = m + 1 by NAT_1:6;
defpred S1[ Nat] means [(H1 \/ H2),phi] is m1 + 1,{[H1,phi]},{(R#1 S)} -derivable ;
A2: [((H1 \/ H2) \/ (H1 \/ H2)),phi] is 1,{[(H1 \/ H2),phi]},{(R#1 S)} -derivable ;
A3: S1[ 0 ] ;
A4: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume  S1[n] ; ::_thesis: S1[n + 1]
then  [(H1 \/ H2),phi] is (n + 1) + 1,{[H1,phi]},{(R#1 S)} \/ {(R#1 S)} -derivable by Lm21, A2;
hence  S1[n + 1] ; ::_thesis: verum
end;
 for n being Nat holds S1[n] from NAT_1:sch_2(A3, A4);
hence  for b1 being set st b1 = [((H1 \/ H2) null m1),phi] holds
b1 is m1,{[H1,phi]},{(R#1 S)} -derivable by A1; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster non empty functional isotone for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
existence
 not for b1 being isotone RuleSet of S holds b1 is empty
proof
take  {(R#0 S)} ; ::_thesis: not {(R#0 S)} is empty
thus  not {(R#0 S)} is empty ; ::_thesis: verum
end;
end;

theorem Th13: :: FOMODEL4:13
for X being set
for S being Language
for phi being wff string of S
for D being RuleSet of S st X is D -inconsistent & D is isotone & R#1 S in D & R#8 S in D holds
xnot phi is X,D -provable
proof
let X be set ; ::_thesis: for S being Language
for phi being wff string of S
for D being RuleSet of S st X is D -inconsistent & D is isotone & R#1 S in D & R#8 S in D holds
xnot phi is X,D -provable
let S be Language; ::_thesis: for phi being wff string of S
for D being RuleSet of S st X is D -inconsistent & D is isotone & R#1 S in D & R#8 S in D holds
xnot phi is X,D -provable
let phi be wff string of S; ::_thesis: for D being RuleSet of S st X is D -inconsistent & D is isotone & R#1 S in D & R#8 S in D holds
xnot phi is X,D -provable
let D be RuleSet of S; ::_thesis: ( X is D -inconsistent & D is isotone & R#1 S in D & R#8 S in D implies xnot phi is X,D -provable )
set N = TheNorSymbOf S;
assume  X is D -inconsistent ; ::_thesis: ( not D is isotone or not R#1 S in D or not R#8 S in D or xnot phi is X,D -provable )
then consider phi1, phi2 being wff string of S such that
A1: ( phi1 is X,D -provable & (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 is X,D -provable ) by Def65;
reconsider psi = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 as wff string of S ;
consider H being set , m being Nat such that
A2: ( H c= X & [H,phi1] is m, {} ,D -derivable ) by A1, Def62;
consider K being set , n being Nat such that
A3: ( K c= X & [K,psi] is n, {} ,D -derivable ) by A1, Def62;
reconsider HHH = H, KKK = K as Subset of X by A2, A3;
reconsider sq1 = [H,phi1], sq2 = [K,psi] as S -sequent-like set by A2, A3;
 ( H = sq1 `1 & K = sq2 `1 ) by MCART_1:7;
then reconsider HH = H, KK = K as S -premises-like set ;
reconsider J = (HH \/ KK) \/ {phi} as S -premises-like set ;
assume A4: ( D is isotone & R#1 S in D & R#8 S in D ) ; ::_thesis: xnot phi is X,D -provable
then reconsider DD = D as non empty isotone RuleSet of S ;
reconsider E1 = R#1 S, E8 = R#8 S as Element of DD by A4;
A5: ( DD \/ {E1} = DD null E1 & DD \/ {E8} = DD null E8 & D \/ D is isotone & (D \/ D) \/ {(R#8 S)} is isotone & not D is empty & HHH \/ KKK c= X ) by A4;
A6: ( [((J \/ J) null (n + 1)),phi1] is n + 1,{[J,phi1]},{(R#1 S)} -derivable & [((J \/ J) null (m + 1)),psi] is m + 1,{[J,psi]},{(R#1 S)} -derivable ) ;
 [(HH \/ (KK \/ {phi})),phi1] is m + 1, {} ,D \/ {(R#1 S)} -derivable by Lm21, A2, A4;
then  [J,phi1] is m + 1, {} ,D -derivable by A5, XBOOLE_1:4;
then A7: [J,phi1] is (m + 1) + (n + 1), {} ,D -derivable by A6, A5, Lm21;
 [(KK \/ (HH \/ {phi})),psi] is n + 1, {} ,D \/ {(R#1 S)} -derivable by Lm21, A3, A4;
then  [J,psi] is n + 1, {} ,D -derivable by A5, XBOOLE_1:4;
then A8: [J,psi] is (n + 1) + (m + 1), {} ,D -derivable by A6, Lm21, A5;
 [((KK \/ HH) null (phi1 ^ phi2)),(xnot phi)] is 1,{[J,phi1],[J,psi]},{(R#8 S)} -derivable ;
then  [(KK \/ HH),(xnot phi)] is ((m + n) + 2) + 1,({} null S) \/ ({} null S),(D \/ D) \/ {(R#8 S)} -derivable by Lm48, A7, A8, A4;
hence  xnot phi is X,D -provable by A5, Def62; ::_thesis: verum
end;

registration
let S be Language;
cluster R#5 S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#5 S holds
b1 is isotone
proof
set R = R#5 S;
set Q = S -sequents ;
now__::_thesis:_for_Seqts,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts_c=_Seqts2_holds_
(R#5_S)_._Seqts_c=_(R#5_S)_._Seqts2
let Seqts, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts c= Seqts2 implies (R#5 S) . Seqts c= (R#5 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume A1: Seqts c= Seqts2 ; ::_thesis: (R#5 S) . Seqts c= (R#5 S) . Seqts2
now__::_thesis:_for_x_being_set_st_x_in_(R#5_S)_._Seqts_holds_
x_in_(R#5_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#5 S) . Seqts implies x in (R#5 S) . Seqts2 )
assume A2: x in (R#5 S) . Seqts ; ::_thesis: x in (R#5 S) . Seqts2
reconsider seqt = x as Element of S -sequents by A2;
 [Seqts,seqt] in P#5 S by A2, Lm29;
then  seqt Rule5 Seqts by Def39;
then consider v1, v2 being literal Element of S, z being set , p being FinSequence such that
A3: ( seqt `1 = z \/ {(<*v1*> ^ p)} & v2 is (z \/ {p}) \/ {(seqt `2)} -absent & [(z \/ {((v1 SubstWith v2) . p)}),(seqt `2)] in Seqts ) by Def26;
 seqt Rule5 Seqts2 by Def26, A1, A3;
then  [Seqts2,seqt] in P#5 S by Def39;
hence  x in (R#5 S) . Seqts2 by Th2; ::_thesis: verum
end;
hence  (R#5 S) . Seqts c= (R#5 S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#5 S holds
b1 is isotone by Def9; ::_thesis: verum
end;
end;

registration
let S be Language;
let l be literal Element of S;
let t be termal string of S;
let phi be wff string of S;
cluster[{((l,t) SubstIn phi)},(<*l*> ^ phi)] -> 1, {} ,{(R#4 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [{((l,t) SubstIn phi)},(<*l*> ^ phi)] holds
b1 is 1, {} ,{(R#4 S)} -derivable
proof
set Q = S -sequents ;
set psi = (l,t) SubstIn phi;
reconsider Sq = [{((l,t) SubstIn phi)},(<*l*> ^ phi)] as S -sequent-like set ;
reconsider SQ = {} null S as S -sequents-like set ;
reconsider seqt = Sq as Element of S -sequents by Def2;
reconsider Seqts = SQ as Element of bool (S -sequents) by Def3;
 ( (seqt `1) \+\ {((l,t) SubstIn phi)} = {} & (seqt `2) \+\ (<*l*> ^ phi) = {} ) ;
then  ( Seqts = {} & seqt `1 = {((l,t) SubstIn phi)} & seqt `2 = <*l*> ^ phi ) by FOMODEL0:29;
then  seqt Rule4 Seqts by Def25;
then  [Seqts,seqt] in P#4 S by Def38;
then  Sq in (R#4 S) . SQ by Th2;
hence  for b1 being set st b1 = [{((l,t) SubstIn phi)},(<*l*> ^ phi)] holds
b1 is 1, {} ,{(R#4 S)} -derivable by Lm50; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster R#4 S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#4 S holds
b1 is isotone
proof
set R = R#4 S;
set Q = S -sequents ;
now__::_thesis:_for_Seqts,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts_c=_Seqts2_holds_
(R#4_S)_._Seqts_c=_(R#4_S)_._Seqts2
let Seqts, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts c= Seqts2 implies (R#4 S) . Seqts c= (R#4 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume  Seqts c= Seqts2 ; ::_thesis: (R#4 S) . Seqts c= (R#4 S) . Seqts2
now__::_thesis:_for_x_being_set_st_x_in_(R#4_S)_._Seqts_holds_
x_in_(R#4_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#4 S) . Seqts implies x in (R#4 S) . Seqts2 )
assume A1: x in (R#4 S) . Seqts ; ::_thesis: x in (R#4 S) . Seqts2
reconsider seqt = x as Element of S -sequents by A1;
 [Seqts,seqt] in P#4 S by A1, Lm29;
then  seqt Rule4 Seqts by Def38;
then consider l being literal Element of S, phi being wff string of S, t being termal string of S such that
A2: ( seqt `1 = {((l,t) SubstIn phi)} & seqt `2 = <*l*> ^ phi ) by Def25;
 seqt Rule4 Seqts2 by A2, Def25;
then  [Seqts2,seqt] in P#4 S by Def38;
hence  x in (R#4 S) . Seqts2 by Th2; ::_thesis: verum
end;
hence  (R#4 S) . Seqts c= (R#4 S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#4 S holds
b1 is isotone by Def9; ::_thesis: verum
end;
end;

Lm51:  for S being Language
for D1 being 0 -ranked 1 -ranked RuleSet of S
for X being b2 -expanded set
for phi being 0wff string of S holds
( (D1 Henkin X) -AtomicEval phi = 1 iff phi in X )
proof
let S be Language; ::_thesis: for D1 being 0 -ranked 1 -ranked RuleSet of S
for X being b1 -expanded set
for phi being 0wff string of S holds
( (D1 Henkin X) -AtomicEval phi = 1 iff phi in X )
let D1 be 0 -ranked 1 -ranked RuleSet of S; ::_thesis: for X being D1 -expanded set
for phi being 0wff string of S holds
( (D1 Henkin X) -AtomicEval phi = 1 iff phi in X )
let X be D1 -expanded set ; ::_thesis: for phi being 0wff string of S holds
( (D1 Henkin X) -AtomicEval phi = 1 iff phi in X )
let phi be 0wff string of S; ::_thesis: ( (D1 Henkin X) -AtomicEval phi = 1 iff phi in X )
 R#0 S in D1 by Def59;
then A1: {(R#0 S)} c= D1 by ZFMISC_1:31;
set TT = AllTermsOf S;
set E = TheEqSymbOf S;
set p = SubTerms phi;
set F = S -firstChar ;
set s = (S -firstChar) . phi;
set n = abs (ar ((S -firstChar) . phi));
set R = (X,D1) -termEq ;
set U = Class ((X,D1) -termEq);
set AF = AtomicFormulasOf S;
set d = (Class ((X,D1) -termEq)) -deltaInterpreter ;
set i = (S,X) -freeInterpreter ;
A2: (abs (ar (TheEqSymbOf S))) - 2 = 0 ;
reconsider I = D1 Henkin X as Element of (Class ((X,D1) -termEq)) -InterpretersOf S ;
set UV = I -TermEval ;
set V = I -AtomicEval phi;
set uv = ((S,X) -freeInterpreter) -TermEval ;
set v = ((S,X) -freeInterpreter) -AtomicEval phi;
set f = (I ===) . ((S -firstChar) . phi);
set G = I . ((S -firstChar) . phi);
set g = ((S,X) -freeInterpreter) . ((S -firstChar) . phi);
set O = OwnSymbolsOf S;
set FF = AllFormulasOf S;
set C = S -multiCat ;
set SS = AllSymbolsOf S;
reconsider pp = SubTerms phi as Element of (abs (ar ((S -firstChar) . phi))) -tuples_on (AllTermsOf S) by FOMODEL0:16;
 pp is Element of Funcs ((Seg (abs (ar ((S -firstChar) . phi)))),(AllTermsOf S)) by FOMODEL0:11;
then reconsider fp = pp as Function of (Seg (abs (ar ((S -firstChar) . phi)))),(AllTermsOf S) ;
A3: 2 -tuples_on (((AllSymbolsOf S) *) \ {{}}) =  {  <*tt1,tt2*> where tt1, tt2 is Element of ((AllSymbolsOf S) *) \ {{}} : verum  }  by FINSEQ_2:99;
 SubTerms phi in (AllTermsOf S) * ;
then reconsider Pp = SubTerms phi as Element of (((AllSymbolsOf S) *) \ {{}}) * ;
A4: phi = <*((S -firstChar) . phi)*> ^ ((S -multiCat) . (SubTerms phi)) by FOMODEL1:def_38;
A5: I -TermEval = (((X,D1) -termEq) -class) * (((S,X) -freeInterpreter) -TermEval) by FOMODEL3:3;
A6: ((S,X) -freeInterpreter) -TermEval = id (AllTermsOf S) by FOMODEL3:4;
 ( (abs (ar ((S -firstChar) . phi))) -tuples_on (AllTermsOf S) c= (AllTermsOf S) * & (AllTermsOf S) * c= (((AllSymbolsOf S) *) \ {{}}) * ) by FINSEQ_2:142;
then  (abs (ar ((S -firstChar) . phi))) -tuples_on (AllTermsOf S) c= (((AllSymbolsOf S) *) \ {{}}) * by XBOOLE_1:1;
then reconsider nc = (((S -firstChar) . phi) -compound) | ((abs (ar ((S -firstChar) . phi))) -tuples_on (AllTermsOf S)) as Function of ((abs (ar ((S -firstChar) . phi))) -tuples_on (AllTermsOf S)),(((AllSymbolsOf S) *) \ {{}}) by FUNCT_2:32;
percases ( (S -firstChar) . phi = TheEqSymbOf S or not (S -firstChar) . phi = TheEqSymbOf S ) ;
supposeA7: (S -firstChar) . phi = TheEqSymbOf S ; ::_thesis: ( (D1 Henkin X) -AtomicEval phi = 1 iff phi in X )
reconsider p1 = SubTerms phi as (0 + 1) + 1 -element Element of (AllTermsOf S) * by A7, A2;
 Pp in 2 -tuples_on (((AllSymbolsOf S) *) \ {{}}) by A2, A7, FOMODEL0:16;
then consider tt11, tt22 being Element of ((AllSymbolsOf S) *) \ {{}} such that
A8: Pp = <*tt11,tt22*> by A3;
A9: (S -multiCat) . <*tt11,tt22*> = tt11 ^ tt22 by FOMODEL0:15;
reconsider p2 = SubTerms phi as (1 + 1) + 0 -element Element of (AllTermsOf S) * by A7, A2;
 ( {(p1 . (0 + 1))} \ (AllTermsOf S) = {} & {(p2 . (1 + 1))} \ (AllTermsOf S) = {} ) ;
then reconsider tt1 = (SubTerms phi) . 1, tt2 = (SubTerms phi) . 2 as Element of AllTermsOf S by ZFMISC_1:60;
reconsider t1 = tt1, t2 = tt2 as termal string of S ;
A10: ( (((X,D1) -termEq) -class) . tt1 = EqClass (((X,D1) -termEq),tt1) & (((X,D1) -termEq) -class) . tt2 = EqClass (((X,D1) -termEq),tt2) ) by FOMODEL3:def_13;
A11: ( tt1 = tt11 & tt2 = tt22 ) by A8, FINSEQ_1:44;
 ( {((id (AllTermsOf S)) . tt1)} \ {tt1} = {} & {((id (AllTermsOf S)) . tt2)} \ {tt2} = {} ) ;
then A12: ( (id (AllTermsOf S)) . tt1 = tt1 & (id (AllTermsOf S)) . tt2 = tt2 ) by ZFMISC_1:15;
 ( (((((X,D1) -termEq) -class) * (((S,X) -freeInterpreter) -TermEval)) . tt1) \+\ ((((X,D1) -termEq) -class) . ((((S,X) -freeInterpreter) -TermEval) . tt1)) = {} & (((((X,D1) -termEq) -class) * (((S,X) -freeInterpreter) -TermEval)) . tt2) \+\ ((((X,D1) -termEq) -class) . ((((S,X) -freeInterpreter) -TermEval) . tt2)) = {} ) ;
then  ( ((((X,D1) -termEq) -class) * (((S,X) -freeInterpreter) -TermEval)) . tt1 = (((X,D1) -termEq) -class) . ((((S,X) -freeInterpreter) -TermEval) . tt1) & ((((X,D1) -termEq) -class) * (((S,X) -freeInterpreter) -TermEval)) . tt2 = (((X,D1) -termEq) -class) . ((((S,X) -freeInterpreter) -TermEval) . tt2) ) by FOMODEL0:29;
then A13: ( I -AtomicEval phi = 1 iff EqClass (((X,D1) -termEq),tt1) = EqClass (((X,D1) -termEq),tt2) ) by A10, A12, A5, A7, A6, FOMODEL2:15;
then A14: ( I -AtomicEval phi = 1 iff [tt1,tt2] in (X,D1) -termEq ) by EQREL_1:35;
A15: (<*(TheEqSymbOf S)*> ^ t1) ^ t2 = phi by A4, A8, A9, A11, A7, FINSEQ_1:32;
thus  ( (D1 Henkin X) -AtomicEval phi = 1 implies phi in X ) ::_thesis: ( phi in X implies (D1 Henkin X) -AtomicEval phi = 1 )
proof
assume  (D1 Henkin X) -AtomicEval phi = 1 ; ::_thesis: phi in X
then  [tt1,tt2] in (X,D1) -termEq by A13, EQREL_1:35;
then consider t11, t22 being termal string of S such that
A16: ( [tt1,tt2] = [t11,t22] & (<*(TheEqSymbOf S)*> ^ t11) ^ t22 is X,D1 -provable ) ;
 ( t11 = tt1 & t22 = tt2 ) by A16, XTUPLE_0:1;
then  <*(TheEqSymbOf S)*> ^ (t11 ^ t22) = phi by A4, A8, A11, A7, FOMODEL0:15;
then  phi is X,D1 -provable by A16, FINSEQ_1:32;
then  {phi} c= X by Def16;
hence  phi in X by ZFMISC_1:31; ::_thesis: verum
end;
assume  phi in X ; ::_thesis: (D1 Henkin X) -AtomicEval phi = 1
then reconsider Xphi = {phi} as Subset of X by ZFMISC_1:31;
 {[{phi},phi]} is {} ,D1 -derivable by Lm18, A1;
then  phi is Xphi,D1 -provable by Def12;
hence  (D1 Henkin X) -AtomicEval phi = 1 by A14, A15; ::_thesis: verum
end;
supposeA17: not (S -firstChar) . phi = TheEqSymbOf S ; ::_thesis: ( (D1 Henkin X) -AtomicEval phi = 1 iff phi in X )
then reconsider o = (S -firstChar) . phi as Element of OwnSymbolsOf S by FOMODEL1:15;
set gg = ((S,X) -freeInterpreter) . o;
 ( (S -firstChar) . phi <> TheEqSymbOf S & I -AtomicEval phi = ((S,X) -freeInterpreter) -AtomicEval phi & ((S,X) -freeInterpreter) -AtomicEval phi = (((S,X) -freeInterpreter) . o) . ((((S,X) -freeInterpreter) -TermEval) * (SubTerms phi)) ) by A17, FOMODEL2:14, FOMODEL3:5;
then I -AtomicEval phi = (((S,X) -freeInterpreter) . o) . ((id (AllTermsOf S)) * fp) by FOMODEL3:4
.= (((S,X) -freeInterpreter) . o) . fp by FUNCT_2:17
.= (X -freeInterpreter o) . (SubTerms phi) by FOMODEL3:def_4
.= ((chi (X,(AtomicFormulasOf S))) * ((o -compound) | ((abs (ar ((S -firstChar) . phi))) -tuples_on (AllTermsOf S)))) . pp by FOMODEL3:def_3
.= (chi (X,(AtomicFormulasOf S))) . (nc . pp) by FUNCT_2:15
.= (chi (X,(AtomicFormulasOf S))) . ((o -compound) . pp) by FUNCT_1:49
.= (chi (X,(AtomicFormulasOf S))) . (o -compound Pp) by FOMODEL3:def_2
.= (chi (X,(AtomicFormulasOf S))) . phi by FOMODEL1:def_38 ;
then  ( I -AtomicEval phi = 1 iff phi in (chi (X,(AtomicFormulasOf S))) " {1} ) by FOMODEL0:25;
then  ( I -AtomicEval phi = 1 iff phi in X /\ (AtomicFormulasOf S) ) by FOMODEL0:24;
then  ( phi in AtomicFormulasOf S & ( I -AtomicEval phi = 1 implies ( phi in X & phi in AtomicFormulasOf S ) ) & ( phi in X & phi in AtomicFormulasOf S implies I -AtomicEval phi = 1 ) ) by XBOOLE_0:def_4;
hence  ( (D1 Henkin X) -AtomicEval phi = 1 iff phi in X ) ; ::_thesis: verum
end;
end;
end;

definition
let S be Language;
let X be set ;
attrX is S -witnessed  means :Def66: :: FOMODEL4:def 66
 for l1 being literal Element of S
for phi1 being wff string of S st <*l1*> ^ phi1 in X holds
ex l2 being literal Element of S st
( (l1,l2) -SymbolSubstIn phi1 in X & not l2 in rng phi1 );
end;

:: deftheorem Def66 defines -witnessed FOMODEL4:def_66_:_
 for S being Language
for X being set holds
( X is S -witnessed iff for l1 being literal Element of S
for phi1 being wff string of S st <*l1*> ^ phi1 in X holds
ex l2 being literal Element of S st
( (l1,l2) -SymbolSubstIn phi1 in X & not l2 in rng phi1 ) );

theorem Th14: :: FOMODEL4:14
for S being Language
for psi being wff string of S
for D1 being 0 -ranked 1 -ranked RuleSet of S
for X being b3 -expanded set st R#1 S in D1 & R#4 S in D1 & R#6 S in D1 & R#7 S in D1 & R#8 S in D1 & X is S -mincover & X is S -witnessed holds
( (D1 Henkin X) -TruthEval psi = 1 iff psi in X )
proof
let S be Language; ::_thesis: for psi being wff string of S
for D1 being 0 -ranked 1 -ranked RuleSet of S
for X being b2 -expanded set st R#1 S in D1 & R#4 S in D1 & R#6 S in D1 & R#7 S in D1 & R#8 S in D1 & X is S -mincover & X is S -witnessed holds
( (D1 Henkin X) -TruthEval psi = 1 iff psi in X )
let psi be wff string of S; ::_thesis: for D1 being 0 -ranked 1 -ranked RuleSet of S
for X being b1 -expanded set st R#1 S in D1 & R#4 S in D1 & R#6 S in D1 & R#7 S in D1 & R#8 S in D1 & X is S -mincover & X is S -witnessed holds
( (D1 Henkin X) -TruthEval psi = 1 iff psi in X )
let D1 be 0 -ranked 1 -ranked RuleSet of S; ::_thesis: for X being D1 -expanded set st R#1 S in D1 & R#4 S in D1 & R#6 S in D1 & R#7 S in D1 & R#8 S in D1 & X is S -mincover & X is S -witnessed holds
( (D1 Henkin X) -TruthEval psi = 1 iff psi in X )
let X be D1 -expanded set ; ::_thesis: ( R#1 S in D1 & R#4 S in D1 & R#6 S in D1 & R#7 S in D1 & R#8 S in D1 & X is S -mincover & X is S -witnessed implies ( (D1 Henkin X) -TruthEval psi = 1 iff psi in X ) )
set TT = AllTermsOf S;
set E = TheEqSymbOf S;
set F = S -firstChar ;
set N = TheNorSymbOf S;
set R = (X,D1) -termEq ;
set U = Class ((X,D1) -termEq);
set L = LettersOf S;
set AF = AtomicFormulasOf S;
set d = (Class ((X,D1) -termEq)) -deltaInterpreter ;
set i = (S,X) -freeInterpreter ;
set II = (Class ((X,D1) -termEq)) -InterpretersOf S;
set D = D1;
set ii = (AllTermsOf S) -InterpretersOf S;
set G0 = R#0 S;
set G1 = R#1 S;
set G2 = R#2 S;
set G4 = R#4 S;
set G6 = R#6 S;
set G7 = R#7 S;
set G8 = R#8 S;
set E0 = {(R#0 S)};
set E1 = {(R#1 S)};
set E2 = {(R#2 S)};
set E4 = {(R#4 S)};
set E6 = {(R#6 S)};
set E7 = {(R#7 S)};
set E8 = {(R#8 S)};
reconsider E0 = {(R#0 S)}, E1 = {(R#1 S)}, E2 = {(R#2 S)}, E4 = {(R#4 S)}, E6 = {(R#6 S)}, E7 = {(R#7 S)}, E8 = {(R#8 S)} as RuleSet of S ;
assume  ( R#1 S in D1 & R#4 S in D1 & R#6 S in D1 & R#7 S in D1 & R#8 S in D1 ) ; ::_thesis: ( not X is S -mincover or not X is S -witnessed or ( (D1 Henkin X) -TruthEval psi = 1 iff psi in X ) )
then  ( R#0 S in D1 & R#1 S in D1 & R#2 S in D1 & R#4 S in D1 & R#6 S in D1 & R#7 S in D1 & R#8 S in D1 ) by Def59;
then reconsider F0 = E0, F1 = E1, F2 = E2, F4 = E4, F6 = E6, F7 = E7, F8 = E8 as Subset of D1 by ZFMISC_1:31;
A1: ( F0 \/ ((F0 \/ F1) \/ F8) c= D1 & F0 \/ F6 c= D1 & F0 c= D1 & F0 \/ (((F0 \/ F1) \/ F8) \/ F7) c= D1 ) ;
reconsider I = D1 Henkin X as Element of (Class ((X,D1) -termEq)) -InterpretersOf S ;
set UV = I -TermEval ;
set uv = ((S,X) -freeInterpreter) -TermEval ;
set O = OwnSymbolsOf S;
set FF = AllFormulasOf S;
set C = S -multiCat ;
set SS = AllSymbolsOf S;
assume A2: ( X is S -mincover & X is S -witnessed ) ; ::_thesis: ( (D1 Henkin X) -TruthEval psi = 1 iff psi in X )
defpred S1[ Nat] means  for phi being wff string of S st phi is $1 -wff holds
( I -TruthEval phi = 1 iff phi in X );
A3: S1[ 0 ]
proof
let phi be wff string of S; ::_thesis: ( phi is 0 -wff implies ( I -TruthEval phi = 1 iff phi in X ) )
assume  phi is 0 -wff ; ::_thesis: ( I -TruthEval phi = 1 iff phi in X )
then reconsider phi0 = phi as 0wff string of S ;
 ( I -AtomicEval phi0 = 1 iff phi0 in X ) by Lm51;
hence  ( I -TruthEval phi = 1 iff phi in X ) ; ::_thesis: verum
end;
A4: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
set Vn = (I,n) -TruthEval ;
assume A5: S1[n] ; ::_thesis: S1[n + 1]
let phi be wff string of S; ::_thesis: ( phi is n + 1 -wff implies ( I -TruthEval phi = 1 iff phi in X ) )
set s = (S -firstChar) . phi;
set V = I -TruthEval phi;
assume A6: phi is n + 1 -wff ; ::_thesis: ( I -TruthEval phi = 1 iff phi in X )
percases ( ( not phi is 0wff & phi is exal ) or ( not phi is 0wff & not phi is exal ) or phi is 0wff ) ;
suppose ( not phi is 0wff & phi is exal ) ; ::_thesis: ( I -TruthEval phi = 1 iff phi in X )
then reconsider phii = phi as non 0wff n + 1 -wff exal string of S by A6;
reconsider phi1 = head phii as n -wff string of S ;
reconsider l = (S -firstChar) . phii as literal Element of S ;
A7: phii = (<*l*> ^ phi1) ^ (tail phii) by FOMODEL2:23
.= <*l*> ^ phi1 ;
hereby  ::_thesis: ( phi in X implies I -TruthEval phi = 1 )
assume  I -TruthEval phi = 1 ; ::_thesis: phi in X
then consider u being Element of Class ((X,D1) -termEq) such that
A8: ((l,u) ReassignIn I) -TruthEval phi1 = 1 by A7, FOMODEL2:19;
consider x being set  such that
A9: ( x in AllTermsOf S & u = Class (((X,D1) -termEq),x) ) by EQREL_1:def_3;
reconsider tt = x as Element of AllTermsOf S by A9;
reconsider psi1 = (l,tt) SubstIn phi1 as n -wff string of S ;
 ((id (AllTermsOf S)) . tt) \+\ tt = {} ;
then  ( (id (AllTermsOf S)) . tt = tt & (((((X,D1) -termEq) -class) * (((S,X) -freeInterpreter) -TermEval)) . tt) \+\ ((((X,D1) -termEq) -class) . ((((S,X) -freeInterpreter) -TermEval) . tt)) = {} ) by FOMODEL0:29;
then A10: ( (((S,X) -freeInterpreter) -TermEval) . tt = tt & ((((X,D1) -termEq) -class) * (((S,X) -freeInterpreter) -TermEval)) . tt = (((X,D1) -termEq) -class) . ((((S,X) -freeInterpreter) -TermEval) . tt) ) by FOMODEL0:29, FOMODEL3:4;
(I -TermEval) . tt = ((((X,D1) -termEq) -class) * (((S,X) -freeInterpreter) -TermEval)) . tt by FOMODEL3:3
.= u by A10, A9, FOMODEL3:def_13 ;
then  1 = I -TruthEval psi1 by A8, FOMODEL3:10;
then  psi1 in X by A5;
then A11: {psi1} c= X by ZFMISC_1:31;
 [{((l,tt) SubstIn phi1)},(<*l*> ^ phi1)] is 1, {} ,{(R#4 S)} -derivable ;
then  ( <*l*> ^ phi1 is X,E4 -provable & F4 c= D1 & E4 is isotone ) by A11, Def62;
then  phii is X,D1 -provable by A7, Lm49;
hence  phi in X by Def61; ::_thesis: verum
end;
assume  phi in X ; ::_thesis: I -TruthEval phi = 1
then consider l2 being literal Element of S such that
A12: ( (l,l2) -SymbolSubstIn phi1 in X & not l2 in rng phi1 ) by A2, Def66, A7;
reconsider psi1 = (l,l2) -SymbolSubstIn phi1 as n -wff string of S ;
consider u being Element of Class ((X,D1) -termEq) such that
A13: ( u = (I . l2) . {} & (l2,u) ReassignIn I = I ) by FOMODEL2:26;
reconsider I2 = (l2,u) ReassignIn I, I1 = (l,u) ReassignIn I as Element of (Class ((X,D1) -termEq)) -InterpretersOf S ;
 I2 -TruthEval psi1 = 1 by A12, A5, A13;
then  I1 -TruthEval phi1 = 1 by A12, FOMODEL3:9;
hence  I -TruthEval phi = 1 by A7, FOMODEL2:19; ::_thesis: verum
end;
suppose ( not phi is 0wff & not phi is exal ) ; ::_thesis: ( I -TruthEval phi = 1 iff phi in X )
then reconsider phii = phi as non 0wff n + 1 -wff non exal string of S by A6;
set phi1 = head phii;
set phi2 = tail phii;
 ((S -firstChar) . phii) \+\ (TheNorSymbOf S) = {} ;
then  (S -firstChar) . phi = TheNorSymbOf S by FOMODEL0:29;
then A14: phi = (<*(TheNorSymbOf S)*> ^ (head phii)) ^ (tail phii) by FOMODEL2:23;
 ( I -TruthEval phi = 1 iff ( I -TruthEval (head phii) = 0 & I -TruthEval (tail phii) = 0 ) ) by A14, FOMODEL2:19;
then  ( I -TruthEval phi = 1 iff ( not I -TruthEval (head phii) = 1 & not I -TruthEval (tail phii) = 1 ) ) by FOMODEL0:39;
then A15: ( I -TruthEval phi = 1 iff ( not head phii in X & not tail phii in X ) ) by A5;
A16: now__::_thesis:_(_xnot_(head_phii)_in_X_&_xnot_(tail_phii)_in_X_implies_phi_in_X_)
assume  ( xnot (head phii) in X & xnot (tail phii) in X ) ; ::_thesis: phi in X
then  ( xnot (head phii) is X,{(R#0 S)} -provable & xnot (tail phii) is X,{(R#0 S)} -provable ) by Th4;
then  ( xnot (head phii) is X,D1 -provable & xnot (tail phii) is X,D1 -provable ) by A1, Lm49;
then  ( xnot (head phii) in X & xnot (tail phii) in X ) by Def61;
then reconsider Y = {(xnot (head phii)),(xnot (tail phii))} as Subset of X by ZFMISC_1:32;
 phi is X null Y,D1 -provable by Lm49, A1, A14;
hence  phi in X by Def61; ::_thesis: verum
end;
now__::_thesis:_(_phi_in_X_implies_(_xnot_(head_phii)_in_X_&_xnot_(tail_phii)_in_X_)_)
reconsider H = {phi} as S -premises-like set ;
assume  phi in X ; ::_thesis: ( xnot (head phii) in X & xnot (tail phii) in X )
then A17: H c= X by ZFMISC_1:31;
A18: [{phi},phi] is 1, {} ,E0 -derivable ;
A19: [(H null (tail phii)),(xnot (head phii))] is 2,{[{phi},phi]},(E0 \/ E1) \/ E8 -derivable by A14;
A20: [(H null (head phii)),(xnot (tail phii))] is 3,{[H,phi]},((E0 \/ E1) \/ E8) \/ E7 -derivable by A14;
 [H,(xnot (head phii))] is 1 + 2, {} ,E0 \/ ((E0 \/ E1) \/ E8) -derivable by A19, Lm21;
then  [H,(xnot (head phii))] is 3, {} ,D1 -derivable by A1, Th11;
then  xnot (head phii) is X,D1 -provable by Def62, A17;
hence  xnot (head phii) in X by Def61; ::_thesis: xnot (tail phii) in X
A21: [H,(xnot (tail phii))] is 1 + 3, {} ,E0 \/ (((E0 \/ E1) \/ E8) \/ E7) -derivable by A18, A20, Lm21;
 [H,(xnot (tail phii))] is 4, {} ,D1 -derivable by A1, A21, Th11;
then  xnot (tail phii) is X,D1 -provable by Def62, A17;
hence  xnot (tail phii) in X by Def61; ::_thesis: verum
end;
hence  ( I -TruthEval phi = 1 iff phi in X ) by A15, A2, A16, FOMODEL2:def_34; ::_thesis: verum
end;
suppose phi is 0wff ; ::_thesis: ( I -TruthEval phi = 1 iff phi in X )
hence  ( I -TruthEval phi = 1 iff phi in X ) by A3; ::_thesis: verum
end;
end;
end;
A22: for n being Nat holds S1[n] from NAT_1:sch_2(A3, A4);
 psi is Depth psi -wff by FOMODEL2:def_31;
hence  ( (D1 Henkin X) -TruthEval psi = 1 iff psi in X ) by A22; ::_thesis: verum
end;

notation
let S be Language;
let D be RuleSet of S;
let X be set ;
antonym D -consistent X for D -inconsistent ;
end;

theorem Th15: :: FOMODEL4:15
for Y being set
for S being Language
for D being RuleSet of S
for X being Subset of Y st X is D -inconsistent holds
Y is D -inconsistent
proof
let Y be set ; ::_thesis: for S being Language
for D being RuleSet of S
for X being Subset of Y st X is D -inconsistent holds
Y is D -inconsistent
let S be Language; ::_thesis: for D being RuleSet of S
for X being Subset of Y st X is D -inconsistent holds
Y is D -inconsistent
let D be RuleSet of S; ::_thesis: for X being Subset of Y st X is D -inconsistent holds
Y is D -inconsistent
let X be Subset of Y; ::_thesis: ( X is D -inconsistent implies Y is D -inconsistent )
set N = TheNorSymbOf S;
assume  X is D -inconsistent ; ::_thesis: Y is D -inconsistent
then consider phi1, phi2 being wff string of S such that
A1: ( phi1 is X,D -provable & (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 is X,D -provable ) by Def65;
thus  Y is D -inconsistent by Def65, A1; ::_thesis: verum
end;

definition
let S be Language;
let D be RuleSet of S;
let X be functional set ;
let phi be Element of ExFormulasOf S;
func(D,phi) AddAsWitnessTo X ->  set  equals :Def67: :: FOMODEL4:def 67
X \/ {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))} if ( X \/ {phi} is D -consistent & (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) <> {} )
otherwise X;
consistency
 for b1 being set holds verum ;
coherence
 ( ( X \/ {phi} is D -consistent & (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) <> {} implies X \/ {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))} is set ) & ( ( not X \/ {phi} is D -consistent or not (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) <> {} ) implies X is set ) ) ;
end;

:: deftheorem Def67 defines AddAsWitnessTo FOMODEL4:def_67_:_
 for S being Language
for D being RuleSet of S
for X being functional set
for phi being Element of ExFormulasOf S holds
( ( X \/ {phi} is D -consistent & (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) <> {} implies (D,phi) AddAsWitnessTo X = X \/ {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))} ) & ( ( not X \/ {phi} is D -consistent or not (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) <> {} ) implies (D,phi) AddAsWitnessTo X = X ) );

registration
let S be Language;
let D be RuleSet of S;
let X be functional set ;
let phi be Element of ExFormulasOf S;
clusterX \ ((D,phi) AddAsWitnessTo X) ->  empty for set ;
coherence
 for b1 being set st b1 = X \ ((D,phi) AddAsWitnessTo X) holds
b1 is empty
proof
set F = S -firstChar ;
set L = LettersOf S;
set Y = (D,phi) AddAsWitnessTo X;
set s1 = (S -firstChar) . phi;
set phi1 = head phi;
set SS = AllSymbolsOf S;
set strings = ((AllSymbolsOf S) *) \ {{}};
set no = SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}));
set s2 = the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})));
set Z = {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))};
defpred S1[] means ( X \/ {phi} is D -consistent & (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) <> {} );
 ( ( S1[] implies ( X null {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))} c= X \/ {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))} & (D,phi) AddAsWitnessTo X = X \/ {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))} ) ) & ( not S1[] implies (D,phi) AddAsWitnessTo X = X null {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))} ) ) by Def67;
hence  for b1 being set st b1 = X \ ((D,phi) AddAsWitnessTo X) holds
b1 is empty ; ::_thesis: verum
end;
end;

registration
let S be Language;
let D be RuleSet of S;
let X be functional set ;
let phi be Element of ExFormulasOf S;
cluster((D,phi) AddAsWitnessTo X) \ X ->  trivial for set ;
coherence
 for b1 being set st b1 = ((D,phi) AddAsWitnessTo X) \ X holds
b1 is trivial
proof
set F = S -firstChar ;
set L = LettersOf S;
set SS = AllSymbolsOf S;
set strings = ((AllSymbolsOf S) *) \ {{}};
set s1 = (S -firstChar) . phi;
set Y = (D,phi) AddAsWitnessTo X;
set phi1 = head phi;
set no = SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}));
set s2 = the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})));
set Z = {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))};
defpred S1[] means ( X \/ {phi} is D -consistent & (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) <> {} );
 ( ( S1[] implies (D,phi) AddAsWitnessTo X = X \/ {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))} ) & ( not S1[] implies (D,phi) AddAsWitnessTo X = X ) ) by Def67;
then  ( ( S1[] implies ((D,phi) AddAsWitnessTo X) \ X = {((((S -firstChar) . phi), the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})))) -SymbolSubstIn (head phi))} \ X ) & ( not S1[] implies ((D,phi) AddAsWitnessTo X) \ X = {} ) ) by XBOOLE_1:40;
hence  for b1 being set st b1 = ((D,phi) AddAsWitnessTo X) \ X holds
b1 is trivial ; ::_thesis: verum
end;
end;

definition
let S be Language;
let D be RuleSet of S;
let X be functional set ;
let phi be Element of ExFormulasOf S;
:: original: AddAsWitnessTo
redefine func(D,phi) AddAsWitnessTo X ->  Subset of (X \/ (AllFormulasOf S));
coherence
 (D,phi) AddAsWitnessTo X is Subset of (X \/ (AllFormulasOf S))
proof
set F = S -firstChar ;
set IT = (D,phi) AddAsWitnessTo X;
set L = LettersOf S;
set l1 = (S -firstChar) . phi;
set phi1 = head phi;
set SS = AllSymbolsOf S;
set strings = ((AllSymbolsOf S) *) \ {{}};
set FF = AllFormulasOf S;
set no = SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}));
set s2 = the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)})));
defpred S1[] means ( X \/ {phi} is D -consistent & (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) <> {} );
percases ( S1[] or not S1[] ) ;
supposeA1: S1[] ; ::_thesis: (D,phi) AddAsWitnessTo X is Subset of (X \/ (AllFormulasOf S))
then reconsider Y = (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) as non empty set ;
 ( the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) in Y & Y c= LettersOf S ) ;
then reconsider l2 = the Element of (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (X \/ {(head phi)}))) as literal Element of AllSymbolsOf S ;
reconsider psi = (((S -firstChar) . phi),l2) -SymbolSubstIn (head phi) as wff string of S ;
reconsider psii = psi as Element of AllFormulasOf S by FOMODEL2:16;
 (D,phi) AddAsWitnessTo X = X \/ {psii} by A1, Def67;
hence  (D,phi) AddAsWitnessTo X is Subset of (X \/ (AllFormulasOf S)) by XBOOLE_1:9; ::_thesis: verum
end;
suppose not S1[] ; ::_thesis: (D,phi) AddAsWitnessTo X is Subset of (X \/ (AllFormulasOf S))
then  (D,phi) AddAsWitnessTo X = X null (AllFormulasOf S) by Def67;
hence  (D,phi) AddAsWitnessTo X is Subset of (X \/ (AllFormulasOf S)) ; ::_thesis: verum
end;
end;
end;
end;

definition
let S be Language;
let D be RuleSet of S;
attrD is Correct  means :Def68: :: FOMODEL4:def 68
 for phi being wff string of S
for X being set st phi is X,D -provable holds
for U being non empty set
for I being Element of U -InterpretersOf S st X is I -satisfied holds
I -TruthEval phi = 1;
end;

:: deftheorem Def68 defines Correct FOMODEL4:def_68_:_
 for S being Language
for D being RuleSet of S holds
( D is Correct iff for phi being wff string of S
for X being set st phi is X,D -provable holds
for U being non empty set
for I being Element of U -InterpretersOf S st X is I -satisfied holds
I -TruthEval phi = 1 );

Lm52:  for X being set
for S being Language
for phi being wff string of S
for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & X \/ {phi} is D -inconsistent holds
xnot phi is X,D -provable
proof
let X be set ; ::_thesis: for S being Language
for phi being wff string of S
for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & X \/ {phi} is D -inconsistent holds
xnot phi is X,D -provable
let S be Language; ::_thesis: for phi being wff string of S
for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & X \/ {phi} is D -inconsistent holds
xnot phi is X,D -provable
let phi be wff string of S; ::_thesis: for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & X \/ {phi} is D -inconsistent holds
xnot phi is X,D -provable
let D be RuleSet of S; ::_thesis: ( D is isotone & R#1 S in D & R#8 S in D & X \/ {phi} is D -inconsistent implies xnot phi is X,D -provable )
set XX = X \/ {phi};
set N = TheNorSymbOf S;
set G1 = R#1 S;
set G8 = R#8 S;
set E1 = {(R#1 S)};
set E8 = {(R#8 S)};
assume A1: ( D is isotone & R#1 S in D & R#8 S in D ) ; ::_thesis: ( not X \/ {phi} is D -inconsistent or xnot phi is X,D -provable )
then reconsider F1 = {(R#1 S)}, F8 = {(R#8 S)} as Subset of D by ZFMISC_1:31;
assume  X \/ {phi} is D -inconsistent ; ::_thesis: xnot phi is X,D -provable
then consider phi1, phi2 being wff string of S such that
A2: ( phi1 is X \/ {phi},D -provable & (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 is X \/ {phi},D -provable ) by Def65;
set nphi1 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2;
consider H1 being set , m1 being Nat such that
A3: ( H1 c= X \/ {phi} & [H1,phi1] is m1, {} ,D -derivable ) by Def62, A2;
consider H2 being set , m2 being Nat such that
A4: ( H2 c= X \/ {phi} & [H2,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] is m2, {} ,D -derivable ) by Def62, A2;
reconsider seqt1 = [H1,phi1], seqt2 = [H2,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] as S -sequent-like set by A3, A4;
 ( (seqt1 `1) \+\ H1 = {} & (seqt2 `1) \+\ H2 = {} ) ;
then reconsider H11 = H1, H22 = H2 as S -premises-like Subset of (X \/ {phi}) by A3, A4, FOMODEL0:29;
reconsider H111 = H11 \ {phi}, H222 = H22 \ {phi} as S -premises-like Subset of X by XBOOLE_1:43;
 ( H11 \ H111 = H11 /\ {phi} & H22 \ H222 = H22 /\ {phi} ) by XBOOLE_1:48;
then reconsider pH1 = H11 \ H111, pH2 = H22 \ H222 as S -premises-like Subset of {phi} ;
reconsider H = H11 \/ H22 as S -premises-like Subset of (X \/ {phi}) ;
reconsider h = H \ {phi} as S -premises-like Subset of X by XBOOLE_1:43;
reconsider hp = H /\ {phi} as S -premises-like Subset of {phi} ;
reconsider Phi = {phi} as S -premises-like set ;
set M = (m1 + m2) + 1;
reconsider hh = h \/ {phi} as S -premises-like set ;
reconsider e = {} null S as S -sequents-like set ;
set x = [hh,phi1];
set y = [hh,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)];
 [((H11 \/ (H22 \/ Phi)) null (m2 + 1)),phi1] is m2 + 1,{[H11,phi1]},{(R#1 S)} -derivable ;
then  [(H11 \/ (H22 \/ {phi})),phi1] is m1 + (m2 + 1), {} ,D \/ {(R#1 S)} -derivable by A3, Lm21, A1;
then  [(H \/ {phi}),phi1] is (m1 + m2) + 1, {} ,D null F1 -derivable by XBOOLE_1:4;
then  [((h \/ hp) \/ {phi}),phi1] is (m1 + m2) + 1, {} ,D -derivable by FOMODEL0:48;
then A5: [(h \/ ({phi} null hp)),phi1] is (m1 + m2) + 1, {} ,D -derivable by XBOOLE_1:4;
 [((H22 \/ (H11 \/ {phi})) null (m1 + 1)),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] is m1 + 1,{[H22,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},{(R#1 S)} -derivable ;
then  [(H22 \/ (H11 \/ {phi})),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] is m2 + (m1 + 1), {} ,D \/ {(R#1 S)} -derivable by A4, Lm21, A1;
then  [(H \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] is (m1 + m2) + 1, {} ,D null F1 -derivable by XBOOLE_1:4;
then  [((h \/ hp) \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] is (m1 + m2) + 1, {} ,D -derivable by FOMODEL0:48;
then A6: [(h \/ ({phi} null hp)),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] is (m1 + m2) + 1, {} ,D -derivable by XBOOLE_1:4;
 [(h null (phi1 ^ phi2)),(xnot phi)] is 1,{[(h \/ {phi}),phi1],[(h \/ {phi}),((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)]},{(R#8 S)} -derivable ;
then  [h,(xnot phi)] is ((m1 + m2) + 1) + 1,e \/ e,(D \/ D) \/ {(R#8 S)} -derivable by A5, A6, Lm48, A1;
then  [h,(xnot phi)] is ((m1 + m2) + 1) + 1, {} ,D null F8 -derivable ;
hence  xnot phi is X,D -provable by Def62; ::_thesis: verum
end;

Lm53:  for X being set
for S being Language
for D being RuleSet of S holds
( X is D -consistent iff for Y being finite Subset of X holds Y is D -consistent )
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S holds
( X is D -consistent iff for Y being finite Subset of X holds Y is D -consistent )
let S be Language; ::_thesis: for D being RuleSet of S holds
( X is D -consistent iff for Y being finite Subset of X holds Y is D -consistent )
let D be RuleSet of S; ::_thesis: ( X is D -consistent iff for Y being finite Subset of X holds Y is D -consistent )
set N = TheNorSymbOf S;
thus  ( X is D -consistent implies for Y being finite Subset of X holds Y is D -consistent ) by Th15; ::_thesis: ( ( for Y being finite Subset of X holds Y is D -consistent ) implies X is D -consistent )
assume A1: for Y being finite Subset of X holds Y is D -consistent ; ::_thesis: X is D -consistent
assume  X is D -inconsistent ; ::_thesis: contradiction
then consider phi1, phi2 being wff string of S such that
A2: ( phi1 is X,D -provable & (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 is X,D -provable ) by Def65;
reconsider phi = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 as non 0wff wff non exal string of S ;
consider H1 being set , m1 being Nat such that
A3: ( H1 c= X & [H1,phi1] is m1, {} ,D -derivable ) by Def62, A2;
consider H2 being set , m2 being Nat such that
A4: ( H2 c= X & [H2,phi] is m2, {} ,D -derivable ) by Def62, A2;
reconsider seqt1 = [H1,phi1], seqt2 = [H2,phi] as S -sequent-like set by A3, A4;
 ( (seqt1 `1) \+\ H1 = {} & (seqt2 `1) \+\ H2 = {} ) ;
then reconsider H11 = H1, H22 = H2 as S -premises-like Subset of X by A3, A4, FOMODEL0:29;
A5: phi1 is H1 null H2,D -provable by A3, Def62;
 phi is H2 null H1,D -provable by A4, Def62;
then  H11 \/ H22 is D -inconsistent by A5, Def65;
hence  contradiction by A1; ::_thesis: verum
end;

Lm54:  for X being set
for S being Language
for D being RuleSet of S st R#0 S in D & X is S -covering & X is D -consistent holds
( X is S -mincover & X is D -expanded )
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S st R#0 S in D & X is S -covering & X is D -consistent holds
( X is S -mincover & X is D -expanded )
let S be Language; ::_thesis: for D being RuleSet of S st R#0 S in D & X is S -covering & X is D -consistent holds
( X is S -mincover & X is D -expanded )
let D be RuleSet of S; ::_thesis: ( R#0 S in D & X is S -covering & X is D -consistent implies ( X is S -mincover & X is D -expanded ) )
set G0 = R#0 S;
set E0 = {(R#0 S)};
assume that
A1: R#0 S in D and
A2: ( X is S -covering & X is D -consistent ) ; ::_thesis: ( X is S -mincover & X is D -expanded )
A3: {(R#0 S)} c= D by A1, ZFMISC_1:31;
A4: for phi being wff string of S holds
( ( phi in X implies not xnot phi in X ) & ( not phi in X implies xnot phi in X ) )
proof
let phi be wff string of S; ::_thesis: ( ( phi in X implies not xnot phi in X ) & ( not phi in X implies xnot phi in X ) )
hereby  ::_thesis: ( not phi in X implies xnot phi in X )
assume  phi in X ; ::_thesis: not xnot phi in X
then  phi is X,{(R#0 S)} -provable by Th4;
then  phi is X,D -provable by A3, Lm49;
then  not xnot phi is X,D -provable by A2, Def65;
then  not xnot phi is X,{(R#0 S)} -provable by A3, Lm49;
hence  not xnot phi in X by Th4; ::_thesis: verum
end;
assume  not phi in X ; ::_thesis: xnot phi in X
hence  xnot phi in X by A2, FOMODEL2:def_40; ::_thesis: verum
end;
then  for phi being wff string of S holds
( phi in X iff not xnot phi in X ) ;
hence  X is S -mincover by FOMODEL2:def_34; ::_thesis: X is D -expanded
now__::_thesis:_for_x_being_set_st_x_is_X,D_-provable_holds_
x_in_X
let x be set ; ::_thesis: ( x is X,D -provable implies x in X )
assume A5: x is X,D -provable ; ::_thesis: x in X
then reconsider phi = x as wff string of S by Lm24;
 not xnot phi is X,D -provable by A5, A2, Def65;
then  not xnot phi is X,{(R#0 S)} -provable by A3, Lm49;
then  not xnot phi in X by Th4;
hence  x in X by A4; ::_thesis: verum
end;
hence  X is D -expanded by Def61; ::_thesis: verum
end;

Lm55:  for X being set
for S being Language
for phi being wff string of S
for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & phi is X,D -provable & X is D -consistent holds
X \/ {phi} is D -consistent
proof
let X be set ; ::_thesis: for S being Language
for phi being wff string of S
for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & phi is X,D -provable & X is D -consistent holds
X \/ {phi} is D -consistent
let S be Language; ::_thesis: for phi being wff string of S
for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & phi is X,D -provable & X is D -consistent holds
X \/ {phi} is D -consistent
let phi be wff string of S; ::_thesis: for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & phi is X,D -provable & X is D -consistent holds
X \/ {phi} is D -consistent
let D be RuleSet of S; ::_thesis: ( D is isotone & R#1 S in D & R#8 S in D & phi is X,D -provable & X is D -consistent implies X \/ {phi} is D -consistent )
assume A1: ( D is isotone & R#1 S in D & R#8 S in D ) ; ::_thesis: ( not phi is X,D -provable or not X is D -consistent or X \/ {phi} is D -consistent )
assume  ( phi is X,D -provable & X is D -consistent ) ; ::_thesis: X \/ {phi} is D -consistent
then  not xnot phi is X,D -provable by Def65;
hence  X \/ {phi} is D -consistent by Lm52, A1; ::_thesis: verum
end;

Lm56:  for U being non empty set
for X being set
for S being Language
for D being RuleSet of S
for I being Element of U -InterpretersOf S st D is Correct & X is I -satisfied holds
X is D -consistent
proof
let U be non empty set ; ::_thesis: for X being set
for S being Language
for D being RuleSet of S
for I being Element of U -InterpretersOf S st D is Correct & X is I -satisfied holds
X is D -consistent
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S
for I being Element of U -InterpretersOf S st D is Correct & X is I -satisfied holds
X is D -consistent
let S be Language; ::_thesis: for D being RuleSet of S
for I being Element of U -InterpretersOf S st D is Correct & X is I -satisfied holds
X is D -consistent
let D be RuleSet of S; ::_thesis: for I being Element of U -InterpretersOf S st D is Correct & X is I -satisfied holds
X is D -consistent
set N = TheNorSymbOf S;
let I be Element of U -InterpretersOf S; ::_thesis: ( D is Correct & X is I -satisfied implies X is D -consistent )
assume A1: ( D is Correct & X is I -satisfied ) ; ::_thesis: X is D -consistent
now__::_thesis:_for_phi1,_phi2_being_wff_string_of_S_holds_
(_not_phi1_is_X,D_-provable_or_not_(<*(TheNorSymbOf_S)*>_^_phi1)_^_phi2_is_X,D_-provable_)
given phi1, phi2 being wff string of S such that A2: ( phi1 is X,D -provable & (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 is X,D -provable ) ; ::_thesis: contradiction
set nphi1 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2;
 ( I -TruthEval phi1 = 1 & I -TruthEval ((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2) = 1 ) by A2, A1, Def68;
hence  contradiction by FOMODEL2:19; ::_thesis: verum
end;
hence  X is D -consistent by Def65; ::_thesis: verum
end;

registration
let S be Language;
let t1, t2 be termal string of S;
cluster(SubTerms ((<*(TheEqSymbOf S)*> ^ t1) ^ t2)) \+\ <*t1,t2*> ->  empty for set ;
coherence
 for b1 being set st b1 = (SubTerms ((<*(TheEqSymbOf S)*> ^ t1) ^ t2)) \+\ <*t1,t2*> holds
b1 is empty
proof
set E = TheEqSymbOf S;
reconsider phi0 = (<*(TheEqSymbOf S)*> ^ t1) ^ t2 as 0wff string of S ;
set C = S -multiCat ;
set F = S -firstChar ;
set ST = SubTerms phi0;
set SS = AllSymbolsOf S;
set TT = AllTermsOf S;
reconsider tt3 = t1, tt4 = t2 as Element of AllTermsOf S by FOMODEL1:def_32;
A1: 2 -tuples_on (AllTermsOf S) =  {  <*tt1,tt2*> where tt1, tt2 is Element of AllTermsOf S : verum  }  by FINSEQ_2:99;
A2: ( phi0 = <*(TheEqSymbOf S)*> ^ (t1 ^ t2) & ((<*(TheEqSymbOf S)*> ^ (t1 ^ t2)) . 1) \+\ (TheEqSymbOf S) = {} ) by FINSEQ_1:32;
then A3: TheEqSymbOf S = phi0 . 1 by FOMODEL0:29
.= (S -firstChar) . phi0 by FOMODEL0:6 ;
then  <*(TheEqSymbOf S)*> ^ ((S -multiCat) . (SubTerms phi0)) = <*(TheEqSymbOf S)*> ^ (t1 ^ t2) by A2, FOMODEL1:def_38;
then A4: (S -multiCat) . (SubTerms phi0) = t1 ^ t2 by FOMODEL0:41;
 ((abs (ar (TheEqSymbOf S))) - 2) + 2 = 0 + 2 ;
then  SubTerms phi0 in 2 -tuples_on (AllTermsOf S) by A3, FOMODEL0:16;
then consider tt1, tt2 being Element of AllTermsOf S such that
A5: SubTerms phi0 = <*tt1,tt2*> by A1;
 ( tt1 is Element of (AllSymbolsOf S) * & tt2 is Element of (AllSymbolsOf S) * & tt3 is Element of (AllSymbolsOf S) * & tt4 is Element of (AllSymbolsOf S) * ) by TARSKI:def_3;
then reconsider tt11 = tt1, tt22 = tt2, tt33 = tt3, tt44 = tt4 as AllSymbolsOf S -valued FinSequence ;
 ((S -multiCat) . <*tt11,tt22*>) \+\ (tt11 ^ tt22) = {} ;
then  tt11 ^ tt22 = tt33 ^ tt44 by A5, A4, FOMODEL0:29;
then  ( tt11 = tt33 & tt22 = tt44 ) by FOMODEL0:def_20;
hence  for b1 being set st b1 = (SubTerms ((<*(TheEqSymbOf S)*> ^ t1) ^ t2)) \+\ <*t1,t2*> holds
b1 is empty by A5, FOMODEL0:29; ::_thesis: verum
end;
end;

Lm57:  for U being non empty set
for S being Language
for t1, t2 being termal string of S
for I being b2,b1 -interpreter-like Function holds
( I -AtomicEval ((<*(TheEqSymbOf S)*> ^ t1) ^ t2) = 1 iff (I -TermEval) . t1 = (I -TermEval) . t2 )
proof
let U be non empty set ; ::_thesis: for S being Language
for t1, t2 being termal string of S
for I being b1,U -interpreter-like Function holds
( I -AtomicEval ((<*(TheEqSymbOf S)*> ^ t1) ^ t2) = 1 iff (I -TermEval) . t1 = (I -TermEval) . t2 )
let S be Language; ::_thesis: for t1, t2 being termal string of S
for I being S,U -interpreter-like Function holds
( I -AtomicEval ((<*(TheEqSymbOf S)*> ^ t1) ^ t2) = 1 iff (I -TermEval) . t1 = (I -TermEval) . t2 )
let t1, t2 be termal string of S; ::_thesis: for I being S,U -interpreter-like Function holds
( I -AtomicEval ((<*(TheEqSymbOf S)*> ^ t1) ^ t2) = 1 iff (I -TermEval) . t1 = (I -TermEval) . t2 )
set E = TheEqSymbOf S;
set phi0 = (<*(TheEqSymbOf S)*> ^ t1) ^ t2;
set ST = SubTerms ((<*(TheEqSymbOf S)*> ^ t1) ^ t2);
set F = S -firstChar ;
 ( (<*(TheEqSymbOf S)*> ^ t1) ^ t2 = <*(TheEqSymbOf S)*> ^ (t1 ^ t2) & ((<*(TheEqSymbOf S)*> ^ (t1 ^ t2)) . 1) \+\ (TheEqSymbOf S) = {} ) by FINSEQ_1:32;
then A1: TheEqSymbOf S = ((<*(TheEqSymbOf S)*> ^ t1) ^ t2) . 1 by FOMODEL0:29
.= (S -firstChar) . ((<*(TheEqSymbOf S)*> ^ t1) ^ t2) by FOMODEL0:6 ;
 (SubTerms ((<*(TheEqSymbOf S)*> ^ t1) ^ t2)) \+\ <*t1,t2*> = {} ;
then  SubTerms ((<*(TheEqSymbOf S)*> ^ t1) ^ t2) = <*t1,t2*> by FOMODEL0:29;
then  ( (SubTerms ((<*(TheEqSymbOf S)*> ^ t1) ^ t2)) . 1 = t1 & (SubTerms ((<*(TheEqSymbOf S)*> ^ t1) ^ t2)) . 2 = t2 ) by FINSEQ_1:44;
hence  for I being S,U -interpreter-like Function holds
( I -AtomicEval ((<*(TheEqSymbOf S)*> ^ t1) ^ t2) = 1 iff (I -TermEval) . t1 = (I -TermEval) . t2 ) by A1, FOMODEL2:15; ::_thesis: verum
end;

definition
let S be Language;
let R be Rule of S;
attrR is Correct  means :Def69: :: FOMODEL4:def 69
 for X being set st X is S -correct holds
R . X is S -correct ;
end;

:: deftheorem Def69 defines Correct FOMODEL4:def_69_:_
 for S being Language
for R being Rule of S holds
( R is Correct iff for X being set st X is S -correct holds
R . X is S -correct );

registration
let S be Language;
clusterS -sequent-like  -> S -null for set ;
coherence
 for b1 being set st b1 is S -sequent-like holds
b1 is S -null ;
end;

Lm58:  for S being Language holds R#0 S is Correct
proof
let S be Language; ::_thesis: R#0 S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#0_S)_._X_is_S_-correct
set f = R#0 S;
set R = P#0 S;
set Q = S -sequents ;
let X be set ; ::_thesis: ( X is S -correct implies (R#0 S) . X is S -correct )
assume  X is S -correct ; ::_thesis: (R#0 S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_phi_being_wff_string_of_S_st_[x,phi]_in_(R#0_S)_._X_holds_
I_-TruthEval_phi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for phi being wff string of S st [x,phi] in (R#0 S) . X holds
I -TruthEval phi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for phi being wff string of S st [x,phi] in (R#0 S) . X holds
I -TruthEval phi = 1
let x be I -satisfied set ; ::_thesis: for phi being wff string of S st [x,phi] in (R#0 S) . X holds
I -TruthEval phi = 1
let phi be wff string of S; ::_thesis: ( [x,phi] in (R#0 S) . X implies I -TruthEval phi = 1 )
set s = [x,phi];
A1: ( dom (P#0 S) c= bool (S -sequents) & [x,phi] `1 = x & [x,phi] `2 = phi ) ;
assume A2: [x,phi] in (R#0 S) . X ; ::_thesis: I -TruthEval phi = 1
then A3: ( [x,phi] in S -sequents & [X,[x,phi]] in P#0 S ) by Lm29;
then  X in dom (P#0 S) by XTUPLE_0:def_12;
then reconsider XX = X as Subset of (S -sequents) ;
reconsider seqt = [x,phi] as Element of S -sequents by A2, Lm29;
 seqt Rule0 XX by A3, Def31;
then  phi in x by A1, Def18;
hence  I -TruthEval phi = 1 by FOMODEL2:def_42; ::_thesis: verum
end;
hence  (R#0 S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#0 S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#0 S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#0 S holds
b1 is Correct by Lm58;
end;

registration
let S be Language;
cluster Relation-like bool (S -sequents) -defined bool (S -sequents) -valued Function-like total quasi_total Correct for Element of Funcs ((bool (S -sequents)),(bool (S -sequents)));
existence
 ex b1 being Rule of S st b1 is Correct
proof
take  R#0 S ; ::_thesis: R#0 S is Correct
thus  R#0 S is Correct ; ::_thesis: verum
end;
end;

Lm59:  for S being Language holds R#1 S is Correct
proof
let S be Language; ::_thesis: R#1 S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#1_S)_._X_is_S_-correct
set f = R#1 S;
set R = P#1 S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
set FF = AllFormulasOf S;
set TT = AllTermsOf S;
set SS = AllSymbolsOf S;
set F = S -firstChar ;
set C = S -multiCat ;
let X be set ; ::_thesis: ( X is S -correct implies (R#1 S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#1 S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#1_S)_._X_holds_
I_-TruthEval_psi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#1 S) . X holds
I -TruthEval psi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#1 S) . X holds
I -TruthEval psi = 1
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#1 S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#1 S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
set TE = I -TermEval ;
set d = U -deltaInterpreter ;
A2: ( ([x,psi] `1) \+\ x = {} & ([x,psi] `2) \+\ psi = {} ) ;
A3: ( dom (P#1 S) c= bool (S -sequents) & [x,psi] `1 = x & [x,psi] `2 = psi ) ;
assume A4: [x,psi] in (R#1 S) . X ; ::_thesis: I -TruthEval psi = 1
then A5: ( [x,psi] in S -sequents & [X,[x,psi]] in P#1 S ) by Lm29;
then  X in dom (P#1 S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A4, Lm29;
 seqt Rule1 Seqts by A5, Def32;
then consider y being set  such that
A6: ( y in Seqts & y `1 c= seqt `1 & seqt `2 = y `2 ) by Def19;
reconsider H = y `1 as Subset of x by A2, A6;
 [H,psi] in Seqts by A6, A3, MCART_1:21;
hence  I -TruthEval psi = 1 by FOMODEL2:def_44; ::_thesis: verum
end;
hence  (R#1 S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#1 S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#1 S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#1 S holds
b1 is Correct by Lm59;
end;

Lm60:  for S being Language holds R#2 S is Correct
proof
let S be Language; ::_thesis: R#2 S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#2_S)_._X_is_S_-correct
set f = R#2 S;
set R = P#2 S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
set FF = AllFormulasOf S;
set TT = AllTermsOf S;
set SS = AllSymbolsOf S;
set F = S -firstChar ;
set C = S -multiCat ;
let X be set ; ::_thesis: ( X is S -correct implies (R#2 S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#2 S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#2_S)_._X_holds_
I_-TruthEval_psi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#2 S) . X holds
I -TruthEval psi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#2 S) . X holds
I -TruthEval psi = 1
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#2 S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#2 S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
set TE = I -TermEval ;
set d = U -deltaInterpreter ;
A2: ( ([x,psi] `1) \+\ x = {} & ([x,psi] `2) \+\ psi = {} ) ;
assume A3: [x,psi] in (R#2 S) . X ; ::_thesis: I -TruthEval psi = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#2 S ) by Lm29;
then  X in dom (P#2 S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A3, Lm29;
 seqt Rule2 Seqts by A4, Def33;
then consider t being termal string of S such that
A5: seqt `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t by Def20;
 (I -TermEval) . t = (I -TermEval) . t ;
then  I -AtomicEval ((<*(TheEqSymbOf S)*> ^ t) ^ t) = 1 by Lm57;
hence  I -TruthEval psi = 1 by A2, A5; ::_thesis: verum
end;
hence  (R#2 S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#2 S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#2 S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#2 S holds
b1 is Correct by Lm60;
end;

Lm61:  for S being Language holds R#3a S is Correct
proof
let S be Language; ::_thesis: R#3a S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#3a_S)_._X_is_S_-correct
set f = R#3a S;
set R = P#3a S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
set FF = AllFormulasOf S;
set TT = AllTermsOf S;
set SS = AllSymbolsOf S;
set F = S -firstChar ;
set C = S -multiCat ;
let X be set ; ::_thesis: ( X is S -correct implies (R#3a S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#3a S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#3a_S)_._X_holds_
I_-TruthEval_psi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#3a S) . X holds
I -TruthEval psi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#3a S) . X holds
I -TruthEval psi = 1
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#3a S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#3a S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
set TE = I -TermEval ;
set d = U -deltaInterpreter ;
A2: ( ([x,psi] `1) \+\ x = {} & ([x,psi] `2) \+\ psi = {} ) ;
assume A3: [x,psi] in (R#3a S) . X ; ::_thesis: I -TruthEval psi = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#3a S ) by Lm29;
then  X in dom (P#3a S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A3, Lm29;
 seqt Rule3a Seqts by A4, Def34;
then consider t, t1, t2 being termal string of S, y being set  such that
A5: ( y in Seqts & seqt `1 = (y `1) \/ {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} & y `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t1 & seqt `2 = (<*(TheEqSymbOf S)*> ^ t) ^ t2 ) by Def21;
reconsider phi1 = (<*(TheEqSymbOf S)*> ^ t) ^ t1, phi2 = (<*(TheEqSymbOf S)*> ^ t) ^ t2, phi = (<*(TheEqSymbOf S)*> ^ t1) ^ t2 as 0wff string of S ;
reconsider gamma = (y `1) null {phi}, z = {phi} null (y `1) as Subset of x by A5, A2;
 [gamma,phi1] in Seqts by A5, MCART_1:21;
then  I -TruthEval phi1 = 1 by FOMODEL2:def_44;
then A6: ( phi2 = psi & (I -TermEval) . t = (I -TermEval) . t1 ) by Lm57, A2, A5;
 z = {phi} ;
then  I -TruthEval phi = 1 by FOMODEL2:27;
then  (I -TermEval) . t2 = (I -TermEval) . t by A6, Lm57;
then  I -AtomicEval phi2 = 1 by Lm57;
hence  I -TruthEval psi = 1 by A2, A5; ::_thesis: verum
end;
hence  (R#3a S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#3a S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#3a S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#3a S holds
b1 is Correct by Lm61;
end;

Lm62:  for S being Language holds R#3b S is Correct
proof
let S be Language; ::_thesis: R#3b S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#3b_S)_._X_is_S_-correct
set f = R#3b S;
set R = P#3b S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
set FF = AllFormulasOf S;
set TT = AllTermsOf S;
set SS = AllSymbolsOf S;
set F = S -firstChar ;
set C = S -multiCat ;
let X be set ; ::_thesis: ( X is S -correct implies (R#3b S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#3b S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#3b_S)_._X_holds_
I_-TruthEval_psi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#3b S) . X holds
I -TruthEval psi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#3b S) . X holds
I -TruthEval psi = 1
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#3b S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#3b S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
set TE = I -TermEval ;
set d = U -deltaInterpreter ;
A2: ( ([x,psi] `1) \+\ x = {} & ([x,psi] `2) \+\ psi = {} ) ;
assume A3: [x,psi] in (R#3b S) . X ; ::_thesis: I -TruthEval psi = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#3b S ) by Lm29;
then  X in dom (P#3b S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A3, Lm29;
 seqt Rule3b Seqts by A4, Def35;
then consider t1, t2 being termal string of S such that
A5: ( seqt `1 = {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} & seqt `2 = (<*(TheEqSymbOf S)*> ^ t2) ^ t1 ) by Def22;
set phi1 = (<*(TheEqSymbOf S)*> ^ t1) ^ t2;
set phi2 = (<*(TheEqSymbOf S)*> ^ t2) ^ t1;
 {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} is I -satisfied by A5, A2;
then  1 = I -AtomicEval ((<*(TheEqSymbOf S)*> ^ t1) ^ t2) by FOMODEL2:27;
then  (I -TermEval) . t1 = (I -TermEval) . t2 by Lm57;
then  ( I -AtomicEval ((<*(TheEqSymbOf S)*> ^ t2) ^ t1) = 1 & (<*(TheEqSymbOf S)*> ^ t2) ^ t1 = psi ) by A5, A2, Lm57;
hence  I -TruthEval psi = 1 ; ::_thesis: verum
end;
hence  (R#3b S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#3b S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#3b S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#3b S holds
b1 is Correct by Lm62;
end;

Lm63:  for S being Language holds R#3d S is Correct
proof
let S be Language; ::_thesis: R#3d S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#3d_S)_._X_is_S_-correct
set f = R#3d S;
set R = P#3d S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
set FF = AllFormulasOf S;
set TT = AllTermsOf S;
set SS = AllSymbolsOf S;
set F = S -firstChar ;
set C = S -multiCat ;
let X be set ; ::_thesis: ( X is S -correct implies (R#3d S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#3d S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#3d_S)_._X_holds_
I_-TruthEval_psi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#3d S) . X holds
I -TruthEval psi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#3d S) . X holds
I -TruthEval psi = 1
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#3d S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#3d S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
set TE = I -TermEval ;
set d = U -deltaInterpreter ;
A2: ( ([x,psi] `1) \+\ x = {} & ([x,psi] `2) \+\ psi = {} ) ;
assume A3: [x,psi] in (R#3d S) . X ; ::_thesis: I -TruthEval psi = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#3d S ) by Lm29;
then  X in dom (P#3d S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A3, Lm29;
 seqt Rule3d Seqts by A4, Def36;
then consider r being low-compounding Element of S, T1, T2 being abs (ar r) -element Element of (AllTermsOf S) *  such that
A5: ( r is operational & seqt `1 =  {  ((<*(TheEqSymbOf S)*> ^ (TT1 . j)) ^ (TT2 . j)) where j is Element of Seg (abs (ar r)), TT1, TT2 is Function of (Seg (abs (ar r))),(((AllSymbolsOf S) *) \ {{}}) : ( TT1 = T1 & TT2 = T2 )  }  & seqt `2 = (<*(TheEqSymbOf S)*> ^ (r -compound T1)) ^ (r -compound T2) ) by Def23;
reconsider t1 = r -compound T1, t2 = r -compound T2 as termal string of S by A5;
 ( (t1 . 1) \+\ r = {} & (t2 . 1) \+\ r = {} ) ;
then  ( t1 . 1 = r & t2 . 1 = r ) by FOMODEL0:29;
then A6: ( (S -firstChar) . t1 = r & (S -firstChar) . t2 = r ) by FOMODEL0:6;
then  ( SubTerms t1 = T1 & SubTerms t2 = T2 ) by FOMODEL1:def_37;
then A7: ( (I -TermEval) . t1 = (I . r) . ((I -TermEval) * T1) & (I -TermEval) . t2 = (I . r) . ((I -TermEval) * T2) ) by A6, FOMODEL2:21;
reconsider Fam =  {  ((<*(TheEqSymbOf S)*> ^ (TT1 . j)) ^ (TT2 . j)) where j is Element of Seg (abs (ar r)), TT1, TT2 is Function of (Seg (abs (ar r))),(((AllSymbolsOf S) *) \ {{}}) : ( TT1 = T1 & TT2 = T2 )  }  null {} as Subset of x by A5, A2;
now__::_thesis:_(_len_((I_-TermEval)_*_T2)_=_len_((I_-TermEval)_*_T1)_&_(_for_k_being_Nat_st_1_<=_k_&_k_<=_len_((I_-TermEval)_*_T1)_holds_
((I_-TermEval)_*_T1)_._k_=_((I_-TermEval)_*_T2)_._k_)_)
set p = (I -TermEval) * T1;
set q = (I -TermEval) * T2;
 len ((I -TermEval) * T1) = abs (ar r) by CARD_1:def_7;
hence  len ((I -TermEval) * T2) = len ((I -TermEval) * T1) by CARD_1:def_7; ::_thesis: for k being Nat st 1 <= k & k <= len ((I -TermEval) * T1) holds
((I -TermEval) * T1) . k = ((I -TermEval) * T2) . k
let k be Nat; ::_thesis: ( 1 <= k & k <= len ((I -TermEval) * T1) implies ((I -TermEval) * T1) . k = ((I -TermEval) * T2) . k )
assume A8: ( 1 <= k & k <= len ((I -TermEval) * T1) ) ; ::_thesis: ((I -TermEval) * T1) . k = ((I -TermEval) * T2) . k
then A9: ( 1 <= k & k <= abs (ar r) ) by CARD_1:def_7;
then reconsider kk = k as Element of Seg (abs (ar r)) by FINSEQ_1:1;
 k - k <= (abs (ar r)) - k by A9, XREAL_1:9;
then reconsider h = (abs (ar r)) - k as Nat ;
reconsider k1 = k as non zero Nat by A8;
 ( dom (T1 null 0) = Seg ((abs (ar r)) + 0) & rng T1 c= ((AllSymbolsOf S) *) \ {{}} & dom (T2 null 0) = Seg ((abs (ar r)) + 0) & rng T2 c= ((AllSymbolsOf S) *) \ {{}} ) by PARTFUN1:def_2, RELAT_1:def_19;
then  ( T1 is Element of Funcs ((Seg (abs (ar r))),(((AllSymbolsOf S) *) \ {{}})) & T2 is Element of Funcs ((Seg (abs (ar r))),(((AllSymbolsOf S) *) \ {{}})) ) by FUNCT_2:def_2;
then reconsider TT1 = T1, TT2 = T2 as Function of (Seg (abs (ar r))),(((AllSymbolsOf S) *) \ {{}}) ;
 ( T1 is k1 + h -element & T2 is k1 + h -element ) ;
then  ( {(T1 . k1)} \ (AllTermsOf S) = {} & {(T2 . k1)} \ (AllTermsOf S) = {} ) ;
then  ( T1 . k in AllTermsOf S & T2 . k in AllTermsOf S ) by ZFMISC_1:60;
then reconsider t1 = T1 . k, t2 = T2 . k as termal string of S ;
reconsider z = (<*(TheEqSymbOf S)*> ^ t1) ^ t2 as 0wff string of S ;
 ( ((I -TermEval) . (TT1 . kk)) \+\ (((I -TermEval) * TT1) . kk) = {} & ((I -TermEval) . (TT2 . kk)) \+\ (((I -TermEval) * TT2) . kk) = {} ) ;
then A10: ( (I -TermEval) . (TT1 . kk) = ((I -TermEval) * TT1) . kk & (I -TermEval) . (TT2 . kk) = ((I -TermEval) * TT2) . kk ) by FOMODEL0:29;
set ST = <*t1,t2*>;
 (<*(TheEqSymbOf S)*> ^ (TT1 . kk)) ^ (TT2 . kk) in Fam ;
then  I -TruthEval z = 1 by FOMODEL2:def_42;
hence  ((I -TermEval) * T1) . k = ((I -TermEval) * T2) . k by A10, Lm57; ::_thesis: verum
end;
then  (I -TermEval) . t1 = (I -TermEval) . t2 by A7, FINSEQ_1:14;
then  ( I -AtomicEval ((<*(TheEqSymbOf S)*> ^ t1) ^ t2) = 1 & psi = (<*(TheEqSymbOf S)*> ^ t1) ^ t2 ) by Lm57, A2, A5;
hence  I -TruthEval psi = 1 ; ::_thesis: verum
end;
hence  (R#3d S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#3d S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#3d S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#3d S holds
b1 is Correct by Lm63;
end;

Lm64:  for S being Language holds R#3e S is Correct
proof
let S be Language; ::_thesis: R#3e S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#3e_S)_._X_is_S_-correct
set f = R#3e S;
set R = P#3e S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
set FF = AllFormulasOf S;
set TT = AllTermsOf S;
set SS = AllSymbolsOf S;
set F = S -firstChar ;
set C = S -multiCat ;
let X be set ; ::_thesis: ( X is S -correct implies (R#3e S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#3e S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#3e_S)_._X_holds_
I_-TruthEval_psi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#3e S) . X holds
b5 -TruthEval b7 = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#3e S) . X holds
b4 -TruthEval b6 = 1
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#3e S) . X holds
b3 -TruthEval b5 = 1
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#3e S) . X implies b2 -TruthEval b4 = 1 )
set s = [x,psi];
set TE = I -TermEval ;
set d = U -deltaInterpreter ;
A2: ( ([x,psi] `1) \+\ x = {} & ([x,psi] `2) \+\ psi = {} ) ;
assume A3: [x,psi] in (R#3e S) . X ; ::_thesis: b2 -TruthEval b4 = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#3e S ) by Lm29;
then  X in dom (P#3e S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A3, Lm29;
 seqt Rule3e Seqts by A4, Def37;
then consider r being relational Element of S, T1, T2 being abs (ar r) -element Element of (AllTermsOf S) *  such that
A5: ( seqt `1 = {(r -compound T1)} \/  {  ((<*(TheEqSymbOf S)*> ^ (TT1 . j)) ^ (TT2 . j)) where j is Element of Seg (abs (ar r)), TT1, TT2 is Function of (Seg (abs (ar r))),(((AllSymbolsOf S) *) \ {{}}) : ( TT1 = T1 & TT2 = T2 )  }  & seqt `2 = r -compound T2 ) by Def24;
reconsider psi0 = psi as 0wff string of S by A2, A5;
reconsider phi1 = r -compound T1 as 0wff string of S ;
reconsider rr = (S -firstChar) . psi0 as relational Element of S ;
reconsider Fam =  {  ((<*(TheEqSymbOf S)*> ^ (TT1 . j)) ^ (TT2 . j)) where j is Element of Seg (abs (ar r)), TT1, TT2 is Function of (Seg (abs (ar r))),(((AllSymbolsOf S) *) \ {{}}) : ( TT1 = T1 & TT2 = T2 )  }  null {phi1} as Subset of x by A5, A2;
 ( ((<*r*> ^ ((S -multiCat) . T1)) . 1) \+\ r = {} & ((<*r*> ^ ((S -multiCat) . T2)) . 1) \+\ r = {} ) ;
then  ( (<*r*> ^ ((S -multiCat) . T1)) . 1 = r & (<*r*> ^ ((S -multiCat) . T2)) . 1 = r & psi0 = <*r*> ^ ((S -multiCat) . T2) ) by A5, A2, FOMODEL0:29;
then A6: ( (S -firstChar) . phi1 = r & rr = r & psi0 = <*r*> ^ ((S -multiCat) . T2) ) by FOMODEL0:6;
then A7: ( T1 = SubTerms phi1 & T2 = SubTerms psi0 ) by FOMODEL1:def_38;
reconsider y = {phi1} null Fam as Subset of x by A5, A2;
A8: {phi1} = y ;
A9: now__::_thesis:_(_len_((I_-TermEval)_*_T2)_=_len_((I_-TermEval)_*_T1)_&_(_for_k_being_Nat_st_1_<=_k_&_k_<=_len_((I_-TermEval)_*_T1)_holds_
((I_-TermEval)_*_T1)_._k_=_((I_-TermEval)_*_T2)_._k_)_)
set p = (I -TermEval) * T1;
set q = (I -TermEval) * T2;
 len ((I -TermEval) * T1) = abs (ar r) by CARD_1:def_7;
hence  len ((I -TermEval) * T2) = len ((I -TermEval) * T1) by CARD_1:def_7; ::_thesis: for k being Nat st 1 <= k & k <= len ((I -TermEval) * T1) holds
((I -TermEval) * T1) . k = ((I -TermEval) * T2) . k
let k be Nat; ::_thesis: ( 1 <= k & k <= len ((I -TermEval) * T1) implies ((I -TermEval) * T1) . k = ((I -TermEval) * T2) . k )
assume A10: ( 1 <= k & k <= len ((I -TermEval) * T1) ) ; ::_thesis: ((I -TermEval) * T1) . k = ((I -TermEval) * T2) . k
then A11: ( 1 <= k & k <= abs (ar r) ) by CARD_1:def_7;
then reconsider kk = k as Element of Seg (abs (ar r)) by FINSEQ_1:1;
 k - k <= (abs (ar r)) - k by A11, XREAL_1:9;
then reconsider h = (abs (ar r)) - k as Nat ;
reconsider k1 = k as non zero Nat by A10;
 ( dom (T1 null 0) = Seg ((abs (ar r)) + 0) & rng T1 c= ((AllSymbolsOf S) *) \ {{}} & dom (T2 null 0) = Seg ((abs (ar r)) + 0) & rng T2 c= ((AllSymbolsOf S) *) \ {{}} ) by PARTFUN1:def_2, RELAT_1:def_19;
then  ( T1 is Element of Funcs ((Seg (abs (ar r))),(((AllSymbolsOf S) *) \ {{}})) & T2 is Element of Funcs ((Seg (abs (ar r))),(((AllSymbolsOf S) *) \ {{}})) ) by FUNCT_2:def_2;
then reconsider TT1 = T1, TT2 = T2 as Function of (Seg (abs (ar r))),(((AllSymbolsOf S) *) \ {{}}) ;
 ( T1 is k1 + h -element & T2 is k1 + h -element ) ;
then  ( {(T1 . k1)} \ (AllTermsOf S) = {} & {(T2 . k1)} \ (AllTermsOf S) = {} ) ;
then  ( T1 . k in AllTermsOf S & T2 . k in AllTermsOf S ) by ZFMISC_1:60;
then reconsider t1 = T1 . k, t2 = T2 . k as termal string of S ;
reconsider z = (<*(TheEqSymbOf S)*> ^ t1) ^ t2 as 0wff string of S ;
 ( ((I -TermEval) . (TT1 . kk)) \+\ (((I -TermEval) * TT1) . kk) = {} & ((I -TermEval) . (TT2 . kk)) \+\ (((I -TermEval) * TT2) . kk) = {} ) ;
then A12: ( (I -TermEval) . (TT1 . kk) = ((I -TermEval) * TT1) . kk & (I -TermEval) . (TT2 . kk) = ((I -TermEval) * TT2) . kk ) by FOMODEL0:29;
set ST = <*t1,t2*>;
 (<*(TheEqSymbOf S)*> ^ (TT1 . kk)) ^ (TT2 . kk) in Fam ;
then  I -TruthEval z = 1 by FOMODEL2:def_42;
hence  ((I -TermEval) * T1) . k = ((I -TermEval) * T2) . k by A12, Lm57; ::_thesis: verum
end;
percases ( rr = TheEqSymbOf S or rr <> TheEqSymbOf S ) ;
suppose rr = TheEqSymbOf S ; ::_thesis: b2 -TruthEval b4 = 1
 I -AtomicEval psi0 = I -AtomicEval phi1 by A6, A9, A7, FINSEQ_1:14;
hence  I -TruthEval psi = 1 by A8, FOMODEL2:27; ::_thesis: verum
end;
suppose rr <> TheEqSymbOf S ; ::_thesis: b2 -TruthEval b4 = 1
 I -AtomicEval psi0 = I -AtomicEval phi1 by A6, A9, A7, FINSEQ_1:14;
hence  I -TruthEval psi = 1 by A8, FOMODEL2:27; ::_thesis: verum
end;
end;
end;
hence  (R#3e S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#3e S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#3e S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#3e S holds
b1 is Correct by Lm64;
end;

Lm65:  for S being Language holds R#4 S is Correct
proof
let S be Language; ::_thesis: R#4 S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#4_S)_._X_is_S_-correct
set f = R#4 S;
set R = P#4 S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
set FF = AllFormulasOf S;
set TT = AllTermsOf S;
set SS = AllSymbolsOf S;
set F = S -firstChar ;
let X be set ; ::_thesis: ( X is S -correct implies (R#4 S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#4 S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#4_S)_._X_holds_
I_-TruthEval_psi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#4 S) . X holds
I -TruthEval psi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#4 S) . X holds
I -TruthEval psi = 1
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#4 S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#4 S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
A2: ( ([x,psi] `1) \+\ x = {} & ([x,psi] `2) \+\ psi = {} ) ;
assume A3: [x,psi] in (R#4 S) . X ; ::_thesis: I -TruthEval psi = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#4 S ) by Lm29;
then  X in dom (P#4 S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A3, Lm29;
 seqt Rule4 Seqts by A4, Def38;
then consider l being literal Element of S, phi being wff string of S, t being termal string of S such that
A5: ( seqt `1 = {((l,t) SubstIn phi)} & seqt `2 = <*l*> ^ phi ) by Def25;
reconsider tt = t as Element of AllTermsOf S by FOMODEL1:def_32;
reconsider phii = (l,tt) SubstIn phi as wff string of S ;
reconsider u = (I -TermEval) . tt as Element of U ;
reconsider I1 = (l,u) ReassignIn I as Element of U -InterpretersOf S ;
A6: ( x = {phii} & psi = <*l*> ^ phi ) by A5, A2;
then 1 = I -TruthEval phii by FOMODEL2:27
.= I1 -TruthEval phi by FOMODEL3:10 ;
hence  I -TruthEval psi = 1 by A6, FOMODEL2:19; ::_thesis: verum
end;
hence  (R#4 S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#4 S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#4 S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#4 S holds
b1 is Correct by Lm65;
end;

Lm66:  for S being Language holds R#5 S is Correct
proof
let S be Language; ::_thesis: R#5 S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#5_S)_._X_is_S_-correct
set f = R#5 S;
set R = P#5 S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
set FF = AllFormulasOf S;
set TT = AllTermsOf S;
set SS = AllSymbolsOf S;
set F = S -firstChar ;
set O = OwnSymbolsOf S;
let X be set ; ::_thesis: ( X is S -correct implies (R#5 S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#5 S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#5_S)_._X_holds_
1_=_I_-TruthEval_psi
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#5 S) . X holds
1 = I -TruthEval psi
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#5 S) . X holds
1 = I -TruthEval psi
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#5 S) . X holds
1 = I -TruthEval psi
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#5 S) . X implies 1 = I -TruthEval psi )
set s = [x,psi];
A2: ( ([x,psi] `1) \+\ x = {} & ([x,psi] `2) \+\ psi = {} ) ;
A3: ( dom (P#5 S) c= bool (S -sequents) & [x,psi] `1 = x & [x,psi] `2 = psi ) ;
assume A4: [x,psi] in (R#5 S) . X ; ::_thesis: 1 = I -TruthEval psi
then A5: ( [x,psi] in S -sequents & [X,[x,psi]] in P#5 S ) by Lm29;
then  X in dom (P#5 S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A4, Lm29;
 seqt Rule5 Seqts by A5, Def39;
then consider v1, v2 being literal Element of S, y being set , p being FinSequence such that
A6: ( seqt `1 = y \/ {(<*v1*> ^ p)} & v2 is (y \/ {p}) \/ {(seqt `2)} -absent & [(y \/ {((v1 SubstWith v2) . p)}),(seqt `2)] in Seqts ) by Def26;
 {(<*v1*> ^ p)} null y c= AllFormulasOf S by A6, Def64;
then  <*v1*> ^ p in AllFormulasOf S by ZFMISC_1:31;
then reconsider phi1 = <*v1*> ^ p as wff string of S ;
 v1 \+\ (phi1 . 1) = {} ;
then A7: v1 = phi1 . 1 by FOMODEL0:29
.= (S -firstChar) . phi1 by FOMODEL0:6 ;
then reconsider phi1 = phi1 as non 0wff wff exal string of S by FOMODEL2:def_32;
reconsider phi = head phi1 as wff string of S ;
 {psi} null (y \/ {phi}) = {psi} ;
then reconsider Psi = {psi} as non empty Subset of ((y \/ {phi}) \/ {psi}) ;
 {phi} null (y \/ {psi}) is Subset of ((y \/ {phi}) \/ {psi}) by XBOOLE_1:4;
then reconsider Phi = {phi} as non empty Subset of ((y \/ {phi}) \/ {psi}) ;
 y \/ ({phi} \/ {psi}) = (y \/ {phi}) \/ {psi} by XBOOLE_1:4;
then  y null ({phi} \/ {psi}) c= (y \/ {phi}) \/ {psi} ;
then reconsider yyy = y as Subset of ((y \/ {phi}) \/ {psi}) ;
A8: phi1 = (<*v1*> ^ phi) ^ (tail phi1) by A7, FOMODEL2:23
.= <*v1*> ^ phi ;
then A9: phi = p by FOMODEL0:41;
then A10: ( v2 is Psi null ((y \/ {phi}) \/ {psi}) -absent & v2 is Phi null ((y \/ {phi}) \/ {psi}) -absent & v2 is yyy null ((y \/ {phi}) \/ {psi}) -absent ) by A6, A3;
reconsider phi2 = (v1,v2) -SymbolSubstIn phi as wff string of S ;
reconsider yy = y null {phi1}, z = {phi1} null y as I -satisfied Subset of x by A6, A2;
 z = {phi1} ;
then  I -TruthEval phi1 = 1 by FOMODEL2:27;
then consider u being Element of U such that
A11: 1 = ((v1,u) ReassignIn I) -TruthEval phi by A8, FOMODEL2:19;
set f2 = v2 .--> ({} .--> u);
reconsider I1 = (v1,u) ReassignIn I, I2 = (v2,u) ReassignIn I as Element of U -InterpretersOf S ;
 not v2 in rng phi by A10, FOMODEL2:28;
then  I2 -TruthEval phi2 = 1 by A11, FOMODEL3:9;
then reconsider z2 = {phi2} as I2 -satisfied set by FOMODEL2:27;
 not v2 in rng psi by A10, FOMODEL2:28;
then  {v2} misses rng psi by ZFMISC_1:50;
then A12: dom (v2 .--> ({} .--> u)) misses rng psi by FUNCOP_1:13;
I2 | (rng psi) = (I | (rng psi)) +* ((v2 .--> ({} .--> u)) | (rng psi)) by FUNCT_4:71
.= (I | (rng psi)) +* {} by A12, RELAT_1:66
.= I | (rng psi) ;
then A13: I | ((rng psi) /\ (OwnSymbolsOf S)) = (I2 | (rng psi)) | (OwnSymbolsOf S) by RELAT_1:71
.= I2 | ((rng psi) /\ (OwnSymbolsOf S)) by RELAT_1:71 ;
 ( v2 is yyy -absent & yy is I -satisfied ) by A9, A6, A3;
then reconsider yyyy = yyy as I2 -satisfied Subset of x by FOMODEL3:14;
reconsider zz = yyyy \/ z2 as I2 -satisfied set ;
 [zz,psi] in Seqts by A6, A9, A3, FOMODEL0:def_23;
hence 1 = I2 -TruthEval psi by FOMODEL2:def_44
.= I -TruthEval psi by A13, FOMODEL3:13 ;
::_thesis: verum
end;
hence  (R#5 S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#5 S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#5 S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#5 S holds
b1 is Correct by Lm66;
end;

Lm67:  for S being Language holds R#6 S is Correct
proof
let S be Language; ::_thesis: R#6 S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#6_S)_._X_is_S_-correct
set f = R#6 S;
set R = P#6 S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
let X be set ; ::_thesis: ( X is S -correct implies (R#6 S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#6 S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#6_S)_._X_holds_
I_-TruthEval_psi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#6 S) . X holds
I -TruthEval psi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#6 S) . X holds
I -TruthEval psi = 1
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#6 S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#6 S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
A2: ( dom (P#6 S) c= bool (S -sequents) & [x,psi] `1 = x & [x,psi] `2 = psi ) ;
assume A3: [x,psi] in (R#6 S) . X ; ::_thesis: I -TruthEval psi = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#6 S ) by Lm29;
then  X in dom (P#6 S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A3, Lm29;
 seqt Rule6 Seqts by A4, Def40;
then consider y1, y2 being set , phi1, phi2 being wff string of S such that
A5: ( y1 in Seqts & y2 in Seqts & y1 `1 = y2 `1 & y2 `1 = seqt `1 & y1 `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi1 & y2 `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi2 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 ) by Def27;
 ( [x,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi1)] in Seqts & [x,((<*(TheNorSymbOf S)*> ^ phi2) ^ phi2)] in Seqts & psi = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 ) by A5, A2, MCART_1:21;
then  ( I -TruthEval ((<*(TheNorSymbOf S)*> ^ phi1) ^ phi1) = 1 & I -TruthEval ((<*(TheNorSymbOf S)*> ^ phi2) ^ phi2) = 1 ) by FOMODEL2:def_44;
then  ( I -TruthEval phi1 = 0 & I -TruthEval phi2 = 0 ) by FOMODEL2:19;
hence  I -TruthEval psi = 1 by A5, A2, FOMODEL2:19; ::_thesis: verum
end;
hence  (R#6 S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#6 S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#6 S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#6 S holds
b1 is Correct by Lm67;
end;

Lm68:  for S being Language holds R#7 S is Correct
proof
let S be Language; ::_thesis: R#7 S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#7_S)_._X_is_S_-correct
set f = R#7 S;
set R = P#7 S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
let X be set ; ::_thesis: ( X is S -correct implies (R#7 S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#7 S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#7_S)_._X_holds_
I_-TruthEval_psi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#7 S) . X holds
I -TruthEval psi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#7 S) . X holds
I -TruthEval psi = 1
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#7 S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#7 S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
A2: ( dom (P#7 S) c= bool (S -sequents) & [x,psi] `1 = x & [x,psi] `2 = psi ) ;
assume A3: [x,psi] in (R#7 S) . X ; ::_thesis: I -TruthEval psi = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#7 S ) by Lm29;
then  X in dom (P#7 S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A3, Lm29;
 seqt Rule7 Seqts by A4, Def41;
then consider y being set , phi1, phi2 being wff string of S such that
A5: ( y in Seqts & y `1 = seqt `1 & y `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1 ) by Def28;
 ( psi = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1 & [x,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] in Seqts ) by A2, A5, MCART_1:21;
then  I -TruthEval ((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2) = 1 by FOMODEL2:def_44;
then  ( I -TruthEval phi1 = 0 & I -TruthEval phi2 = 0 ) by FOMODEL2:19;
hence  I -TruthEval psi = 1 by A2, A5, FOMODEL2:19; ::_thesis: verum
end;
hence  (R#7 S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#7 S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#7 S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#7 S holds
b1 is Correct by Lm68;
end;

Lm69:  for S being Language holds R#8 S is Correct
proof
let S be Language; ::_thesis: R#8 S is Correct
now__::_thesis:_for_X_being_set_st_X_is_S_-correct_holds_
(R#8_S)_._X_is_S_-correct
set f = R#8 S;
set R = P#8 S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
let X be set ; ::_thesis: ( X is S -correct implies (R#8 S) . X is S -correct )
assume A1: X is S -correct ; ::_thesis: (R#8 S) . X is S -correct
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_x_being_b2_-satisfied_set_
for_psi_being_wff_string_of_S_st_[x,psi]_in_(R#8_S)_._X_holds_
I_-TruthEval_psi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for x being b1 -satisfied set
for psi being wff string of S st [x,psi] in (R#8 S) . X holds
I -TruthEval psi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#8 S) . X holds
I -TruthEval psi = 1
let x be I -satisfied set ; ::_thesis: for psi being wff string of S st [x,psi] in (R#8 S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; ::_thesis: ( [x,psi] in (R#8 S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
A2: ( ([x,psi] `1) \+\ x = {} & ([x,psi] `2) \+\ psi = {} ) ;
assume A3: [x,psi] in (R#8 S) . X ; ::_thesis: I -TruthEval psi = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#8 S ) by Lm29;
then  X in dom (P#8 S) by XTUPLE_0:def_12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A3, Lm29;
 seqt Rule8 Seqts by A4, Def42;
then consider y1, y2 being set , phi, phi1, phi2 being wff string of S such that
A5: ( y1 in Seqts & y2 in Seqts & y1 `1 = y2 `1 & y1 `2 = phi1 & y2 `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & {phi} \/ (seqt `1) = y1 `1 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi) ^ phi ) by Def29;
reconsider Seqts = Seqts as non empty Subset of (S -sequents) by A5;
reconsider seqt1 = y1, seqt2 = y2 as Element of Seqts by A5;
reconsider H = seqt1 `1 as S -premises-like set ;
A6: ( {phi} \/ x = H & psi = (<*(TheNorSymbOf S)*> ^ phi) ^ phi ) by A5, A2;
now__::_thesis:_not_I_-TruthEval_phi_=_1
assume  I -TruthEval phi = 1 ; ::_thesis: contradiction
then reconsider H1 = {phi} as I -satisfied set by FOMODEL2:27;
 H1 \/ x is I -satisfied ;
then reconsider H2 = H as I -satisfied set by A5, A2;
 ( [H2,phi1] in Seqts & [H2,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] in Seqts ) by A5, MCART_1:21;
then  ( I -TruthEval phi1 = 1 & I -TruthEval ((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2) = 1 ) by FOMODEL2:def_44;
hence  contradiction by FOMODEL2:19; ::_thesis: verum
end;
then  I -TruthEval phi = 0 by FOMODEL0:39;
hence  I -TruthEval psi = 1 by A6, FOMODEL2:19; ::_thesis: verum
end;
hence  (R#8 S) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
hence  R#8 S is Correct by Def69; ::_thesis: verum
end;

registration
let S be Language;
cluster R#8 S ->  Correct for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#8 S holds
b1 is Correct by Lm69;
end;

theorem Th16: :: FOMODEL4:16
for S being Language
for D being RuleSet of S st ( for R being Rule of S st R in D holds
R is Correct ) holds
D is Correct
proof
let S be Language; ::_thesis: for D being RuleSet of S st ( for R being Rule of S st R in D holds
R is Correct ) holds
D is Correct
let D be RuleSet of S; ::_thesis: ( ( for R being Rule of S st R in D holds
R is Correct ) implies D is Correct )
set Q = S -sequents ;
set O = OneStep D;
 {} null S is S -correct ;
then reconsider e = {} null (S -sequents) as S -correct Subset of (S -sequents) ;
A1: dom (OneStep D) = bool (S -sequents) by FUNCT_2:def_1;
reconsider RO = rng (OneStep D) as Subset of (bool (S -sequents)) by RELAT_1:def_19;
assume A2: for R being Rule of S st R in D holds
R is Correct ; ::_thesis: D is Correct
defpred S1[ Nat] means  for X being S -correct Subset of (S -sequents) holds (($1,D) -derivables) . X is S -correct ;
A3: S1[ 0 ]
proof
set f = (0,D) -derivables ;
A4: (0,D) -derivables = id (field (OneStep D)) by FUNCT_7:68
.= id ((bool (S -sequents)) \/ RO) by A1
.= id (bool (S -sequents)) ;
let X be S -correct Subset of (S -sequents); ::_thesis: ((0,D) -derivables) . X is S -correct
 ((id (bool (S -sequents))) . X) \+\ X = {} ;
hence  ((0,D) -derivables) . X is S -correct by A4, FOMODEL0:29; ::_thesis: verum
end;
A5: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A6: S1[n] ; ::_thesis: S1[n + 1]
let X be S -correct Subset of (S -sequents); ::_thesis: (((n + 1),D) -derivables) . X is S -correct
set DM = ((n + 1),D) -derivables ;
set Dm = (n,D) -derivables ;
A7: dom ((n,D) -derivables) = bool (S -sequents) by FUNCT_2:def_1;
reconsider oldSeqs = ((n,D) -derivables) . X as S -correct Subset of (S -sequents) by A6;
A8: ((n + 1),D) -derivables = (OneStep D) * ((n,D) -derivables) by FUNCT_7:71;
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S
for_H_being_b2_-satisfied_set_
for_phi_being_wff_string_of_S_st_[H,phi]_in_(((n_+_1),D)_-derivables)_._X_holds_
I_-TruthEval_phi_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S
for H being b1 -satisfied set
for phi being wff string of S st [H,phi] in (((n + 1),D) -derivables) . X holds
I -TruthEval phi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: for H being I -satisfied set
for phi being wff string of S st [H,phi] in (((n + 1),D) -derivables) . X holds
I -TruthEval phi = 1
let H be I -satisfied set ; ::_thesis: for phi being wff string of S st [H,phi] in (((n + 1),D) -derivables) . X holds
I -TruthEval phi = 1
let phi be wff string of S; ::_thesis: ( [H,phi] in (((n + 1),D) -derivables) . X implies I -TruthEval phi = 1 )
assume A9: [H,phi] in (((n + 1),D) -derivables) . X ; ::_thesis: I -TruthEval phi = 1
set Fam =  {  (R .: {oldSeqs}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  ;
(((n + 1),D) -derivables) . X = (OneStep D) . oldSeqs by A7, A8, FUNCT_1:13
.= union (union  {  (R .: {oldSeqs}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  ) by Lm6 ;
then consider x being set  such that
A10: ( [H,phi] in x & x in union  {  (R .: {oldSeqs}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  ) by A9, TARSKI:def_4;
consider y being set  such that
A11: ( x in y & y in  {  (R .: {oldSeqs}) where R is Subset of [:(bool (S -sequents)),(bool (S -sequents)):] : R in D  }  ) by A10, TARSKI:def_4;
consider R being Subset of [:(bool (S -sequents)),(bool (S -sequents)):] such that
A12: ( y = R .: {oldSeqs} & R in D ) by A11;
reconsider RR = R as Correct Rule of S by A2, A12;
reconsider newSeqs = RR . oldSeqs as S -correct Subset of (S -sequents) by Def69;
 dom RR = bool (S -sequents) by FUNCT_2:def_1;
then  ( y = Im (R,oldSeqs) & Im (RR,oldSeqs) = {(RR . oldSeqs)} ) by A12, FUNCT_1:59;
then  [H,phi] in newSeqs by A10, A11, TARSKI:def_1;
hence  I -TruthEval phi = 1 by FOMODEL2:def_44; ::_thesis: verum
end;
hence  (((n + 1),D) -derivables) . X is S -correct by FOMODEL2:def_44; ::_thesis: verum
end;
A13: for n being Nat holds S1[n] from NAT_1:sch_2(A3, A5);
now__::_thesis:_for_phi_being_wff_string_of_S
for_X_being_set_st_phi_is_X,D_-provable_holds_
for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S_st_X_is_I_-satisfied_holds_
I_-TruthEval_phi_=_1
let phi be wff string of S; ::_thesis: for X being set st phi is X,D -provable holds
for U being non empty set
for I being Element of U -InterpretersOf S st X is I -satisfied holds
I -TruthEval phi = 1
let X be set ; ::_thesis: ( phi is X,D -provable implies for U being non empty set
for I being Element of U -InterpretersOf S st X is I -satisfied holds
I -TruthEval phi = 1 )
assume  phi is X,D -provable ; ::_thesis: for U being non empty set
for I being Element of U -InterpretersOf S st X is I -satisfied holds
I -TruthEval phi = 1
then consider H being set , m being Nat such that
A14: ( H c= X & [H,phi] is m, {} ,D -derivable ) by Def62;
reconsider HH = H as Subset of X by A14;
reconsider seqt = [H,phi] as Element of S -sequents by Def2, A14;
reconsider okSeqs = ((m,D) -derivables) . e as S -correct Subset of (S -sequents) by A13;
hereby  ::_thesis: verum
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S st X is I -satisfied holds
I -TruthEval phi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: ( X is I -satisfied implies I -TruthEval phi = 1 )
assume  X is I -satisfied ; ::_thesis: I -TruthEval phi = 1
then reconsider XX = X as I -satisfied set ;
reconsider HHH = HH as I -satisfied Subset of XX ;
 [HHH,phi] in okSeqs by A14, Def7;
hence  I -TruthEval phi = 1 by FOMODEL2:def_44; ::_thesis: verum
end;
end;
hence  D is Correct by Def68; ::_thesis: verum
end;

registration
let S be Language;
let R be Correct Rule of S;
cluster{R} ->  Correct for RuleSet of S;
coherence
 for b1 being RuleSet of S st b1 = {R} holds
b1 is Correct
proof
set D = {R};
 for P being Rule of S st P in {R} holds
P is Correct by TARSKI:def_1;
hence  for b1 being RuleSet of S st b1 = {R} holds
b1 is Correct by Th16; ::_thesis: verum
end;
end;

registration
let S be Language;
clusterS -rules  ->  Correct for RuleSet of S;
coherence
 for b1 being RuleSet of S st b1 = S -rules holds
b1 is Correct
proof
set A = {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)};
set B = {(R#5 S),(R#6 S),(R#7 S),(R#8 S)};
set IT = S -rules ;
now__::_thesis:_for_P_being_Rule_of_S_st_P_in_S_-rules_holds_
P_is_Correct
let P be Rule of S; ::_thesis: ( P in S -rules implies P is Correct )
assume  P in S -rules ; ::_thesis: P is Correct
then  ( P in {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} or P in {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} ) by XBOOLE_0:def_3;
hence  P is Correct by ENUMSET1:def_2, ENUMSET1:def_6; ::_thesis: verum
end;
hence  for b1 being RuleSet of S st b1 = S -rules holds
b1 is Correct by Th16; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster R#9 S ->  isotone for Rule of S;
coherence
 for b1 being Rule of S st b1 = R#9 S holds
b1 is isotone
proof
set R = R#9 S;
set Q = S -sequents ;
now__::_thesis:_for_Seqts,_Seqts2_being_Subset_of_(S_-sequents)_st_Seqts_c=_Seqts2_holds_
(R#9_S)_._Seqts_c=_(R#9_S)_._Seqts2
let Seqts, Seqts2 be Subset of (S -sequents); ::_thesis: ( Seqts c= Seqts2 implies (R#9 S) . Seqts c= (R#9 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume A1: Seqts c= Seqts2 ; ::_thesis: (R#9 S) . Seqts c= (R#9 S) . Seqts2
now__::_thesis:_for_x_being_set_st_x_in_(R#9_S)_._Seqts_holds_
x_in_(R#9_S)_._Seqts2
let x be set ; ::_thesis: ( x in (R#9 S) . Seqts implies x in (R#9 S) . Seqts2 )
assume A2: x in (R#9 S) . Seqts ; ::_thesis: x in (R#9 S) . Seqts2
then A3: ( x in S -sequents & [Seqts,x] in P#9 S ) by Lm29;
reconsider seqt = x as Element of S -sequents by A2;
 seqt Rule9 Seqts by A3, Def43;
then consider y being set , phi being wff string of S such that
A4: ( y in Seqts & seqt `2 = phi & y `1 = seqt `1 & y `2 = xnot (xnot phi) ) by Def30;
 seqt Rule9 Seqts2 by A4, Def30, A1;
then  [Seqts2,seqt] in P#9 S by Def43;
hence  x in (R#9 S) . Seqts2 by Th2; ::_thesis: verum
end;
hence  (R#9 S) . Seqts c= (R#9 S) . Seqts2 by TARSKI:def_3; ::_thesis: verum
end;
hence  for b1 being Rule of S st b1 = R#9 S holds
b1 is isotone by Def9; ::_thesis: verum
end;
end;

registration
let S be Language;
let H be S -premises-like set ;
let phi be wff string of S;
cluster[H,phi] null 1 -> 1,{[H,(xnot (xnot phi))]},{(R#9 S)} -derivable for set ;
coherence
 for b1 being set st b1 = [H,phi] null 1 holds
b1 is 1,{[H,(xnot (xnot phi))]},{(R#9 S)} -derivable
proof
set N = TheNorSymbOf S;
set nphi = xnot phi;
set phii = xnot (xnot phi);
set y = [H,(xnot (xnot phi))];
set SQ = {[H,(xnot (xnot phi))]};
set Sq = [H,phi];
set Q = S -sequents ;
reconsider seqt = [H,phi] as Element of S -sequents by Def2;
reconsider Seqts = {[H,(xnot (xnot phi))]} as Element of bool (S -sequents) by Def3;
 ( (seqt `2) \+\ phi = {} & ([H,(xnot (xnot phi))] `1) \+\ H = {} & (seqt `1) \+\ H = {} & ([H,(xnot (xnot phi))] `2) \+\ (xnot (xnot phi)) = {} ) ;
then  ( seqt `2 = phi & [H,(xnot (xnot phi))] `1 = H & seqt `1 = H & [H,(xnot (xnot phi))] `2 = xnot (xnot phi) ) by FOMODEL0:29;
then  ( [H,(xnot (xnot phi))] `1 = seqt `1 & seqt `2 = phi & [H,(xnot (xnot phi))] `2 = xnot (xnot phi) & [H,(xnot (xnot phi))] in Seqts ) by TARSKI:def_1;
then  seqt Rule9 Seqts by Def30;
then  [Seqts,seqt] in P#9 S by Def43;
then  [H,phi] in (R#9 S) . {[H,(xnot (xnot phi))]} by Th2;
hence  for b1 being set st b1 = [H,phi] null 1 holds
b1 is 1,{[H,(xnot (xnot phi))]},{(R#9 S)} -derivable by Lm50; ::_thesis: verum
end;
end;

registration
let X be set ;
let S be Language;
cluster non empty Relation-like NAT -defined AtomicFormulaSymbolsOf S -valued Function-like finite FinSequence-like FinSubsequence-like countable V209() 0wff 0 -wff wff X -implied for Element of ((AllSymbolsOf S) *) \ {{}};
existence
 ex b1 being 0 -wff string of S st b1 is X -implied
proof
set E = TheEqSymbOf S;
set t = the termal string of S;
set phi = (<*(TheEqSymbOf S)*> ^ the termal string of S) ^ the termal string of S;
take  (<*(TheEqSymbOf S)*> ^ the termal string of S) ^ the termal string of S ; ::_thesis: (<*(TheEqSymbOf S)*> ^ the termal string of S) ^ the termal string of S is X -implied
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S_st_X_is_I_-satisfied_holds_
I_-TruthEval_((<*(TheEqSymbOf_S)*>_^_the_termal_string_of_S)_^_the_termal_string_of_S)_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S st X is I -satisfied holds
I -TruthEval ((<*(TheEqSymbOf S)*> ^ the termal string of S) ^ the termal string of S) = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: ( X is I -satisfied implies I -TruthEval ((<*(TheEqSymbOf S)*> ^ the termal string of S) ^ the termal string of S) = 1 )
assume  X is I -satisfied ; ::_thesis: I -TruthEval ((<*(TheEqSymbOf S)*> ^ the termal string of S) ^ the termal string of S) = 1
set TE = I -TermEval ;
 (I -TermEval) . the termal string of S = (I -TermEval) . the termal string of S ;
hence  I -TruthEval ((<*(TheEqSymbOf S)*> ^ the termal string of S) ^ the termal string of S) = 1 by Lm57; ::_thesis: verum
end;
hence  (<*(TheEqSymbOf S)*> ^ the termal string of S) ^ the termal string of S is X -implied by FOMODEL2:def_45; ::_thesis: verum
end;
end;

registration
let X be set ;
let S be Language;
cluster non empty Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like countable V209() wff X -implied for Element of ((AllSymbolsOf S) *) \ {{}};
existence
 ex b1 being wff string of S st b1 is X -implied
proof
take  the 0 -wff X -implied string of S ; ::_thesis: the 0 -wff X -implied string of S is X -implied
thus  the 0 -wff X -implied string of S is X -implied ; ::_thesis: verum
end;
end;

registration
let S be Language;
let X be set ;
let phi be wff X -implied string of S;
cluster xnot (xnot phi) ->  wff X -implied for wff string of S;
coherence
 for b1 being wff string of S st b1 = xnot (xnot phi) holds
b1 is X -implied
proof
now__::_thesis:_for_U_being_non_empty_set_
for_I_being_Element_of_U_-InterpretersOf_S_st_X_is_I_-satisfied_holds_
I_-TruthEval_(xnot_(xnot_phi))_=_1
let U be non empty set ; ::_thesis: for I being Element of U -InterpretersOf S st X is I -satisfied holds
I -TruthEval (xnot (xnot phi)) = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; ::_thesis: ( X is I -satisfied implies I -TruthEval (xnot (xnot phi)) = 1 )
set v = I -TruthEval phi;
set phi1 = xnot phi;
set phi2 = xnot (xnot phi);
set v1 = I -TruthEval (xnot phi);
set v2 = I -TruthEval (xnot (xnot phi));
 ( ('not' (I -TruthEval phi)) \+\ (I -TruthEval (xnot phi)) = {} & ('not' (I -TruthEval (xnot phi))) \+\ (I -TruthEval (xnot (xnot phi))) = {} ) ;
then A1: ( I -TruthEval (xnot phi) = 'not' (I -TruthEval phi) & I -TruthEval (xnot (xnot phi)) = 'not' (I -TruthEval (xnot phi)) ) by FOMODEL0:29;
assume  X is I -satisfied ; ::_thesis: I -TruthEval (xnot (xnot phi)) = 1
hence  I -TruthEval (xnot (xnot phi)) = 1 by A1, FOMODEL2:def_45; ::_thesis: verum
end;
hence  for b1 being wff string of S st b1 = xnot (xnot phi) holds
b1 is X -implied by FOMODEL2:def_45; ::_thesis: verum
end;
end;

definition
let X be set ;
let S be Language;
let phi be wff string of S;
attrphi is X -provable  means :Def70: :: FOMODEL4:def 70
phi is X,{(R#9 S)} \/ (S -rules) -provable ;
end;

:: deftheorem Def70 defines -provable FOMODEL4:def_70_:_
 for X being set
for S being Language
for phi being wff string of S holds
( phi is X -provable iff phi is X,{(R#9 S)} \/ (S -rules) -provable );

begin

definition
let X be functional set ;
let S be Language;
let D be RuleSet of S;
let num be Function of NAT,(ExFormulasOf S);
set SS = AllSymbolsOf S;
set EF = ExFormulasOf S;
set FF = AllFormulasOf S;
set Y = X \/ (AllFormulasOf S);
set DD = bool (X \/ (AllFormulasOf S));
func(D,num) AddWitnessesTo X ->  Function of NAT,(bool (X \/ (AllFormulasOf S))) means :Def71: :: FOMODEL4:def 71
( it . 0 = X & ( for mm being Element of NAT holds it . (mm + 1) = (D,(num . mm)) AddAsWitnessTo (it . mm) ) );
existence
 ex b1 being Function of NAT,(bool (X \/ (AllFormulasOf S))) st
( b1 . 0 = X & ( for mm being Element of NAT holds b1 . (mm + 1) = (D,(num . mm)) AddAsWitnessTo (b1 . mm) ) )
proof
reconsider Z = X null (AllFormulasOf S) as Element of bool (X \/ (AllFormulasOf S)) ;
deffunc H1( Nat, Element of bool (X \/ (AllFormulasOf S))) ->  Element of bool (X \/ (AllFormulasOf S)) = (X \/ (AllFormulasOf S)) typed/\ ((D,(num . $1)) AddAsWitnessTo $2);
consider f being Function of NAT,(bool (X \/ (AllFormulasOf S))) such that
A1: ( f . 0 = Z & ( for n being Nat holds f . (n + 1) = H1(n,f . n) ) ) from NAT_1:sch_12();
take  f ; ::_thesis: ( f . 0 = X & ( for mm being Element of NAT holds f . (mm + 1) = (D,(num . mm)) AddAsWitnessTo (f . mm) ) )
now__::_thesis:_for_n_being_Nat_holds_f_._(n_+_1)_=_(D,(num_._n))_AddAsWitnessTo_(f_._n)
let n be Nat; ::_thesis: f . (n + 1) = (D,(num . n)) AddAsWitnessTo (f . n)
reconsider nn = n as Element of NAT by ORDINAL1:def_12;
A2: ( (D,(num . nn)) AddAsWitnessTo (f . nn) c= (AllFormulasOf S) \/ (f . nn) & (AllFormulasOf S) \/ (f . nn) c= (AllFormulasOf S) \/ (X \/ (AllFormulasOf S)) ) by XBOOLE_1:9;
(AllFormulasOf S) \/ (X \/ (AllFormulasOf S)) = ((AllFormulasOf S) \/ (AllFormulasOf S)) \/ X by XBOOLE_1:4
.= X \/ (AllFormulasOf S) ;
then reconsider A = (D,(num . nn)) AddAsWitnessTo (f . nn) as Subset of (X \/ (AllFormulasOf S)) by A2, XBOOLE_1:1;
 f . (n + 1) = A null (X \/ (AllFormulasOf S)) by A1;
hence  f . (n + 1) = (D,(num . n)) AddAsWitnessTo (f . n) ; ::_thesis: verum
end;
hence  ( f . 0 = X & ( for mm being Element of NAT holds f . (mm + 1) = (D,(num . mm)) AddAsWitnessTo (f . mm) ) ) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of NAT,(bool (X \/ (AllFormulasOf S))) st b1 . 0 = X & ( for mm being Element of NAT holds b1 . (mm + 1) = (D,(num . mm)) AddAsWitnessTo (b1 . mm) ) & b2 . 0 = X & ( for mm being Element of NAT holds b2 . (mm + 1) = (D,(num . mm)) AddAsWitnessTo (b2 . mm) ) holds
b1 = b2
proof
deffunc H1( Nat, Element of bool (X \/ (AllFormulasOf S))) ->  Subset of ($2 \/ (AllFormulasOf S)) = (D,(num . $1)) AddAsWitnessTo $2;
let IT1, IT2 be Function of NAT,(bool (X \/ (AllFormulasOf S))); ::_thesis: ( IT1 . 0 = X & ( for mm being Element of NAT holds IT1 . (mm + 1) = (D,(num . mm)) AddAsWitnessTo (IT1 . mm) ) & IT2 . 0 = X & ( for mm being Element of NAT holds IT2 . (mm + 1) = (D,(num . mm)) AddAsWitnessTo (IT2 . mm) ) implies IT1 = IT2 )
assume that
A3: IT1 . 0 = X and
A4: for mm being Element of NAT holds IT1 . (mm + 1) = H1(mm,IT1 . mm) and
A5: IT2 . 0 = X and
A6: for mm being Element of NAT holds IT2 . (mm + 1) = H1(mm,IT2 . mm) ; ::_thesis: IT1 = IT2
A7: for m being Nat holds IT1 . (m + 1) = H1(m,IT1 . m)
proof
let m be Nat; ::_thesis: IT1 . (m + 1) = H1(m,IT1 . m)
reconsider mm = m as Element of NAT by ORDINAL1:def_12;
 IT1 . (mm + 1) = H1(mm,IT1 . mm) by A4;
hence  IT1 . (m + 1) = H1(m,IT1 . m) ; ::_thesis: verum
end;
A8: for m being Nat holds IT2 . (m + 1) = H1(m,IT2 . m)
proof
let m be Nat; ::_thesis: IT2 . (m + 1) = H1(m,IT2 . m)
reconsider mm = m as Element of NAT by ORDINAL1:def_12;
 IT2 . (mm + 1) = H1(mm,IT2 . mm) by A6;
hence  IT2 . (m + 1) = H1(m,IT2 . m) ; ::_thesis: verum
end;
A9: dom IT1 = NAT by FUNCT_2:def_1;
A10: dom IT2 = NAT by FUNCT_2:def_1;
thus  IT1 = IT2 from NAT_1:sch_15(A9, A3, A7, A10, A5, A8); ::_thesis: verum
end;
end;

:: deftheorem Def71 defines AddWitnessesTo FOMODEL4:def_71_:_
 for X being functional set
for S being Language
for D being RuleSet of S
for num being Function of NAT,(ExFormulasOf S)
for b5 being Function of NAT,(bool (X \/ (AllFormulasOf S))) holds
( b5 = (D,num) AddWitnessesTo X iff ( b5 . 0 = X & ( for mm being Element of NAT holds b5 . (mm + 1) = (D,(num . mm)) AddAsWitnessTo (b5 . mm) ) ) );

notation
let X be functional set ;
let S be Language;
let D be RuleSet of S;
let num be Function of NAT,(ExFormulasOf S);
synonym (D,num) addw X for (D,num) AddWitnessesTo X;
end;

Lm70:  for X being set
for S being Language
for phi being wff string of S
for l1, l2 being literal Element of S
for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & X \/ {((l1,l2) -SymbolSubstIn phi)} is D -inconsistent & l2 is X \/ {phi} -absent holds
X \/ {(<*l1*> ^ phi)} is D -inconsistent
proof
let X be set ; ::_thesis: for S being Language
for phi being wff string of S
for l1, l2 being literal Element of S
for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & X \/ {((l1,l2) -SymbolSubstIn phi)} is D -inconsistent & l2 is X \/ {phi} -absent holds
X \/ {(<*l1*> ^ phi)} is D -inconsistent
let S be Language; ::_thesis: for phi being wff string of S
for l1, l2 being literal Element of S
for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & X \/ {((l1,l2) -SymbolSubstIn phi)} is D -inconsistent & l2 is X \/ {phi} -absent holds
X \/ {(<*l1*> ^ phi)} is D -inconsistent
let phi be wff string of S; ::_thesis: for l1, l2 being literal Element of S
for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & X \/ {((l1,l2) -SymbolSubstIn phi)} is D -inconsistent & l2 is X \/ {phi} -absent holds
X \/ {(<*l1*> ^ phi)} is D -inconsistent
let l1, l2 be literal Element of S; ::_thesis: for D being RuleSet of S st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & X \/ {((l1,l2) -SymbolSubstIn phi)} is D -inconsistent & l2 is X \/ {phi} -absent holds
X \/ {(<*l1*> ^ phi)} is D -inconsistent
let D be RuleSet of S; ::_thesis: ( D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & X \/ {((l1,l2) -SymbolSubstIn phi)} is D -inconsistent & l2 is X \/ {phi} -absent implies X \/ {(<*l1*> ^ phi)} is D -inconsistent )
set E = TheEqSymbOf S;
set L = LettersOf S;
set SS = AllSymbolsOf S;
set Q = S -sequents ;
set psi = (l1,l2) -SymbolSubstIn phi;
set E1 = R#1 S;
set E2 = R#2 S;
set E5 = R#5 S;
set ll = the Element of (rng phi) /\ (LettersOf S);
 the Element of (rng phi) /\ (LettersOf S) in LettersOf S by TARSKI:def_3;
then reconsider l = the Element of (rng phi) /\ (LettersOf S) as literal Element of S ;
reconsider t = <*l*> as termal string of S ;
set N = TheNorSymbOf S;
reconsider yes = (<*(TheEqSymbOf S)*> ^ t) ^ t as 0wff string of S ;
A1: rng yes = (rng (<*(TheEqSymbOf S)*> ^ t)) \/ (rng t) by FINSEQ_1:31
.= ((rng <*(TheEqSymbOf S)*>) \/ (rng t)) \/ (rng t) by FINSEQ_1:31
.= (rng <*(TheEqSymbOf S)*>) \/ ((rng t) \/ (rng t)) by XBOOLE_1:4
.= {(TheEqSymbOf S)} \/ (rng t) by FINSEQ_1:38
.= {(TheEqSymbOf S)} \/ {l} by FINSEQ_1:38 ;
reconsider lll = the Element of (rng phi) /\ (LettersOf S) as Element of rng phi by XBOOLE_0:def_4;
A2: {lll} \/ {(TheEqSymbOf S),(TheNorSymbOf S)} c= (rng phi) \/ {(TheEqSymbOf S),(TheNorSymbOf S)} by XBOOLE_1:9;
reconsider no = xnot yes as wff string of S ;
A3: rng no = (rng (<*(TheNorSymbOf S)*> ^ yes)) \/ (rng yes) by FINSEQ_1:31
.= ((rng <*(TheNorSymbOf S)*>) \/ (rng yes)) \/ (rng yes) by FINSEQ_1:31
.= (rng <*(TheNorSymbOf S)*>) \/ ((rng yes) \/ (rng yes)) by XBOOLE_1:4
.= {(TheNorSymbOf S)} \/ (rng yes) by FINSEQ_1:38
.= ({(TheEqSymbOf S)} \/ {(TheNorSymbOf S)}) \/ {l} by A1, XBOOLE_1:4
.= {(TheEqSymbOf S),(TheNorSymbOf S)} \/ {l} by ENUMSET1:1 ;
assume A4: ( D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & X \/ {((l1,l2) -SymbolSubstIn phi)} is D -inconsistent ) ; ::_thesis: ( not l2 is X \/ {phi} -absent or X \/ {(<*l1*> ^ phi)} is D -inconsistent )
then  no is X \/ {((l1,l2) -SymbolSubstIn phi)},D -provable by Th13;
then consider H3 being set , m being Nat such that
A5: ( H3 c= X \/ {((l1,l2) -SymbolSubstIn phi)} & [H3,no] is m, {} ,D -derivable ) by Def62;
reconsider seqt1 = [H3,no] as Element of S -sequents by Def2, A5;
 (seqt1 `1) \+\ H3 = {} ;
then reconsider H33 = H3 as S -premises-like Subset of (X \/ {((l1,l2) -SymbolSubstIn phi)}) by A5, FOMODEL0:29;
reconsider H1 = H33 /\ X as S -premises-like Subset of X ;
reconsider H2 = H33 /\ {((l1,l2) -SymbolSubstIn phi)} as S -premises-like Subset of {((l1,l2) -SymbolSubstIn phi)} ;
 ( {phi} null X c= X \/ {phi} & X null {phi} c= X \/ {phi} ) ;
then reconsider XX = X, Phi = {phi} as Subset of (X \/ {phi}) ;
reconsider H11 = H1 as S -premises-like Subset of XX ;
reconsider NO = {no}, Phii = {phi} as Subset of (((AllSymbolsOf S) *) \ {{}}) ;
assume A6: l2 is X \/ {phi} -absent ; ::_thesis: X \/ {(<*l1*> ^ phi)} is D -inconsistent
then A7: ( l2 is XX -absent & l2 is Phi -absent ) ;
then  not l2 in SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ Phii) by FOMODEL2:def_38;
then  ( not l2 in rng phi & not l2 in {(TheEqSymbOf S),(TheNorSymbOf S)} ) by FOMODEL0:45, TARSKI:def_2;
then  not l2 in rng no by A3, A2, XBOOLE_0:def_3;
then  not l2 in SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ NO) by FOMODEL0:45;
then reconsider ln = l2 as {no} -absent Element of S by FOMODEL2:def_38;
reconsider lN = ln as literal {phi} \/ {no} -absent Element of S by A7;
 lN is H11 -absent by A6;
then reconsider lx = lN as literal H11 \/ ({phi} \/ {no}) -absent Element of S ;
 H11 \/ ({phi} \/ {no}) = (H1 \/ {phi}) \/ {no} by XBOOLE_1:4;
then  lx is (H1 \/ {phi}) \/ {no} -absent ;
then reconsider l22 = l2 as literal (H1 \/ {phi}) \/ {no} -absent Element of S ;
reconsider F2 = {(R#2 S)}, F1 = {(R#1 S)}, F5 = {(R#5 S)} as Subset of D by A4, ZFMISC_1:31;
A8: ( D \/ (F1 \/ F5) = D & F2 c= D & {} c= X \/ {(<*l1*> ^ phi)} & H1 \/ {(<*l1*> ^ phi)} c= X \/ {(<*l1*> ^ phi)} ) by XBOOLE_1:2, XBOOLE_1:9;
A9: H33 null (X \/ {((l1,l2) -SymbolSubstIn phi)}) = H1 \/ H2 by XBOOLE_1:23;
A10: [((H1 \/ {(<*l1*> ^ phi)}) null l22),no] is 1,{[(H1 \/ {((l1,l2) -SymbolSubstIn phi)}),no]},{(R#5 S)} -derivable ;
 [((H1 \/ H2) \/ {((l1,l2) -SymbolSubstIn phi)}),no] is 1,{[(H1 \/ H2),no]},{(R#1 S)} -derivable ;
then  [(H1 \/ ({((l1,l2) -SymbolSubstIn phi)} null H2)),no] is 1,{[H33,no]},{(R#1 S)} -derivable by A9, XBOOLE_1:4;
then  [(H1 \/ {(<*l1*> ^ phi)}),no] is 1 + 1,{[H33,no]},{(R#1 S)} \/ {(R#5 S)} -derivable by A10, Lm21;
then  [(H1 \/ {(<*l1*> ^ phi)}),no] is m + 2, {} ,D -derivable by A8, A4, Lm21, A5;
then A11: no is X \/ {(<*l1*> ^ phi)},D -provable by A8, Def62;
set seqt2 = [{},yes];
 ( {[{},((<*(TheEqSymbOf S)*> ^ t) ^ t)]} is {(R#2 S)} -derivable & ([{},yes] `1) \+\ {} = {} & ([{},yes] `2) \+\ yes = {} ) ;
then  yes is {} ,{(R#2 S)} -provable by Def12;
then  yes is X \/ {(<*l1*> ^ phi)},D -provable by A8, Lm49;
hence  X \/ {(<*l1*> ^ phi)} is D -inconsistent by A11, Def65; ::_thesis: verum
end;

theorem Th17: :: FOMODEL4:17
for k, m being Nat
for S being Language
for D being RuleSet of S
for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent holds
( ((D,num) addw X) . k c= ((D,num) addw X) . (k + m) & (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . m) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . m is D -consistent )
proof
let k, m be Nat; ::_thesis: for S being Language
for D being RuleSet of S
for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent holds
( ((D,num) addw X) . k c= ((D,num) addw X) . (k + m) & (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . m) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . m is D -consistent )
let S be Language; ::_thesis: for D being RuleSet of S
for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent holds
( ((D,num) addw X) . k c= ((D,num) addw X) . (k + m) & (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . m) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . m is D -consistent )
let D be RuleSet of S; ::_thesis: for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent holds
( ((D,num) addw X) . k c= ((D,num) addw X) . (k + m) & (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . m) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . m is D -consistent )
let X be functional set ; ::_thesis: for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent holds
( ((D,num) addw X) . k c= ((D,num) addw X) . (k + m) & (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . m) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . m is D -consistent )
set L = LettersOf S;
set F = S -firstChar ;
set FF = AllFormulasOf S;
set SS = AllSymbolsOf S;
set strings = ((AllSymbolsOf S) *) \ {{}};
set EF = ExFormulasOf S;
let num be Function of NAT,(ExFormulasOf S); ::_thesis: ( D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent implies ( ((D,num) addw X) . k c= ((D,num) addw X) . (k + m) & (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . m) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . m is D -consistent ) )
set f = (D,num) addw X;
assume A1: ( D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D ) ; ::_thesis: ( not (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite or not X is D -consistent or ( ((D,num) addw X) . k c= ((D,num) addw X) . (k + m) & (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . m) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . m is D -consistent ) )
assume A2: ( (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent ) ; ::_thesis: ( ((D,num) addw X) . k c= ((D,num) addw X) . (k + m) & (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . m) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . m is D -consistent )
defpred S1[ Nat] means ( ((D,num) addw X) . k c= ((D,num) addw X) . (k + $1) & (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . $1) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . $1 is D -consistent );
A3: S1[ 0 ] by A2, Def71;
A4: for m being Nat st S1[m] holds
S1[m + 1]
proof
let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] )
reconsider mk = k + m, MM = m + 1, mm = m as Element of NAT by ORDINAL1:def_12;
reconsider phii = num . mm as Element of ExFormulasOf S ;
reconsider phi = num . mm as wff exal string of S by TARSKI:def_3;
reconsider phi1 = head phi as wff string of S ;
reconsider l1 = (S -firstChar) . phi as literal Element of S ;
A5: phi = (<*l1*> ^ phi1) ^ (tail phi) by FOMODEL2:23
.= <*l1*> ^ phi1 ;
reconsider fmk = (D,(num . mk)) AddAsWitnessTo (((D,num) addw X) . mk) as Subset of ((((D,num) addw X) . mk) \/ (AllFormulasOf S)) ;
reconsider fmm = (D,(num . mm)) AddAsWitnessTo (((D,num) addw X) . mm) as Subset of ((((D,num) addw X) . mm) \/ (AllFormulasOf S)) ;
 (((D,num) addw X) . mk) \ fmk = {} ;
then  ((D,num) addw X) . mk c= fmk by XBOOLE_1:37;
then A6: ( ((D,num) addw X) . mk c= ((D,num) addw X) . (mk + 1) & ((D,num) addw X) . MM = fmm ) by Def71;
assume A7: S1[m] ; ::_thesis: S1[m + 1]
hence  ((D,num) addw X) . k c= ((D,num) addw X) . (k + (m + 1)) by A6, XBOOLE_1:1; ::_thesis: ( (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . (m + 1)) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . (m + 1) is D -consistent )
 (((D,num) addw X) . mm) \ fmm = {} ;
then reconsider fm = ((D,num) addw X) . mm as functional Subset of fmm by XBOOLE_1:37;
reconsider sm = fm /\ (((AllSymbolsOf S) *) \ {{}}) as Subset of (fmm /\ (((AllSymbolsOf S) *) \ {{}})) by XBOOLE_1:26;
reconsider t = fmm \ (((D,num) addw X) . mm) as trivial set ;
reconsider i = (LettersOf S) \ (SymbolsOf sm) as infinite set by A7;
reconsider T = t /\ (((AllSymbolsOf S) *) \ {{}}) as functional finite FinSequence-membered set ;
 fmm = fm \/ t by XBOOLE_1:45;
then  SymbolsOf (fmm /\ (((AllSymbolsOf S) *) \ {{}})) = SymbolsOf (sm \/ T) by XBOOLE_1:23
.= (SymbolsOf sm) \/ (SymbolsOf T) by FOMODEL0:47 ;
then  (LettersOf S) \ (SymbolsOf (fmm /\ (((AllSymbolsOf S) *) \ {{}}))) = i \ (SymbolsOf T) by XBOOLE_1:41;
hence  (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . (m + 1)) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite by Def71; ::_thesis: ((D,num) addw X) . (m + 1) is D -consistent
reconsider LF = (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (fm \/ {(head phii)}))) as Subset of (LettersOf S) ;
percases ( ( fm \/ {phii} is D -consistent & LF <> {} ) or not fm \/ {phii} is D -consistent or not LF <> {} ) ;
supposeA8: ( fm \/ {phii} is D -consistent & LF <> {} ) ; ::_thesis: ((D,num) addw X) . (m + 1) is D -consistent
then reconsider LF = LF as non empty Subset of (LettersOf S) ;
set ll2 = the Element of LF;
reconsider l2 = the Element of LF as literal Element of S by TARSKI:def_3;
 not the Element of LF in SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (fm \/ {(head phii)})) by XBOOLE_0:def_5;
then  ( fm \/ {(<*l1*> ^ phi1)} is D -consistent & l2 is fm \/ {phi1} -absent ) by A8, A5, FOMODEL2:def_38;
then A9: fm \/ {((l1,l2) -SymbolSubstIn phi1)} is D -consistent by Lm70, A1;
thus  ((D,num) addw X) . (m + 1) is D -consistent by A8, Def67, A9, A6; ::_thesis: verum
end;
suppose ( not fm \/ {phii} is D -consistent or not LF <> {} ) ; ::_thesis: ((D,num) addw X) . (m + 1) is D -consistent
hence  ((D,num) addw X) . (m + 1) is D -consistent by A7, A6, Def67; ::_thesis: verum
end;
end;
end;
 for n being Nat holds S1[n] from NAT_1:sch_2(A3, A4);
hence  ( ((D,num) addw X) . k c= ((D,num) addw X) . (k + m) & (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . m) /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & ((D,num) addw X) . m is D -consistent ) ; ::_thesis: verum
end;

definition
let X be functional set ;
let S be Language;
let D be RuleSet of S;
let num be Function of NAT,(ExFormulasOf S);
funcX WithWitnessesFrom (D,num) ->  Subset of (X \/ (AllFormulasOf S)) equals :: FOMODEL4:def 72
 union (rng ((D,num) AddWitnessesTo X));
coherence
 union (rng ((D,num) AddWitnessesTo X)) is Subset of (X \/ (AllFormulasOf S))
proof
set FF = AllFormulasOf S;
set Y = X \/ (AllFormulasOf S);
set f = (D,num) AddWitnessesTo X;
reconsider F = rng ((D,num) AddWitnessesTo X) as Subset of (bool (X \/ (AllFormulasOf S))) by RELAT_1:def_19;
 (union F) \ (X \/ (AllFormulasOf S)) = {} ;
hence  union (rng ((D,num) AddWitnessesTo X)) is Subset of (X \/ (AllFormulasOf S)) ; ::_thesis: verum
end;
end;

:: deftheorem  defines WithWitnessesFrom FOMODEL4:def_72_:_
 for X being functional set
for S being Language
for D being RuleSet of S
for num being Function of NAT,(ExFormulasOf S) holds X WithWitnessesFrom (D,num) = union (rng ((D,num) AddWitnessesTo X));

notation
let X be functional set ;
let S be Language;
let D be RuleSet of S;
let num be Function of NAT,(ExFormulasOf S);
synonym X addw (D,num) for X WithWitnessesFrom (D,num);
end;

Lm71:  for S being Language
for D being RuleSet of S
for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#2 S in D & R#5 S in D & R#8 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent holds
X addw (D,num) is D -consistent
proof
let S be Language; ::_thesis: for D being RuleSet of S
for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#2 S in D & R#5 S in D & R#8 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent holds
X addw (D,num) is D -consistent
let D be RuleSet of S; ::_thesis: for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#2 S in D & R#5 S in D & R#8 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent holds
X addw (D,num) is D -consistent
let X be functional set ; ::_thesis: for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#2 S in D & R#5 S in D & R#8 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent holds
X addw (D,num) is D -consistent
set EF = ExFormulasOf S;
set L = LettersOf S;
set G1 = R#1 S;
let num be Function of NAT,(ExFormulasOf S); ::_thesis: ( D is isotone & R#1 S in D & R#2 S in D & R#5 S in D & R#8 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent implies X addw (D,num) is D -consistent )
set XX = X addw (D,num);
set f = (D,num) addw X;
set G2 = R#2 S;
set G5 = R#5 S;
set G8 = R#8 S;
set SS = AllSymbolsOf S;
set strings = ((AllSymbolsOf S) *) \ {{}};
assume A1: ( D is isotone & R#1 S in D & R#2 S in D & R#5 S in D & R#8 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D -consistent ) ; ::_thesis: X addw (D,num) is D -consistent
A2: for nn, mm being Element of NAT st mm in dom ((D,num) addw X) & nn in dom ((D,num) addw X) & nn < mm holds
((D,num) addw X) . nn c= ((D,num) addw X) . mm
proof
let nn, mm be Element of NAT ; ::_thesis: ( mm in dom ((D,num) addw X) & nn in dom ((D,num) addw X) & nn < mm implies ((D,num) addw X) . nn c= ((D,num) addw X) . mm )
set m = mm;
set n = nn;
assume  ( mm in dom ((D,num) addw X) & nn in dom ((D,num) addw X) & nn < mm ) ; ::_thesis: ((D,num) addw X) . nn c= ((D,num) addw X) . mm
then  nn - nn <= mm - nn by XREAL_1:9;
then  0 <= mm - nn ;
then reconsider k = mm - nn as Nat ;
 ((D,num) addw X) . nn c= ((D,num) addw X) . (nn + k) by Th17, A1;
hence  ((D,num) addw X) . nn c= ((D,num) addw X) . mm ; ::_thesis: verum
end;
now__::_thesis:_for_Y_being_finite_Subset_of_(X_addw_(D,num))_holds_Y_is_D_-consistent
let Y be finite Subset of (X addw (D,num)); ::_thesis: Y is D -consistent
consider kk being Element of NAT  such that
A3: Y c= ((D,num) addw X) . kk by A2, HENMODEL:3;
 ((D,num) addw X) . kk is D -consistent by A1, Th17;
hence  Y is D -consistent by A3, Th15; ::_thesis: verum
end;
hence  X addw (D,num) is D -consistent by Lm53; ::_thesis: verum
end;

registration
let X be functional set ;
let S be Language;
let D be RuleSet of S;
let num be Function of NAT,(ExFormulasOf S);
clusterX \ (X addw (D,num)) ->  empty for set ;
coherence
 for b1 being set st b1 = X \ (X addw (D,num)) holds
b1 is empty
proof
set XX = X addw (D,num);
set f = (D,num) addw X;
set Y = rng ((D,num) addw X);
reconsider ff = (D,num) addw X as Function of NAT,(rng ((D,num) addw X)) by FUNCT_2:6;
 ff . 0 = X by Def71;
then  X c= X addw (D,num) by ZFMISC_1:74;
hence  for b1 being set st b1 = X \ (X addw (D,num)) holds
b1 is empty ; ::_thesis: verum
end;
end;

theorem Th18: :: FOMODEL4:18
for Z being set
for S being Language
for D being RuleSet of S
for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X addw (D,num) c= Z & Z is D -consistent & rng num = ExFormulasOf S holds
Z is S -witnessed
proof
let Z be set ; ::_thesis: for S being Language
for D being RuleSet of S
for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X addw (D,num) c= Z & Z is D -consistent & rng num = ExFormulasOf S holds
Z is S -witnessed
let S be Language; ::_thesis: for D being RuleSet of S
for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X addw (D,num) c= Z & Z is D -consistent & rng num = ExFormulasOf S holds
Z is S -witnessed
let D be RuleSet of S; ::_thesis: for X being functional set
for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X addw (D,num) c= Z & Z is D -consistent & rng num = ExFormulasOf S holds
Z is S -witnessed
let X be functional set ; ::_thesis: for num being Function of NAT,(ExFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X addw (D,num) c= Z & Z is D -consistent & rng num = ExFormulasOf S holds
Z is S -witnessed
set L = LettersOf S;
set F = S -firstChar ;
set EF = ExFormulasOf S;
let num be Function of NAT,(ExFormulasOf S); ::_thesis: ( D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X addw (D,num) c= Z & Z is D -consistent & rng num = ExFormulasOf S implies Z is S -witnessed )
set f = (D,num) addw X;
set Y = X addw (D,num);
set SS = AllSymbolsOf S;
 X \ (X addw (D,num)) = {} ;
then A1: X c= X addw (D,num) by XBOOLE_1:37;
assume A2: ( D is isotone & R#1 S in D & R#8 S in D & R#2 S in D & R#5 S in D & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite ) ; ::_thesis: ( not X addw (D,num) c= Z or not Z is D -consistent or not rng num = ExFormulasOf S or Z is S -witnessed )
assume A3: ( X addw (D,num) c= Z & Z is D -consistent ) ; ::_thesis: ( not rng num = ExFormulasOf S or Z is S -witnessed )
then  ( X c= Z & Z is D -consistent ) by A1, XBOOLE_1:1;
then A4: X is D -consistent by Th15;
assume A5: rng num = ExFormulasOf S ; ::_thesis: Z is S -witnessed
set strings = ((AllSymbolsOf S) *) \ {{}};
 for l1 being literal Element of S
for phi1 being wff string of S st <*l1*> ^ phi1 in Z holds
ex l2 being literal Element of S st
( (l1,l2) -SymbolSubstIn phi1 in Z & not l2 in rng phi1 )
proof
let l1 be literal Element of S; ::_thesis: for phi1 being wff string of S st <*l1*> ^ phi1 in Z holds
ex l2 being literal Element of S st
( (l1,l2) -SymbolSubstIn phi1 in Z & not l2 in rng phi1 )
let phi1 be wff string of S; ::_thesis: ( <*l1*> ^ phi1 in Z implies ex l2 being literal Element of S st
( (l1,l2) -SymbolSubstIn phi1 in Z & not l2 in rng phi1 ) )
set phi = <*l1*> ^ phi1;
 ( <*l1*> ^ phi1 = (<*l1*> ^ phi1) ^ {} & not <*l1*> ^ phi1 is 0wff ) ;
then A6: ( l1 = (S -firstChar) . (<*l1*> ^ phi1) & phi1 = head (<*l1*> ^ phi1) ) by FOMODEL2:23;
 <*l1*> ^ phi1 in ExFormulasOf S ;
then reconsider phii = <*l1*> ^ phi1 as Element of ExFormulasOf S ;
consider x being set  such that
A7: ( x in dom num & num . x = phii ) by A5, FUNCT_1:def_3;
reconsider mm = x as Element of NAT by A7;
reconsider MM = mm + 1 as Element of NAT by ORDINAL1:def_12;
reconsider Xm = ((D,num) addw X) . mm as functional set ;
set no = SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ ((((D,num) addw X) . mm) \/ {phi1}));
reconsider T = (((AllSymbolsOf S) *) \ {{}}) /\ {phi1} as finite FinSequence-membered Subset of {phi1} ;
reconsider t = SymbolsOf T as finite set ;
reconsider i = (LettersOf S) \ (SymbolsOf ((((D,num) addw X) . mm) /\ (((AllSymbolsOf S) *) \ {{}}))) as infinite Subset of (LettersOf S) by Th17, A2, A4;
A8: SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ ((((D,num) addw X) . mm) \/ {phi1})) = SymbolsOf (((((AllSymbolsOf S) *) \ {{}}) /\ (((D,num) addw X) . mm)) \/ ((((AllSymbolsOf S) *) \ {{}}) /\ {phi1})) by XBOOLE_1:23
.= (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ (((D,num) addw X) . mm))) \/ (SymbolsOf T) by FOMODEL0:47 ;
then  (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ ((((D,num) addw X) . mm) \/ {phi1}))) = i \ t by XBOOLE_1:41;
then reconsider yes = (LettersOf S) \ (SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ ((((D,num) addw X) . mm) \/ {phi1}))) as non empty Subset of (LettersOf S) ;
set ll2 = the Element of yes;
reconsider l2 = the Element of yes as literal Element of S by TARSKI:def_3;
set psi1 = (l1,l2) -SymbolSubstIn phi1;
 dom ((D,num) addw X) = NAT by FUNCT_2:def_1;
then A9: ( ((D,num) addw X) . mm in rng ((D,num) addw X) & ((D,num) addw X) . MM in rng ((D,num) addw X) ) by FUNCT_1:def_3;
then  ((D,num) addw X) . mm c= X addw (D,num) by ZFMISC_1:74;
then A10: ((D,num) addw X) . mm c= Z by A3, XBOOLE_1:1;
assume  <*l1*> ^ phi1 in Z ; ::_thesis: ex l2 being literal Element of S st
( (l1,l2) -SymbolSubstIn phi1 in Z & not l2 in rng phi1 )
then  {(<*l1*> ^ phi1)} c= Z by ZFMISC_1:31;
then  (((D,num) addw X) . mm) \/ {(<*l1*> ^ phi1)} c= Z by A10, XBOOLE_1:8;
then  (((D,num) addw X) . mm) \/ {(<*l1*> ^ phi1)} is D -consistent by A3, Th15;
then (((D,num) addw X) . mm) \/ {((l1,l2) -SymbolSubstIn phi1)} = (D,phii) AddAsWitnessTo (((D,num) addw X) . mm) by Def67, A6
.= ((D,num) addw X) . (mm + 1) by Def71, A7 ;
then  {((l1,l2) -SymbolSubstIn phi1)} null (((D,num) addw X) . mm) c= ((D,num) addw X) . MM ;
then  (l1,l2) -SymbolSubstIn phi1 in ((D,num) addw X) . MM by ZFMISC_1:31;
then A11: (l1,l2) -SymbolSubstIn phi1 in X addw (D,num) by A9, TARSKI:def_4;
take  l2 ; ::_thesis: ( (l1,l2) -SymbolSubstIn phi1 in Z & not l2 in rng phi1 )
thus  (l1,l2) -SymbolSubstIn phi1 in Z by A3, A11; ::_thesis: not l2 in rng phi1
 not l2 in SymbolsOf ((((AllSymbolsOf S) *) \ {{}}) /\ ((((D,num) addw X) . mm) \/ {phi1})) by XBOOLE_0:def_5;
then  not l2 in SymbolsOf {phi1} by A8, XBOOLE_0:def_3;
hence  not l2 in rng phi1 by FOMODEL0:45; ::_thesis: verum
end;
hence  Z is S -witnessed by Def66; ::_thesis: verum
end;

begin

definition
let X be set ;
let S be Language;
let D be RuleSet of S;
let phi be Element of AllFormulasOf S;
func(D,phi) AddFormulaTo X ->  set  equals :Def73: :: FOMODEL4:def 73
X \/ {phi} if  not xnot phi is X,D -provable
otherwise X \/ {(xnot phi)};
consistency
 for b1 being set holds verum ;
coherence
 ( ( not xnot phi is X,D -provable implies X \/ {phi} is set ) & ( xnot phi is X,D -provable implies X \/ {(xnot phi)} is set ) ) ;
end;

:: deftheorem Def73 defines AddFormulaTo FOMODEL4:def_73_:_
 for X being set
for S being Language
for D being RuleSet of S
for phi being Element of AllFormulasOf S holds
( ( not xnot phi is X,D -provable implies (D,phi) AddFormulaTo X = X \/ {phi} ) & ( xnot phi is X,D -provable implies (D,phi) AddFormulaTo X = X \/ {(xnot phi)} ) );

definition
let X be set ;
let S be Language;
let D be RuleSet of S;
let phi be Element of AllFormulasOf S;
:: original: AddFormulaTo
redefine func(D,phi) AddFormulaTo X ->  Subset of (X \/ (AllFormulasOf S));
coherence
 (D,phi) AddFormulaTo X is Subset of (X \/ (AllFormulasOf S))
proof
set F = S -firstChar ;
set IT = (D,phi) AddFormulaTo X;
set FF = AllFormulasOf S;
reconsider Y = X \/ (AllFormulasOf S) as non empty set ;
reconsider XX = X null (AllFormulasOf S) as Subset of Y ;
reconsider FFF = (AllFormulasOf S) null X as non empty Subset of Y ;
 ( xnot phi is Element of FFF & phi is Element of FFF ) by FOMODEL2:16;
then reconsider phii = phi, psii = xnot phi as Element of Y ;
reconsider Phi = {phii}, Psi = {psii} as Subset of Y ;
defpred S1[] means  xnot phi is X,D -provable ;
 ( ( not S1[] implies (D,phi) AddFormulaTo X = XX \/ Phi ) & ( S1[] implies (D,phi) AddFormulaTo X = XX \/ Psi ) ) by Def73;
hence  (D,phi) AddFormulaTo X is Subset of (X \/ (AllFormulasOf S)) ; ::_thesis: verum
end;
end;

registration
let X be set ;
let S be Language;
let D be RuleSet of S;
let phi be Element of AllFormulasOf S;
clusterX \ ((D,phi) AddFormulaTo X) ->  empty ;
coherence
 X \ ((D,phi) AddFormulaTo X) is empty
proof
set Y = (D,phi) AddFormulaTo X;
set psi = xnot phi;
defpred S1[] means  xnot phi is X,D -provable ;
 ( ( not S1[] implies (D,phi) AddFormulaTo X = X \/ {phi} ) & ( S1[] implies (D,phi) AddFormulaTo X = X \/ {(xnot phi)} ) ) by Def73;
then  ( X null {phi} c= (D,phi) AddFormulaTo X or X null {(xnot phi)} c= (D,phi) AddFormulaTo X ) ;
hence  X \ ((D,phi) AddFormulaTo X) is empty ; ::_thesis: verum
end;
end;

definition
let X be set ;
let S be Language;
let D be RuleSet of S;
let num be Function of NAT,(AllFormulasOf S);
set SS = AllSymbolsOf S;
set FF = AllFormulasOf S;
set Y = X \/ (AllFormulasOf S);
set DD = bool (X \/ (AllFormulasOf S));
func(D,num) AddFormulasTo X ->  Function of NAT,(bool (X \/ (AllFormulasOf S))) means :Def74: :: FOMODEL4:def 74
( it . 0 = X & ( for m being Nat holds it . (m + 1) = (D,(num . m)) AddFormulaTo (it . m) ) );
existence
 ex b1 being Function of NAT,(bool (X \/ (AllFormulasOf S))) st
( b1 . 0 = X & ( for m being Nat holds b1 . (m + 1) = (D,(num . m)) AddFormulaTo (b1 . m) ) )
proof
reconsider Z = X null (AllFormulasOf S) as Element of bool (X \/ (AllFormulasOf S)) ;
deffunc H1( Nat, Element of bool (X \/ (AllFormulasOf S))) ->  Element of bool (X \/ (AllFormulasOf S)) = (X \/ (AllFormulasOf S)) typed/\ ((D,(num . $1)) AddFormulaTo $2);
consider f being Function of NAT,(bool (X \/ (AllFormulasOf S))) such that
A1: ( f . 0 = Z & ( for n being Nat holds f . (n + 1) = H1(n,f . n) ) ) from NAT_1:sch_12();
take  f ; ::_thesis: ( f . 0 = X & ( for m being Nat holds f . (m + 1) = (D,(num . m)) AddFormulaTo (f . m) ) )
now__::_thesis:_for_n_being_Nat_holds_f_._(n_+_1)_=_(D,(num_._n))_AddFormulaTo_(f_._n)
let n be Nat; ::_thesis: f . (n + 1) = (D,(num . n)) AddFormulaTo (f . n)
reconsider nn = n as Element of NAT by ORDINAL1:def_12;
A2: ( (D,(num . nn)) AddFormulaTo (f . nn) c= (AllFormulasOf S) \/ (f . nn) & (AllFormulasOf S) \/ (f . nn) c= (AllFormulasOf S) \/ (X \/ (AllFormulasOf S)) ) by XBOOLE_1:9;
(AllFormulasOf S) \/ (X \/ (AllFormulasOf S)) = ((AllFormulasOf S) \/ (AllFormulasOf S)) \/ X by XBOOLE_1:4
.= X \/ (AllFormulasOf S) ;
then reconsider A = (D,(num . nn)) AddFormulaTo (f . nn) as Subset of (X \/ (AllFormulasOf S)) by A2, XBOOLE_1:1;
 f . (n + 1) = A null (X \/ (AllFormulasOf S)) by A1;
hence  f . (n + 1) = (D,(num . n)) AddFormulaTo (f . n) ; ::_thesis: verum
end;
hence  ( f . 0 = X & ( for m being Nat holds f . (m + 1) = (D,(num . m)) AddFormulaTo (f . m) ) ) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of NAT,(bool (X \/ (AllFormulasOf S))) st b1 . 0 = X & ( for m being Nat holds b1 . (m + 1) = (D,(num . m)) AddFormulaTo (b1 . m) ) & b2 . 0 = X & ( for m being Nat holds b2 . (m + 1) = (D,(num . m)) AddFormulaTo (b2 . m) ) holds
b1 = b2
proof
deffunc H1( Nat, Element of bool (X \/ (AllFormulasOf S))) ->  Subset of ($2 \/ (AllFormulasOf S)) = (D,(num . $1)) AddFormulaTo $2;
let IT1, IT2 be Function of NAT,(bool (X \/ (AllFormulasOf S))); ::_thesis: ( IT1 . 0 = X & ( for m being Nat holds IT1 . (m + 1) = (D,(num . m)) AddFormulaTo (IT1 . m) ) & IT2 . 0 = X & ( for m being Nat holds IT2 . (m + 1) = (D,(num . m)) AddFormulaTo (IT2 . m) ) implies IT1 = IT2 )
assume that
A3: IT1 . 0 = X and
A4: for m being Nat holds IT1 . (m + 1) = H1(m,IT1 . m) and
A5: IT2 . 0 = X and
A6: for m being Nat holds IT2 . (m + 1) = H1(m,IT2 . m) ; ::_thesis: IT1 = IT2
A7: for m being Nat holds IT1 . (m + 1) = H1(m,IT1 . m) by A4;
A8: for m being Nat holds IT2 . (m + 1) = H1(m,IT2 . m) by A6;
A9: dom IT1 = NAT by FUNCT_2:def_1;
A10: dom IT2 = NAT by FUNCT_2:def_1;
thus  IT1 = IT2 from NAT_1:sch_15(A9, A3, A7, A10, A5, A8); ::_thesis: verum
end;
end;

:: deftheorem Def74 defines AddFormulasTo FOMODEL4:def_74_:_
 for X being set
for S being Language
for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S)
for b5 being Function of NAT,(bool (X \/ (AllFormulasOf S))) holds
( b5 = (D,num) AddFormulasTo X iff ( b5 . 0 = X & ( for m being Nat holds b5 . (m + 1) = (D,(num . m)) AddFormulaTo (b5 . m) ) ) );

Lm72:  for k, m being Nat
for X being set
for S being Language
for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & X is D -consistent holds
( ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + m) & ((D,num) AddFormulasTo X) . m is D -consistent )
proof
let k, m be Nat; ::_thesis: for X being set
for S being Language
for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & X is D -consistent holds
( ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + m) & ((D,num) AddFormulasTo X) . m is D -consistent )
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & X is D -consistent holds
( ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + m) & ((D,num) AddFormulasTo X) . m is D -consistent )
let S be Language; ::_thesis: for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & X is D -consistent holds
( ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + m) & ((D,num) AddFormulasTo X) . m is D -consistent )
let D be RuleSet of S; ::_thesis: for num being Function of NAT,(AllFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & X is D -consistent holds
( ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + m) & ((D,num) AddFormulasTo X) . m is D -consistent )
set FF = AllFormulasOf S;
let num be Function of NAT,(AllFormulasOf S); ::_thesis: ( D is isotone & R#1 S in D & R#8 S in D & X is D -consistent implies ( ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + m) & ((D,num) AddFormulasTo X) . m is D -consistent ) )
set f = (D,num) AddFormulasTo X;
assume A1: ( D is isotone & R#1 S in D & R#8 S in D ) ; ::_thesis: ( not X is D -consistent or ( ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + m) & ((D,num) AddFormulasTo X) . m is D -consistent ) )
assume A2: X is D -consistent ; ::_thesis: ( ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + m) & ((D,num) AddFormulasTo X) . m is D -consistent )
defpred S1[ Nat] means ( ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + $1) & ((D,num) AddFormulasTo X) . $1 is D -consistent );
A3: S1[ 0 ] by A2, Def74;
A4: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
set fkn1 = (D,(num . (k + n))) AddFormulaTo (((D,num) AddFormulasTo X) . (k + n));
set fkn = ((D,num) AddFormulasTo X) . (k + n);
assume A5: S1[n] ; ::_thesis: S1[n + 1]
A6: (((D,num) AddFormulasTo X) . (k + n)) \ ((D,(num . (k + n))) AddFormulaTo (((D,num) AddFormulasTo X) . (k + n))) = {} ;
((D,num) AddFormulasTo X) . (k + (n + 1)) = ((D,num) AddFormulasTo X) . ((k + n) + 1)
.= (D,(num . (k + n))) AddFormulaTo (((D,num) AddFormulasTo X) . (k + n)) by Def74 ;
then  ((D,num) AddFormulasTo X) . (k + n) c= ((D,num) AddFormulasTo X) . (k + (n + 1)) by A6, XBOOLE_1:37;
hence  ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + (n + 1)) by A5, XBOOLE_1:1; ::_thesis: ((D,num) AddFormulasTo X) . (n + 1) is D -consistent
reconsider phii = num . n as Element of AllFormulasOf S ;
reconsider phi = phii as wff string of S ;
reconsider psi = xnot phi as wff string of S ;
reconsider psii = psi as Element of AllFormulasOf S by FOMODEL2:16;
set fn = ((D,num) AddFormulasTo X) . n;
set fN = (D,(num . n)) AddFormulaTo (((D,num) AddFormulasTo X) . n);
defpred S2[] means  xnot phii is ((D,num) AddFormulasTo X) . n,D -provable ;
A7: ((D,num) AddFormulasTo X) . (n + 1) = (D,(num . n)) AddFormulaTo (((D,num) AddFormulasTo X) . n) by Def74;
percases ( not S2[] or S2[] ) ;
supposeA8: not S2[] ; ::_thesis: ((D,num) AddFormulasTo X) . (n + 1) is D -consistent
then  (((D,num) AddFormulasTo X) . n) \/ {phii} is D -consistent by A1, Lm52;
hence  ((D,num) AddFormulasTo X) . (n + 1) is D -consistent by A7, A8, Def73; ::_thesis: verum
end;
supposeA9: S2[] ; ::_thesis: ((D,num) AddFormulasTo X) . (n + 1) is D -consistent
then  (((D,num) AddFormulasTo X) . n) \/ {(xnot phii)} is D -consistent by Lm55, A1, A5;
hence  ((D,num) AddFormulasTo X) . (n + 1) is D -consistent by A7, A9, Def73; ::_thesis: verum
end;
end;
end;
 for n being Nat holds S1[n] from NAT_1:sch_2(A3, A4);
hence  ( ((D,num) AddFormulasTo X) . k c= ((D,num) AddFormulasTo X) . (k + m) & ((D,num) AddFormulasTo X) . m is D -consistent ) ; ::_thesis: verum
end;

definition
let X be set ;
let S be Language;
let D be RuleSet of S;
let num be Function of NAT,(AllFormulasOf S);
func(D,num) CompletionOf X ->  Subset of (X \/ (AllFormulasOf S)) equals :: FOMODEL4:def 75
 union (rng ((D,num) AddFormulasTo X));
coherence
 union (rng ((D,num) AddFormulasTo X)) is Subset of (X \/ (AllFormulasOf S))
proof
set FF = AllFormulasOf S;
set Y = X \/ (AllFormulasOf S);
set f = (D,num) AddFormulasTo X;
reconsider F = rng ((D,num) AddFormulasTo X) as Subset of (bool (X \/ (AllFormulasOf S))) by RELAT_1:def_19;
 (union F) \ (X \/ (AllFormulasOf S)) = {} ;
hence  union (rng ((D,num) AddFormulasTo X)) is Subset of (X \/ (AllFormulasOf S)) ; ::_thesis: verum
end;
end;

:: deftheorem  defines CompletionOf FOMODEL4:def_75_:_
 for X being set
for S being Language
for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) holds (D,num) CompletionOf X = union (rng ((D,num) AddFormulasTo X));

registration
let X be set ;
let S be Language;
let D be RuleSet of S;
let num be Function of NAT,(AllFormulasOf S);
clusterX \ ((D,num) CompletionOf X) ->  empty for set ;
coherence
 for b1 being set st b1 = X \ ((D,num) CompletionOf X) holds
b1 is empty
proof
set f = (D,num) AddFormulasTo X;
set XX = (D,num) CompletionOf X;
reconsider ff = (D,num) AddFormulasTo X as Function of NAT,(rng ((D,num) AddFormulasTo X)) by FUNCT_2:6;
 ff . 0 in rng ((D,num) AddFormulasTo X) ;
then  ((D,num) AddFormulasTo X) . 0 c= (D,num) CompletionOf X by ZFMISC_1:74;
then  X c= (D,num) CompletionOf X by Def74;
hence  for b1 being set st b1 = X \ ((D,num) CompletionOf X) holds
b1 is empty ; ::_thesis: verum
end;
end;

Lm73:  for X being set
for S being Language
for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) st rng num = AllFormulasOf S holds
(D,num) CompletionOf X is S -covering
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) st rng num = AllFormulasOf S holds
(D,num) CompletionOf X is S -covering
let S be Language; ::_thesis: for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) st rng num = AllFormulasOf S holds
(D,num) CompletionOf X is S -covering
let D be RuleSet of S; ::_thesis: for num being Function of NAT,(AllFormulasOf S) st rng num = AllFormulasOf S holds
(D,num) CompletionOf X is S -covering
set FF = AllFormulasOf S;
let num be Function of NAT,(AllFormulasOf S); ::_thesis: ( rng num = AllFormulasOf S implies (D,num) CompletionOf X is S -covering )
set XX = (D,num) CompletionOf X;
set f = (D,num) AddFormulasTo X;
assume A1: rng num = AllFormulasOf S ; ::_thesis: (D,num) CompletionOf X is S -covering
reconsider ff = (D,num) AddFormulasTo X as Function of NAT,(rng ((D,num) AddFormulasTo X)) by FUNCT_2:6;
hereby  :: according to FOMODEL2:def_40 ::_thesis: verum
let phi be wff string of S; ::_thesis: ( phi in (D,num) CompletionOf X or xnot phi in (D,num) CompletionOf X )
reconsider phii = phi as Element of AllFormulasOf S by FOMODEL2:16;
consider x being set  such that
A2: ( x in dom num & num . x = phii ) by A1, FUNCT_1:def_3;
reconsider mm = x as Element of NAT by A2;
reconsider MM = mm + 1 as Element of NAT by ORDINAL1:def_12;
 ((D,num) AddFormulasTo X) . (mm + 1) = (D,(num . mm)) AddFormulaTo (((D,num) AddFormulasTo X) . mm) by Def74;
then  ( ((D,num) AddFormulasTo X) . (mm + 1) = (((D,num) AddFormulasTo X) . mm) \/ {phi} or ((D,num) AddFormulasTo X) . (mm + 1) = (((D,num) AddFormulasTo X) . mm) \/ {(xnot phi)} ) by A2, Def73;
then A3: ( {phi} null (((D,num) AddFormulasTo X) . mm) c= ((D,num) AddFormulasTo X) . MM or {(xnot phi)} null (((D,num) AddFormulasTo X) . mm) c= ((D,num) AddFormulasTo X) . MM ) ;
 ff . MM is Element of rng ((D,num) AddFormulasTo X) ;
then  ((D,num) AddFormulasTo X) . (mm + 1) c= (D,num) CompletionOf X by ZFMISC_1:74;
then  ( {phi} c= (D,num) CompletionOf X or {(xnot phi)} c= (D,num) CompletionOf X ) by A3, XBOOLE_1:1;
hence  ( phi in (D,num) CompletionOf X or xnot phi in (D,num) CompletionOf X ) by ZFMISC_1:31; ::_thesis: verum
end;
end;

Lm74:  for X being set
for S being Language
for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & X is D -consistent holds
(D,num) CompletionOf X is D -consistent
proof
let X be set ; ::_thesis: for S being Language
for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & X is D -consistent holds
(D,num) CompletionOf X is D -consistent
let S be Language; ::_thesis: for D being RuleSet of S
for num being Function of NAT,(AllFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & X is D -consistent holds
(D,num) CompletionOf X is D -consistent
let D be RuleSet of S; ::_thesis: for num being Function of NAT,(AllFormulasOf S) st D is isotone & R#1 S in D & R#8 S in D & X is D -consistent holds
(D,num) CompletionOf X is D -consistent
set EF = ExFormulasOf S;
set L = LettersOf S;
set FF = AllFormulasOf S;
let num be Function of NAT,(AllFormulasOf S); ::_thesis: ( D is isotone & R#1 S in D & R#8 S in D & X is D -consistent implies (D,num) CompletionOf X is D -consistent )
set XX = (D,num) CompletionOf X;
set G1 = R#1 S;
set G8 = R#8 S;
set SS = AllSymbolsOf S;
set strings = ((AllSymbolsOf S) *) \ {{}};
set f = (D,num) AddFormulasTo X;
assume A1: ( D is isotone & R#1 S in D & R#8 S in D & X is D -consistent ) ; ::_thesis: (D,num) CompletionOf X is D -consistent
A2: for nn, mm being Element of NAT st mm in dom ((D,num) AddFormulasTo X) & nn in dom ((D,num) AddFormulasTo X) & nn < mm holds
((D,num) AddFormulasTo X) . nn c= ((D,num) AddFormulasTo X) . mm
proof
let nn, mm be Element of NAT ; ::_thesis: ( mm in dom ((D,num) AddFormulasTo X) & nn in dom ((D,num) AddFormulasTo X) & nn < mm implies ((D,num) AddFormulasTo X) . nn c= ((D,num) AddFormulasTo X) . mm )
set m = mm;
set n = nn;
assume  ( mm in dom ((D,num) AddFormulasTo X) & nn in dom ((D,num) AddFormulasTo X) & nn < mm ) ; ::_thesis: ((D,num) AddFormulasTo X) . nn c= ((D,num) AddFormulasTo X) . mm
then  nn - nn <= mm - nn by XREAL_1:9;
then  0 <= mm - nn ;
then reconsider k = mm - nn as Nat ;
 ((D,num) AddFormulasTo X) . nn c= ((D,num) AddFormulasTo X) . (nn + k) by A1, Lm72;
hence  ((D,num) AddFormulasTo X) . nn c= ((D,num) AddFormulasTo X) . mm ; ::_thesis: verum
end;
now__::_thesis:_for_Y_being_finite_Subset_of_((D,num)_CompletionOf_X)_holds_Y_is_D_-consistent
let Y be finite Subset of ((D,num) CompletionOf X); ::_thesis: Y is D -consistent
consider kk being Element of NAT  such that
A3: Y c= ((D,num) AddFormulasTo X) . kk by A2, HENMODEL:3;
 ((D,num) AddFormulasTo X) . kk is D -consistent by A1, Lm72;
hence  Y is D -consistent by A3, Th15; ::_thesis: verum
end;
hence  (D,num) CompletionOf X is D -consistent by Lm53; ::_thesis: verum
end;

theorem :: FOMODEL4:19
for y, X being set
for S being Language
for R being Relation of (bool (S -sequents)),(S -sequents) holds
( y in (FuncRule R) . X iff ( y in S -sequents & [X,y] in R ) ) by Lm29;

 Lm75:  for S being Language
for D2 being 2 -ranked RuleSet of S
for X being functional set st AllFormulasOf S is countable & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D2 -consistent & D2 is isotone holds
ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied
proof
let S be Language; ::_thesis: for D2 being 2 -ranked RuleSet of S
for X being functional set st AllFormulasOf S is countable & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D2 -consistent & D2 is isotone holds
ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied
let D2 be 2 -ranked RuleSet of S; ::_thesis: for X being functional set st AllFormulasOf S is countable & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D2 -consistent & D2 is isotone holds
ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied
let X be functional set ; ::_thesis: ( AllFormulasOf S is countable & (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D2 -consistent & D2 is isotone implies ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied )
set EF = ExFormulasOf S;
set FF = AllFormulasOf S;
set G0 = R#0 S;
set G2 = R#2 S;
set G3a = R#3a S;
set G3b = R#3b S;
set G3d = R#3d S;
set G3e = R#3e S;
set G1 = R#1 S;
set G4 = R#4 S;
set G5 = R#5 S;
set G6 = R#6 S;
set G7 = R#7 S;
set G8 = R#8 S;
set L = LettersOf S;
set SS = AllSymbolsOf S;
set strings = ((AllSymbolsOf S) *) \ {{}};
set no = SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}));
set D = D2;
reconsider D1 = D2 as 0 -ranked 1 -ranked RuleSet of S ;
assume  AllFormulasOf S is countable ; ::_thesis: ( not (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite or not X is D2 -consistent or not D2 is isotone or ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied )
then reconsider FFC = AllFormulasOf S as countable set ;
consider numaa being Function of NAT,(((AllSymbolsOf S) *) \ {{}}) such that
A1: FFC = rng numaa by SUPINF_2:def_8;
reconsider numa = numaa as Function of NAT,(AllFormulasOf S) by A1, FUNCT_2:6;
 (ExFormulasOf S) \ FFC = {} ;
then reconsider EFC = ExFormulasOf S as Subset of FFC by XBOOLE_1:37;
consider numee being Function of NAT,FFC such that
A2: EFC = rng numee by SUPINF_2:def_8;
reconsider nume = numee as Function of NAT,(ExFormulasOf S) by A2, FUNCT_2:6;
assume A3: ( (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite & X is D2 -consistent ) ; ::_thesis: ( not D2 is isotone or ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied )
A4: ( R#0 S in D1 & R#1 S in D2 & R#2 S in D1 & R#4 S in D2 & R#5 S in D2 & R#6 S in D2 & R#7 S in D2 & R#8 S in D2 ) by Def59;
set X1 = X addw (D1,nume);
set X2 = (D1,numa) CompletionOf (X addw (D1,nume));
A5: (D1,numa) CompletionOf (X addw (D1,nume)) is S -covering by Lm73, A1;
assume A6: D2 is isotone ; ::_thesis: ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied
then  X addw (D1,nume) is D2 -consistent by Lm71, A3, A4;
then A7: (D1,numa) CompletionOf (X addw (D1,nume)) is D2 -consistent by Lm74, A6, A4;
then A8: ( (D1,numa) CompletionOf (X addw (D1,nume)) is S -mincover & (D1,numa) CompletionOf (X addw (D1,nume)) is D2 -expanded ) by A4, Lm54, A5;
reconsider X22 = (D1,numa) CompletionOf (X addw (D1,nume)) as D1 -expanded set by A4, Lm54, A5, A7;
take U = Class ((X22,D1) -termEq); ::_thesis: ex I being Element of U -InterpretersOf S st X is I -satisfied
reconsider I = D1 Henkin X22 as Element of U -InterpretersOf S ;
take  I ; ::_thesis: X is I -satisfied
 ( X \ (X addw (D1,nume)) = {} & (X addw (D1,nume)) \ ((D1,numa) CompletionOf (X addw (D1,nume))) = {} ) ;
then A9: ( X c= X addw (D1,nume) & X addw (D1,nume) c= X22 ) by XBOOLE_1:37;
then A10: X22 is S -witnessed by Th18, A3, A4, A6, A7, A2;
hereby  :: according to FOMODEL2:def_42 ::_thesis: verum
let phi be wff string of S; ::_thesis: ( phi in X implies I -TruthEval phi = 1 )
assume  phi in X ; ::_thesis: I -TruthEval phi = 1
then  phi in X22 by A9, TARSKI:def_3;
hence  I -TruthEval phi = 1 by A10, Th14, A8, A4; ::_thesis: verum
end;
end;

Lm76:  for S being Language
for D2 being 2 -ranked RuleSet of S
for X being functional set st X is finite & AllFormulasOf S is countable & X is D2 -consistent & D2 is isotone holds
ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied
proof
let S be Language; ::_thesis: for D2 being 2 -ranked RuleSet of S
for X being functional set st X is finite & AllFormulasOf S is countable & X is D2 -consistent & D2 is isotone holds
ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied
let D2 be 2 -ranked RuleSet of S; ::_thesis: for X being functional set st X is finite & AllFormulasOf S is countable & X is D2 -consistent & D2 is isotone holds
ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied
let X be functional set ; ::_thesis: ( X is finite & AllFormulasOf S is countable & X is D2 -consistent & D2 is isotone implies ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied )
set SS = AllSymbolsOf S;
set L = LettersOf S;
set strings = ((AllSymbolsOf S) *) \ {{}};
set FF = AllFormulasOf S;
set no = SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}));
assume  X is finite ; ::_thesis: ( not AllFormulasOf S is countable or not X is D2 -consistent or not D2 is isotone or ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied )
then reconsider XS = X /\ (((AllSymbolsOf S) *) \ {{}}) as finite FinSequence-membered set ;
 SymbolsOf XS is finite ;
then A1: (LettersOf S) \ (SymbolsOf (X /\ (((AllSymbolsOf S) *) \ {{}}))) is infinite ;
assume  ( AllFormulasOf S is countable & X is D2 -consistent & D2 is isotone ) ; ::_thesis: ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied
hence  ex U being non empty set ex I being Element of U -InterpretersOf S st X is I -satisfied by A1, Lm75; ::_thesis: verum
end;

Lm77:  for U being non empty set
for X being set
for S1, S2 being Language
for I1 being Element of U -InterpretersOf S1
for I2 being Element of U -InterpretersOf S2 st S2 is S1 -extending & X c= AllFormulasOf S1 & I1 | (OwnSymbolsOf S1) = I2 | (OwnSymbolsOf S1) holds
( X is I1 -satisfied iff X is I2 -satisfied )
proof
let U be non empty set ; ::_thesis: for X being set
for S1, S2 being Language
for I1 being Element of U -InterpretersOf S1
for I2 being Element of U -InterpretersOf S2 st S2 is S1 -extending & X c= AllFormulasOf S1 & I1 | (OwnSymbolsOf S1) = I2 | (OwnSymbolsOf S1) holds
( X is I1 -satisfied iff X is I2 -satisfied )
let X be set ; ::_thesis: for S1, S2 being Language
for I1 being Element of U -InterpretersOf S1
for I2 being Element of U -InterpretersOf S2 st S2 is S1 -extending & X c= AllFormulasOf S1 & I1 | (OwnSymbolsOf S1) = I2 | (OwnSymbolsOf S1) holds
( X is I1 -satisfied iff X is I2 -satisfied )
let S1, S2 be Language; ::_thesis: for I1 being Element of U -InterpretersOf S1
for I2 being Element of U -InterpretersOf S2 st S2 is S1 -extending & X c= AllFormulasOf S1 & I1 | (OwnSymbolsOf S1) = I2 | (OwnSymbolsOf S1) holds
( X is I1 -satisfied iff X is I2 -satisfied )
set II1 = U -InterpretersOf S1;
set II2 = U -InterpretersOf S2;
set a1 = the adicity of S1;
set a2 = the adicity of S2;
set O1 = OwnSymbolsOf S1;
set E1 = TheEqSymbOf S1;
set E2 = TheEqSymbOf S2;
set N1 = TheNorSymbOf S1;
set N2 = TheNorSymbOf S2;
set FF1 = AllFormulasOf S1;
set AS1 = AtomicFormulaSymbolsOf S1;
 ( dom the adicity of S1 = AtomicFormulaSymbolsOf S1 & OwnSymbolsOf S1 c= AtomicFormulaSymbolsOf S1 ) by FOMODEL1:1, FUNCT_2:def_1;
then reconsider O11 = OwnSymbolsOf S1 as Subset of (dom the adicity of S1) ;
let I1 be Element of U -InterpretersOf S1; ::_thesis: for I2 being Element of U -InterpretersOf S2 st S2 is S1 -extending & X c= AllFormulasOf S1 & I1 | (OwnSymbolsOf S1) = I2 | (OwnSymbolsOf S1) holds
( X is I1 -satisfied iff X is I2 -satisfied )
let I2 be Element of U -InterpretersOf S2; ::_thesis: ( S2 is S1 -extending & X c= AllFormulasOf S1 & I1 | (OwnSymbolsOf S1) = I2 | (OwnSymbolsOf S1) implies ( X is I1 -satisfied iff X is I2 -satisfied ) )
assume A1: S2 is S1 -extending ; ::_thesis: ( not X c= AllFormulasOf S1 or not I1 | (OwnSymbolsOf S1) = I2 | (OwnSymbolsOf S1) or ( X is I1 -satisfied iff X is I2 -satisfied ) )
then  the adicity of S1 c= the adicity of S2 by FOMODEL1:def_41;
then  the adicity of S1 | (OwnSymbolsOf S1) = ( the adicity of S2 | (dom the adicity of S1)) | (OwnSymbolsOf S1) by GRFUNC_1:23
.= the adicity of S2 | (O11 null (dom the adicity of S1)) by RELAT_1:71 ;
then A2: ( the adicity of S1 | (OwnSymbolsOf S1) = the adicity of S2 | (OwnSymbolsOf S1) & TheNorSymbOf S1 = TheNorSymbOf S2 & TheEqSymbOf S1 = TheEqSymbOf S2 ) by A1, FOMODEL1:def_41;
assume A3: ( X c= AllFormulasOf S1 & I1 | (OwnSymbolsOf S1) = I2 | (OwnSymbolsOf S1) ) ; ::_thesis: ( X is I1 -satisfied iff X is I2 -satisfied )
hereby  ::_thesis: ( X is I2 -satisfied implies X is I1 -satisfied )
assume A4: X is I1 -satisfied ; ::_thesis: X is I2 -satisfied
now__::_thesis:_for_phi2_being_wff_string_of_S2_st_phi2_in_X_holds_
I2_-TruthEval_phi2_=_1
let phi2 be wff string of S2; ::_thesis: ( phi2 in X implies I2 -TruthEval phi2 = 1 )
assume A5: phi2 in X ; ::_thesis: I2 -TruthEval phi2 = 1
then  phi2 in AllFormulasOf S1 by A3;
then reconsider phi1 = phi2 as wff string of S1 ;
consider phi22 being wff string of S2 such that
A6: ( phi1 = phi22 & I1 -TruthEval phi1 = I2 -TruthEval phi22 ) by A2, A3, FOMODEL3:12;
thus  I2 -TruthEval phi2 = 1 by A6, A5, A4, FOMODEL2:def_42; ::_thesis: verum
end;
hence  X is I2 -satisfied by FOMODEL2:def_42; ::_thesis: verum
end;
assume A7: X is I2 -satisfied ; ::_thesis: X is I1 -satisfied
now__::_thesis:_for_phi1_being_wff_string_of_S1_st_phi1_in_X_holds_
I1_-TruthEval_phi1_=_1
let phi1 be wff string of S1; ::_thesis: ( phi1 in X implies I1 -TruthEval phi1 = 1 )
assume A8: phi1 in X ; ::_thesis: I1 -TruthEval phi1 = 1
consider phi2 being wff string of S2 such that
A9: ( phi1 = phi2 & I1 -TruthEval phi1 = I2 -TruthEval phi2 ) by A2, A3, FOMODEL3:12;
thus  I1 -TruthEval phi1 = 1 by A9, A8, A7, FOMODEL2:def_42; ::_thesis: verum
end;
hence  X is I1 -satisfied by FOMODEL2:def_42; ::_thesis: verum
end;

registration
let S be Language;
let r1, r2 be isotone Rule of S;
cluster{r1,r2} ->  isotone for RuleSet of S;
coherence
 for b1 being RuleSet of S st b1 = {r1,r2} holds
b1 is isotone
proof
 {r1,r2} = {r1} \/ {r2} by ENUMSET1:1;
hence  for b1 being RuleSet of S st b1 = {r1,r2} holds
b1 is isotone ; ::_thesis: verum
end;
end;

registration
let S be Language;
let r1, r2, r3, r4 be isotone Rule of S;
clusterK336(r1,r2,r3,r4) ->  isotone for RuleSet of S;
coherence
 for b1 being RuleSet of S st b1 = {r1,r2,r3,r4} holds
b1 is isotone
proof
 {r1,r2,r3,r4} = {r1,r2} \/ {r3,r4} by ENUMSET1:5;
hence  for b1 being RuleSet of S st b1 = {r1,r2,r3,r4} holds
b1 is isotone ; ::_thesis: verum
end;
end;

registration
let S be Language;
clusterS -rules  ->  isotone for RuleSet of S;
coherence
 for b1 being RuleSet of S st b1 = S -rules holds
b1 is isotone
proof
set A = {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)};
set B = {(R#5 S),(R#6 S),(R#7 S),(R#8 S)};
set IT = S -rules ;
 {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} = {(R#0 S),(R#1 S),(R#2 S),(R#3a S)} \/ {(R#3b S),(R#3d S),(R#3e S),(R#4 S)} by ENUMSET1:25;
then reconsider AA = {(R#0 S),(R#1 S),(R#2 S),(R#3a S),(R#3b S),(R#3d S),(R#3e S),(R#4 S)} as isotone RuleSet of S ;
 AA \/ {(R#5 S),(R#6 S),(R#7 S),(R#8 S)} is isotone ;
hence  for b1 being RuleSet of S st b1 = S -rules holds
b1 is isotone ; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster functional isotone Correct for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
existence
 ex b1 being isotone RuleSet of S st b1 is Correct
proof
take  S -rules ; ::_thesis: S -rules is Correct
thus  S -rules is Correct ; ::_thesis: verum
end;
end;

registration
let S be Language;
cluster functional isotone 2 -ranked Correct for Element of bool (Funcs ((bool (S -sequents)),(bool (S -sequents))));
existence
 ex b1 being isotone Correct RuleSet of S st b1 is 2 -ranked
proof
take  S -rules ; ::_thesis: S -rules is 2 -ranked
thus  S -rules is 2 -ranked ; ::_thesis: verum
end;
end;

registration
let S be countable Language;
cluster AllFormulasOf S ->  countable ;
coherence
 AllFormulasOf S is countable ;
end;

theorem Th20: :: FOMODEL4:20
for Z being set
for S being countable Language
for D being RuleSet of S st D is 2 -ranked & D is isotone & D is Correct & Z is D -consistent & Z c= AllFormulasOf S holds
ex U being non empty set ex I being Element of U -InterpretersOf S st Z is I -satisfied
proof
let Z be set ; ::_thesis: for S being countable Language
for D being RuleSet of S st D is 2 -ranked & D is isotone & D is Correct & Z is D -consistent & Z c= AllFormulasOf S holds
ex U being non empty set ex I being Element of U -InterpretersOf S st Z is I -satisfied
let S be countable Language; ::_thesis: for D being RuleSet of S st D is 2 -ranked & D is isotone & D is Correct & Z is D -consistent & Z c= AllFormulasOf S holds
ex U being non empty set ex I being Element of U -InterpretersOf S st Z is I -satisfied
set S1 = S;
let D be RuleSet of S; ::_thesis: ( D is 2 -ranked & D is isotone & D is Correct & Z is D -consistent & Z c= AllFormulasOf S implies ex U being non empty set ex I being Element of U -InterpretersOf S st Z is I -satisfied )
set FF1 = AllFormulasOf S;
assume A1: ( D is 2 -ranked & D is isotone & D is Correct & Z is D -consistent & Z c= AllFormulasOf S ) ; ::_thesis: ex U being non empty set ex I being Element of U -InterpretersOf S st Z is I -satisfied
then reconsider X = Z as Subset of (AllFormulasOf S) ;
set S2 = S addLettersNotIn X;
set O1 = OwnSymbolsOf S;
set O2 = OwnSymbolsOf (S addLettersNotIn X);
set FF2 = AllFormulasOf (S addLettersNotIn X);
set SS1 = AllSymbolsOf S;
set SS2 = AllSymbolsOf (S addLettersNotIn X);
set strings2 = ((AllSymbolsOf (S addLettersNotIn X)) *) \ {{}};
set L2 = LettersOf (S addLettersNotIn X);
reconsider D1 = D as 2 -ranked Correct RuleSet of S by A1;
 (OwnSymbolsOf S) \ (OwnSymbolsOf (S addLettersNotIn X)) = {} ;
then reconsider O11 = OwnSymbolsOf S as non empty Subset of (OwnSymbolsOf (S addLettersNotIn X)) by XBOOLE_1:37;
reconsider D2 = (S addLettersNotIn X) -rules as isotone 2 -ranked Correct RuleSet of (S addLettersNotIn X) ;
reconsider sub1 = X /\ (((AllSymbolsOf (S addLettersNotIn X)) *) \ {{}}) as Subset of X ;
reconsider sub2 = SymbolsOf sub1 as Subset of (SymbolsOf X) by FOMODEL0:46;
reconsider inf = (LettersOf (S addLettersNotIn X)) \ (SymbolsOf X) as Subset of ((LettersOf (S addLettersNotIn X)) \ sub2) by XBOOLE_1:34;
A2: (LettersOf (S addLettersNotIn X)) \ (sub2 null inf) is infinite ;
now__::_thesis:_for_Y_being_finite_Subset_of_X_holds_Y_is_D2_-consistent
let Y be finite Subset of X; ::_thesis: Y is D2 -consistent
reconsider YY = Y as functional set ;
reconsider YYY = YY as functional Subset of (AllFormulasOf S) by XBOOLE_1:1;
 ( YY is finite & AllFormulasOf S is countable & YY is D1 -consistent & D1 is isotone ) by A1, Th15;
then consider U being non empty set  such that
A3: ex I1 being Element of U -InterpretersOf S st YY is I1 -satisfied by Lm76;
set II1 = U -InterpretersOf S;
set II2 = U -InterpretersOf (S addLettersNotIn X);
set I02 = the S addLettersNotIn X,U -interpreter-like Function;
consider I1 being Element of U -InterpretersOf S such that
A4: YYY is I1 -satisfied by A3;
reconsider I2 = ( the S addLettersNotIn X,U -interpreter-like Function +* I1) | (OwnSymbolsOf (S addLettersNotIn X)) as Element of U -InterpretersOf (S addLettersNotIn X) by FOMODEL2:2;
I2 | (OwnSymbolsOf S) = ( the S addLettersNotIn X,U -interpreter-like Function +* I1) | (O11 null (OwnSymbolsOf (S addLettersNotIn X))) by RELAT_1:71
.= ( the S addLettersNotIn X,U -interpreter-like Function | (OwnSymbolsOf S)) +* (I1 | (OwnSymbolsOf S)) by FUNCT_4:71
.= I1 | (OwnSymbolsOf S) ;
then  YYY is I2 -satisfied by A4, Lm77;
hence  Y is D2 -consistent by Lm56; ::_thesis: verum
end;
then  X is D2 -consistent by Lm53;
then consider U being non empty set  such that
A5: ex I being Element of U -InterpretersOf (S addLettersNotIn X) st X is I -satisfied by A2, Lm75;
set II1 = U -InterpretersOf S;
set II2 = U -InterpretersOf (S addLettersNotIn X);
consider I2 being Element of U -InterpretersOf (S addLettersNotIn X) such that
A6: X is I2 -satisfied by A5;
take  U ; ::_thesis: ex I being Element of U -InterpretersOf S st Z is I -satisfied
reconsider I1 = I2 | (OwnSymbolsOf S) as Element of U -InterpretersOf S by FOMODEL2:2;
take  I1 ; ::_thesis: Z is I1 -satisfied
 I1 | (OwnSymbolsOf S) = (I2 | (OwnSymbolsOf S)) null (OwnSymbolsOf S) ;
hence  Z is I1 -satisfied by A6, Lm77; ::_thesis: verum
end;

Lm78:  for Z being set
for S being countable Language
for phi being wff string of S st Z c= AllFormulasOf S & xnot phi is Z -implied holds
xnot phi is Z,S -rules -provable
proof
let Z be set ; ::_thesis: for S being countable Language
for phi being wff string of S st Z c= AllFormulasOf S & xnot phi is Z -implied holds
xnot phi is Z,S -rules -provable
let S be countable Language; ::_thesis: for phi being wff string of S st Z c= AllFormulasOf S & xnot phi is Z -implied holds
xnot phi is Z,S -rules -provable
set D = S -rules ;
set FF = AllFormulasOf S;
let phi be wff string of S; ::_thesis: ( Z c= AllFormulasOf S & xnot phi is Z -implied implies xnot phi is Z,S -rules -provable )
assume  Z c= AllFormulasOf S ; ::_thesis: ( not xnot phi is Z -implied or xnot phi is Z,S -rules -provable )
then reconsider X = Z as Subset of (AllFormulasOf S) ;
set psi = xnot phi;
 phi in AllFormulasOf S by FOMODEL2:16;
then reconsider Phi = {phi} as non empty Subset of (AllFormulasOf S) by ZFMISC_1:31;
reconsider Y = X \/ Phi as non empty Subset of (AllFormulasOf S) ;
reconsider XX = X null Phi as Subset of Y ;
reconsider Phii = Phi null X as non empty Subset of Y ;
assume A1: xnot phi is Z -implied ; ::_thesis: xnot phi is Z,S -rules -provable
assume  not xnot phi is Z,S -rules -provable ; ::_thesis: contradiction
then  ( S -rules is isotone & R#1 S in S -rules & R#8 S in S -rules & not xnot phi is Z,S -rules -provable ) by Def59;
then consider U being non empty set , I being Element of U -InterpretersOf S such that
A2: Y is I -satisfied by Th20, Lm52;
 (I -TruthEval (xnot phi)) \+\ ('not' (I -TruthEval phi)) = {} ;
then A3: I -TruthEval (xnot phi) = 'not' (I -TruthEval phi) by FOMODEL0:29;
A4: Y /\ XX is I -satisfied by A2;
 phi in Phii by TARSKI:def_1;
then  1 = 'not' (I -TruthEval (xnot phi)) by A3, A2, FOMODEL2:def_42;
hence  contradiction by A4, A1, FOMODEL2:def_45; ::_thesis: verum
end;


theorem :: FOMODEL4:21
for X being set
for C being countable Language
for phi being wff string of C st X c= AllFormulasOf C & phi is X -implied holds
phi is X -provable
proof
let X be set ; ::_thesis: for C being countable Language
for phi being wff string of C st X c= AllFormulasOf C & phi is X -implied holds
phi is X -provable
let C be countable Language; ::_thesis: for phi being wff string of C st X c= AllFormulasOf C & phi is X -implied holds
phi is X -provable
let phi be wff string of C; ::_thesis: ( X c= AllFormulasOf C & phi is X -implied implies phi is X -provable )
reconsider S = C as Language ;
reconsider DD = {(R#9 S)} as RuleSet of S ;
set FF = AllFormulasOf C;
set D = C -rules ;
assume  X c= AllFormulasOf C ; ::_thesis: ( not phi is X -implied or phi is X -provable )
then reconsider Y = X as Subset of (AllFormulasOf C) ;
assume  phi is X -implied ; ::_thesis: phi is X -provable
then reconsider phii = phi as wff X -implied string of C ;
set psi = xnot (xnot phii);
 xnot (xnot phii) is Y,C -rules -provable by Lm78;
then consider H being set , m being Nat such that
A1: ( H c= Y & [H,(xnot (xnot phii))] is m, {} ,C -rules -derivable ) by Def62;
reconsider seqt = [H,(xnot (xnot phii))] as C -sequent-like set by A1;
A2: (seqt `1) \+\ H = {} ;
reconsider HH = H as S -premises-like set by A2, FOMODEL0:29;
reconsider HC = H as C -premises-like set by A2, FOMODEL0:29;
reconsider a = phi as wff string of S ;
 [HC,phi] null 1 is 1,{[HC,(xnot (xnot phi))]},{(R#9 C)} -derivable ;
then  [HC,phi] is m + 1, {} ,(C -rules) \/ {(R#9 C)} -derivable by Lm21, A1;
then  phi is Y,(C -rules) \/ {(R#9 C)} -provable by A1, Def62;
hence  phi is X -provable by Def70; ::_thesis: verum
end;