:: SUBLEMMA semantic presentation

begin


theorem Th1: :: SUBLEMMA:1
for f, g, h, h1, h2 being Function st dom h1 c= dom h & dom h2 c= dom h holds
(f +* g) +* h = ((f +* h1) +* (g +* h2)) +* h
proof
let f, g, h, h1, h2 be Function; ::_thesis: ( dom h1 c= dom h & dom h2 c= dom h implies (f +* g) +* h = ((f +* h1) +* (g +* h2)) +* h )
assume that
A1: dom h1 c= dom h and
A2: dom h2 c= dom h ; ::_thesis: (f +* g) +* h = ((f +* h1) +* (g +* h2)) +* h
 ( dom (f +* h1) = (dom f) \/ (dom h1) & dom (g +* h2) = (dom g) \/ (dom h2) ) by FUNCT_4:def_1;
then  dom ((f +* h1) +* (g +* h2)) = ((dom f) \/ (dom h1)) \/ ((dom g) \/ (dom h2)) by FUNCT_4:def_1;
then  dom (((f +* h1) +* (g +* h2)) +* h) = (((dom f) \/ (dom h1)) \/ ((dom g) \/ (dom h2))) \/ (dom h) by FUNCT_4:def_1;
then  dom (((f +* h1) +* (g +* h2)) +* h) = ((dom f) \/ (dom h1)) \/ (((dom g) \/ (dom h2)) \/ (dom h)) by XBOOLE_1:4;
then  dom (((f +* h1) +* (g +* h2)) +* h) = ((dom f) \/ (dom h1)) \/ ((dom g) \/ ((dom h2) \/ (dom h))) by XBOOLE_1:4;
then  dom (((f +* h1) +* (g +* h2)) +* h) = ((dom f) \/ (dom h1)) \/ ((dom g) \/ (dom h)) by A2, XBOOLE_1:12;
then  dom (((f +* h1) +* (g +* h2)) +* h) = (((dom f) \/ (dom h1)) \/ (dom h)) \/ (dom g) by XBOOLE_1:4;
then  dom (((f +* h1) +* (g +* h2)) +* h) = ((dom f) \/ ((dom h1) \/ (dom h))) \/ (dom g) by XBOOLE_1:4;
then A3: dom (((f +* h1) +* (g +* h2)) +* h) = ((dom f) \/ (dom h)) \/ (dom g) by A1, XBOOLE_1:12;
A4: for b being set st b in dom ((f +* g) +* h) holds
((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b
proof
let b be set ; ::_thesis: ( b in dom ((f +* g) +* h) implies ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b )
assume  b in dom ((f +* g) +* h) ; ::_thesis: ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b
A5: now__::_thesis:_(_not_b_in_dom_h_implies_((f_+*_g)_+*_h)_._b_=_(((f_+*_h1)_+*_(g_+*_h2))_+*_h)_._b_)
assume A6: not b in dom h ; ::_thesis: ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b
then A7: (((f +* h1) +* (g +* h2)) +* h) . b = ((f +* h1) +* (g +* h2)) . b by FUNCT_4:11;
A8: ((f +* g) +* h) . b = (f +* g) . b by A6, FUNCT_4:11;
A9: now__::_thesis:_(_b_in_dom_g_implies_((f_+*_g)_+*_h)_._b_=_(((f_+*_h1)_+*_(g_+*_h2))_+*_h)_._b_)
A10: not b in dom h2 by A2, A6;
assume A11: b in dom g ; ::_thesis: ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b
 dom g c= (dom g) \/ (dom h2) by XBOOLE_1:7;
then  b in (dom g) \/ (dom h2) by A11;
then  b in dom (g +* h2) by FUNCT_4:def_1;
then A12: (((f +* h1) +* (g +* h2)) +* h) . b = (g +* h2) . b by A7, FUNCT_4:13;
 ((f +* g) +* h) . b = g . b by A8, A11, FUNCT_4:13;
hence  ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b by A12, A10, FUNCT_4:11; ::_thesis: verum
end;
now__::_thesis:_(_not_b_in_dom_g_implies_((f_+*_g)_+*_h)_._b_=_(((f_+*_h1)_+*_(g_+*_h2))_+*_h)_._b_)
A13: not b in dom h1 by A1, A6;
assume A14: not b in dom g ; ::_thesis: ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b
 not b in dom h2 by A2, A6;
then  not b in (dom g) \/ (dom h2) by A14, XBOOLE_0:def_3;
then  not b in dom (g +* h2) by FUNCT_4:def_1;
then A15: (((f +* h1) +* (g +* h2)) +* h) . b = (f +* h1) . b by A7, FUNCT_4:11;
 ((f +* g) +* h) . b = f . b by A8, A14, FUNCT_4:11;
hence  ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b by A15, A13, FUNCT_4:11; ::_thesis: verum
end;
hence  ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b by A9; ::_thesis: verum
end;
now__::_thesis:_(_b_in_dom_h_implies_((f_+*_g)_+*_h)_._b_=_(((f_+*_h1)_+*_(g_+*_h2))_+*_h)_._b_)
assume A16: b in dom h ; ::_thesis: ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b
then  ((f +* g) +* h) . b = h . b by FUNCT_4:13;
hence  ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b by A16, FUNCT_4:13; ::_thesis: verum
end;
hence  ((f +* g) +* h) . b = (((f +* h1) +* (g +* h2)) +* h) . b by A5; ::_thesis: verum
end;
 dom (f +* g) = (dom f) \/ (dom g) by FUNCT_4:def_1;
then  dom ((f +* g) +* h) = ((dom f) \/ (dom g)) \/ (dom h) by FUNCT_4:def_1;
hence  (f +* g) +* h = ((f +* h1) +* (g +* h2)) +* h by A3, A4, FUNCT_1:2, XBOOLE_1:4; ::_thesis: verum
end;

theorem Th2: :: SUBLEMMA:2
for Al being QC-alphabet
for x being bound_QC-variable of Al
for vS1 being Function st x in dom vS1 holds
(vS1 | ((dom vS1) \ {x})) +* (x .--> (vS1 . x)) = vS1
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for vS1 being Function st x in dom vS1 holds
(vS1 | ((dom vS1) \ {x})) +* (x .--> (vS1 . x)) = vS1
let x be bound_QC-variable of Al; ::_thesis: for vS1 being Function st x in dom vS1 holds
(vS1 | ((dom vS1) \ {x})) +* (x .--> (vS1 . x)) = vS1
let vS1 be Function; ::_thesis: ( x in dom vS1 implies (vS1 | ((dom vS1) \ {x})) +* (x .--> (vS1 . x)) = vS1 )
assume  x in dom vS1 ; ::_thesis: (vS1 | ((dom vS1) \ {x})) +* (x .--> (vS1 . x)) = vS1
then  ( ((dom vS1) \ {x}) \/ {x} = (dom vS1) \/ {x} & {x} c= dom vS1 ) by XBOOLE_1:39, ZFMISC_1:31;
then  ((dom vS1) \ {x}) \/ {x} = dom vS1 by XBOOLE_1:12;
hence  (vS1 | ((dom vS1) \ {x})) +* (x .--> (vS1 . x)) = vS1 by FUNCT_7:14; ::_thesis: verum
end;

definition
let Al be QC-alphabet ;
let A be non empty set ;
mode Val_Sub of A,Al is PartFunc of (bound_QC-variables Al),A;
end;


notation
let Al be QC-alphabet ;
let A be non empty set ;
let v be Element of Valuations_in (Al,A);
let vS be Val_Sub of A,Al;
synonym v . vS for Al +* A;
end;

definition
let Al be QC-alphabet ;
let A be non empty set ;
let v be Element of Valuations_in (Al,A);
let vS be Val_Sub of A,Al;
:: original: .
redefine funcv . vS ->  Element of Valuations_in (Al,A);
coherence
 . is Element of Valuations_in (Al,A)
proof
 v is Element of Funcs ((bound_QC-variables Al),A) by VALUAT_1:def_1;
then  ex f being Function st
( v = f & dom f = bound_QC-variables Al & rng f c= A ) by FUNCT_2:def_2;
then  dom (v +* vS) = (bound_QC-variables Al) \/ (dom vS) by FUNCT_4:def_1;
then A1: dom (v +* vS) = bound_QC-variables Al by XBOOLE_1:12;
 rng (v +* vS) c= (rng v) \/ (rng vS) by FUNCT_4:17;
then  v +* vS in Funcs ((bound_QC-variables Al),A) by A1, FUNCT_2:def_2;
hence  . is Element of Valuations_in (Al,A) by VALUAT_1:def_1; ::_thesis: verum
end;
end;

definition
let Al be QC-alphabet ;
let S be Element of CQC-Sub-WFF Al;
:: original: `1
redefine funcS `1  ->  Element of CQC-WFF Al;
coherence
 S `1 is Element of CQC-WFF Al
proof
 S in CQC-Sub-WFF Al ;
then  S in  {  S1 where S1 is Element of QC-Sub-WFF Al : S1 `1 is Element of CQC-WFF Al  }  by SUBSTUT1:def_39;
then  ex S1 being Element of QC-Sub-WFF Al st
( S = S1 & S1 `1 is Element of CQC-WFF Al ) ;
hence  S `1 is Element of CQC-WFF Al ; ::_thesis: verum
end;
end;

definition
let Al be QC-alphabet ;
let S be Element of CQC-Sub-WFF Al;
let A be non empty set ;
let v be Element of Valuations_in (Al,A);
func Val_S (v,S) ->  Val_Sub of A,Al equals :: SUBLEMMA:def 1
(@ (S `2)) * v;
coherence
 (@ (S `2)) * v is Val_Sub of A,Al ;
end;

:: deftheorem  defines Val_S SUBLEMMA:def_1_:_
 for Al being QC-alphabet
for S being Element of CQC-Sub-WFF Al
for A being non empty set
for v being Element of Valuations_in (Al,A) holds Val_S (v,S) = (@ (S `2)) * v;

theorem Th3: :: SUBLEMMA:3
for Al being QC-alphabet
for S being Element of CQC-Sub-WFF Al st S is Al -Sub_VERUM holds
CQC_Sub S = VERUM Al
proof
let Al be QC-alphabet ; ::_thesis: for S being Element of CQC-Sub-WFF Al st S is Al -Sub_VERUM holds
CQC_Sub S = VERUM Al
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( S is Al -Sub_VERUM implies CQC_Sub S = VERUM Al )
 ex F being Function of (QC-Sub-WFF Al),(QC-WFF Al) st
( CQC_Sub S = F . S & ( for S9 being Element of QC-Sub-WFF Al holds
( ( S9 is Al -Sub_VERUM implies F . S9 = VERUM Al ) & ( S9 is Sub_atomic implies F . S9 = (the_pred_symbol_of (S9 `1)) ! (CQC_Subst ((Sub_the_arguments_of S9),(S9 `2))) ) & ( S9 is Sub_negative implies F . S9 = 'not' (F . (Sub_the_argument_of S9)) ) & ( S9 is Sub_conjunctive implies F . S9 = (F . (Sub_the_left_argument_of S9)) '&' (F . (Sub_the_right_argument_of S9)) ) & ( S9 is Sub_universal implies F . S9 = Quant (S9,(F . (Sub_the_scope_of S9))) ) ) ) ) by SUBSTUT1:def_38;
hence  ( S is Al -Sub_VERUM implies CQC_Sub S = VERUM Al ) ; ::_thesis: verum
end;

definition
let Al be QC-alphabet ;
let S be Element of CQC-Sub-WFF Al;
let A be non empty set ;
let v be Element of Valuations_in (Al,A);
let J be interpretation of Al,A;
predJ,v |= S means :Def2: :: SUBLEMMA:def 2
J,v |= S `1 ;
end;

:: deftheorem Def2 defines |= SUBLEMMA:def_2_:_
 for Al being QC-alphabet
for S being Element of CQC-Sub-WFF Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for J being interpretation of Al,A holds
( J,v |= S iff J,v |= S `1 );

theorem Th4: :: SUBLEMMA:4
for Al being QC-alphabet
for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al st S is Al -Sub_VERUM holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
proof
let Al be QC-alphabet ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al st S is Al -Sub_VERUM holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al st S is Al -Sub_VERUM holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
let J be interpretation of Al,A; ::_thesis: for S being Element of CQC-Sub-WFF Al st S is Al -Sub_VERUM holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( S is Al -Sub_VERUM implies for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) )
assume A1: S is Al -Sub_VERUM ; ::_thesis: for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
 ex Sub being CQC_Substitution of Al st S = [(VERUM Al),Sub] by A1, SUBSTUT1:def_19;
then  S `1 = VERUM Al by MCART_1:7;
then  ( J,v . (Val_S (v,S)) |= VERUM Al iff J,v . (Val_S (v,S)) |= S ) by Def2;
hence  ( J,v |= CQC_Sub S implies J,v . (Val_S (v,S)) |= S ) by VALUAT_1:32; ::_thesis: ( J,v . (Val_S (v,S)) |= S implies J,v |= CQC_Sub S )
 ( J,v . (Val_S (v,S)) |= S implies J,v |= VERUM Al ) by VALUAT_1:32;
hence  ( J,v . (Val_S (v,S)) |= S implies J,v |= CQC_Sub S ) by A1, Th3; ::_thesis: verum
end;

theorem Th5: :: SUBLEMMA:5
for Al being QC-alphabet
for k, i being Element of NAT
for ll being CQC-variable_list of k,Al st i in dom ll holds
ll . i is bound_QC-variable of Al
proof
let Al be QC-alphabet ; ::_thesis: for k, i being Element of NAT
for ll being CQC-variable_list of k,Al st i in dom ll holds
ll . i is bound_QC-variable of Al
let k, i be Element of NAT ; ::_thesis: for ll being CQC-variable_list of k,Al st i in dom ll holds
ll . i is bound_QC-variable of Al
let ll be CQC-variable_list of k,Al; ::_thesis: ( i in dom ll implies ll . i is bound_QC-variable of Al )
assume  i in dom ll ; ::_thesis: ll . i is bound_QC-variable of Al
then A1: ll . i in rng ll by FUNCT_1:3;
 rng ll c= bound_QC-variables Al by RELAT_1:def_19;
hence  ll . i is bound_QC-variable of Al by A1; ::_thesis: verum
end;

theorem Th6: :: SUBLEMMA:6
for Al being QC-alphabet
for S being Element of CQC-Sub-WFF Al st S is Sub_atomic holds
CQC_Sub S = (the_pred_symbol_of (S `1)) ! (CQC_Subst ((Sub_the_arguments_of S),(S `2)))
proof
let Al be QC-alphabet ; ::_thesis: for S being Element of CQC-Sub-WFF Al st S is Sub_atomic holds
CQC_Sub S = (the_pred_symbol_of (S `1)) ! (CQC_Subst ((Sub_the_arguments_of S),(S `2)))
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( S is Sub_atomic implies CQC_Sub S = (the_pred_symbol_of (S `1)) ! (CQC_Subst ((Sub_the_arguments_of S),(S `2))) )
 ex F being Function of (QC-Sub-WFF Al),(QC-WFF Al) st
( CQC_Sub S = F . S & ( for S9 being Element of QC-Sub-WFF Al holds
( ( S9 is Al -Sub_VERUM implies F . S9 = VERUM Al ) & ( S9 is Sub_atomic implies F . S9 = (the_pred_symbol_of (S9 `1)) ! (CQC_Subst ((Sub_the_arguments_of S9),(S9 `2))) ) & ( S9 is Sub_negative implies F . S9 = 'not' (F . (Sub_the_argument_of S9)) ) & ( S9 is Sub_conjunctive implies F . S9 = (F . (Sub_the_left_argument_of S9)) '&' (F . (Sub_the_right_argument_of S9)) ) & ( S9 is Sub_universal implies F . S9 = Quant (S9,(F . (Sub_the_scope_of S9))) ) ) ) ) by SUBSTUT1:def_38;
hence  ( S is Sub_atomic implies CQC_Sub S = (the_pred_symbol_of (S `1)) ! (CQC_Subst ((Sub_the_arguments_of S),(S `2))) ) ; ::_thesis: verum
end;

theorem :: SUBLEMMA:7
for Al being QC-alphabet
for k being Element of NAT
for P, P9 being QC-pred_symbol of k,Al
for ll, ll9 being CQC-variable_list of k,Al
for Sub, Sub9 being CQC_Substitution of Al st Sub_the_arguments_of (Sub_P (P,ll,Sub)) = Sub_the_arguments_of (Sub_P (P9,ll9,Sub9)) holds
ll = ll9
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for P, P9 being QC-pred_symbol of k,Al
for ll, ll9 being CQC-variable_list of k,Al
for Sub, Sub9 being CQC_Substitution of Al st Sub_the_arguments_of (Sub_P (P,ll,Sub)) = Sub_the_arguments_of (Sub_P (P9,ll9,Sub9)) holds
ll = ll9
let k be Element of NAT ; ::_thesis: for P, P9 being QC-pred_symbol of k,Al
for ll, ll9 being CQC-variable_list of k,Al
for Sub, Sub9 being CQC_Substitution of Al st Sub_the_arguments_of (Sub_P (P,ll,Sub)) = Sub_the_arguments_of (Sub_P (P9,ll9,Sub9)) holds
ll = ll9
let P, P9 be QC-pred_symbol of k,Al; ::_thesis: for ll, ll9 being CQC-variable_list of k,Al
for Sub, Sub9 being CQC_Substitution of Al st Sub_the_arguments_of (Sub_P (P,ll,Sub)) = Sub_the_arguments_of (Sub_P (P9,ll9,Sub9)) holds
ll = ll9
let ll, ll9 be CQC-variable_list of k,Al; ::_thesis: for Sub, Sub9 being CQC_Substitution of Al st Sub_the_arguments_of (Sub_P (P,ll,Sub)) = Sub_the_arguments_of (Sub_P (P9,ll9,Sub9)) holds
ll = ll9
let Sub, Sub9 be CQC_Substitution of Al; ::_thesis: ( Sub_the_arguments_of (Sub_P (P,ll,Sub)) = Sub_the_arguments_of (Sub_P (P9,ll9,Sub9)) implies ll = ll9 )
assume A1: Sub_the_arguments_of (Sub_P (P,ll,Sub)) = Sub_the_arguments_of (Sub_P (P9,ll9,Sub9)) ; ::_thesis: ll = ll9
consider k1 being Element of NAT , P1 being QC-pred_symbol of k1,Al, ll1 being QC-variable_list of k1,Al, e1 being Element of vSUB Al such that
A2: Sub_the_arguments_of (Sub_P (P,ll,Sub)) = ll1 and
A3: Sub_P (P,ll,Sub) = Sub_P (P1,ll1,e1) by SUBSTUT1:def_29;
A4: ( P ! ll = <*P*> ^ ll & P1 ! ll1 = <*P1*> ^ ll1 ) by QC_LANG1:8;
 Sub_P (P,ll,Sub) = [(P ! ll),Sub] by SUBSTUT1:9;
then  [(P ! ll),Sub] = [(P1 ! ll1),e1] by A3, SUBSTUT1:9;
then A5: <*P1*> ^ ll1 = <*P*> ^ ll by A4, XTUPLE_0:1;
 ( (<*P1*> ^ ll1) . 1 = P1 & (<*P*> ^ ll) . 1 = P ) by FINSEQ_1:41;
then A6: ll1 = ll by A5, FINSEQ_1:33;
consider k2 being Element of NAT , P2 being QC-pred_symbol of k2,Al, ll2 being QC-variable_list of k2,Al, e2 being Element of vSUB Al such that
A7: Sub_the_arguments_of (Sub_P (P9,ll9,Sub9)) = ll2 and
A8: Sub_P (P9,ll9,Sub9) = Sub_P (P2,ll2,e2) by SUBSTUT1:def_29;
A9: ( P9 ! ll9 = <*P9*> ^ ll9 & P2 ! ll2 = <*P2*> ^ ll2 ) by QC_LANG1:8;
 Sub_P (P9,ll9,Sub9) = [(P9 ! ll9),Sub9] by SUBSTUT1:9;
then  [(P9 ! ll9),Sub9] = [(P2 ! ll2),e2] by A8, SUBSTUT1:9;
then A10: <*P2*> ^ ll2 = <*P9*> ^ ll9 by A9, XTUPLE_0:1;
 ( (<*P2*> ^ ll2) . 1 = P2 & (<*P9*> ^ ll9) . 1 = P9 ) by FINSEQ_1:41;
hence  ll = ll9 by A1, A2, A7, A6, A10, FINSEQ_1:33; ::_thesis: verum
end;

definition
let k be Element of NAT ;
let Al be QC-alphabet ;
let P be QC-pred_symbol of k,Al;
let ll be CQC-variable_list of k,Al;
let Sub be CQC_Substitution of Al;
:: original: Sub_P
redefine func Sub_P (P,ll,Sub) ->  Element of CQC-Sub-WFF Al;
coherence
 Sub_P (P,ll,Sub) is Element of CQC-Sub-WFF Al
proof
set X =  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  ;
  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  = CQC-Sub-WFF Al by SUBSTUT1:def_39;
then A1: for G being Element of QC-Sub-WFF Al st G `1 is Element of CQC-WFF Al holds
G in CQC-Sub-WFF Al ;
 Sub_P (P,ll,Sub) = [(P ! ll),Sub] by SUBSTUT1:9;
then  ( (Sub_P (P,ll,Sub)) `1 = P ! ll & P ! ll in CQC-WFF Al ) by MCART_1:7;
hence  Sub_P (P,ll,Sub) is Element of CQC-Sub-WFF Al by A1; ::_thesis: verum
end;
end;

theorem Th8: :: SUBLEMMA:8
for Al being QC-alphabet
for k being Element of NAT
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds CQC_Sub (Sub_P (P,ll,Sub)) = P ! (CQC_Subst (ll,Sub))
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds CQC_Sub (Sub_P (P,ll,Sub)) = P ! (CQC_Subst (ll,Sub))
let k be Element of NAT ; ::_thesis: for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds CQC_Sub (Sub_P (P,ll,Sub)) = P ! (CQC_Subst (ll,Sub))
let P be QC-pred_symbol of k,Al; ::_thesis: for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds CQC_Sub (Sub_P (P,ll,Sub)) = P ! (CQC_Subst (ll,Sub))
let ll be CQC-variable_list of k,Al; ::_thesis: for Sub being CQC_Substitution of Al holds CQC_Sub (Sub_P (P,ll,Sub)) = P ! (CQC_Subst (ll,Sub))
let Sub be CQC_Substitution of Al; ::_thesis: CQC_Sub (Sub_P (P,ll,Sub)) = P ! (CQC_Subst (ll,Sub))
A1: P ! ll is atomic by QC_LANG1:def_18;
A2: Sub_P (P,ll,Sub) = [(P ! ll),Sub] by SUBSTUT1:9;
then A3: (Sub_P (P,ll,Sub)) `2 = Sub by MCART_1:7;
 (Sub_P (P,ll,Sub)) `1 = P ! ll by A2, MCART_1:7;
then  ( Sub_the_arguments_of (Sub_P (P,ll,Sub)) = ll & the_pred_symbol_of ((Sub_P (P,ll,Sub)) `1) = P ) by A1, QC_LANG1:def_22, SUBSTUT1:def_29;
hence  CQC_Sub (Sub_P (P,ll,Sub)) = P ! (CQC_Subst (ll,Sub)) by A3, Th6; ::_thesis: verum
end;

theorem :: SUBLEMMA:9
for Al being QC-alphabet
for k being Element of NAT
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds P ! (CQC_Subst (ll,Sub)) is Element of CQC-WFF Al
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds P ! (CQC_Subst (ll,Sub)) is Element of CQC-WFF Al
let k be Element of NAT ; ::_thesis: for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds P ! (CQC_Subst (ll,Sub)) is Element of CQC-WFF Al
let P be QC-pred_symbol of k,Al; ::_thesis: for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds P ! (CQC_Subst (ll,Sub)) is Element of CQC-WFF Al
let ll be CQC-variable_list of k,Al; ::_thesis: for Sub being CQC_Substitution of Al holds P ! (CQC_Subst (ll,Sub)) is Element of CQC-WFF Al
let Sub be CQC_Substitution of Al; ::_thesis: P ! (CQC_Subst (ll,Sub)) is Element of CQC-WFF Al
 CQC_Sub (Sub_P (P,ll,Sub)) = P ! (CQC_Subst (ll,Sub)) by Th8;
hence  P ! (CQC_Subst (ll,Sub)) is Element of CQC-WFF Al ; ::_thesis: verum
end;

theorem Th10: :: SUBLEMMA:10
for Al being QC-alphabet
for k being Element of NAT
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds CQC_Subst (ll,Sub) is CQC-variable_list of k,Al
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds CQC_Subst (ll,Sub) is CQC-variable_list of k,Al
let k be Element of NAT ; ::_thesis: for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds CQC_Subst (ll,Sub) is CQC-variable_list of k,Al
let ll be CQC-variable_list of k,Al; ::_thesis: for Sub being CQC_Substitution of Al holds CQC_Subst (ll,Sub) is CQC-variable_list of k,Al
let Sub be CQC_Substitution of Al; ::_thesis: CQC_Subst (ll,Sub) is CQC-variable_list of k,Al
reconsider ll = ll as FinSequence of bound_QC-variables Al by SUBSTUT1:34;
reconsider s = CQC_Subst (ll,Sub) as FinSequence of bound_QC-variables Al ;
A1: s = CQC_Subst ((@ ll),Sub) by SUBSTUT1:def_5;
 len ll = k by CARD_1:def_7;
then  len (@ ll) = k by SUBSTUT1:def_4;
then  len s = k by A1, SUBSTUT1:def_3;
then  s is CQC-variable_list of k,Al by SUBSTUT1:34;
hence  CQC_Subst (ll,Sub) is CQC-variable_list of k,Al by A1, SUBSTUT1:def_4; ::_thesis: verum
end;

registration
let Al be QC-alphabet ;
let k be Element of NAT ;
let ll be CQC-variable_list of k,Al;
let Sub be CQC_Substitution of Al;
cluster CQC_Subst (ll,Sub) ->  bound_QC-variables Al -valued k -element ;
coherence
 ( CQC_Subst (ll,Sub) is bound_QC-variables Al -valued & CQC_Subst (ll,Sub) is k -element ) by Th10;
end;

theorem Th11: :: SUBLEMMA:11
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st not x in dom (S `2) holds
(v . (Val_S (v,S))) . x = v . x
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st not x in dom (S `2) holds
(v . (Val_S (v,S))) . x = v . x
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st not x in dom (S `2) holds
(v . (Val_S (v,S))) . x = v . x
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st not x in dom (S `2) holds
(v . (Val_S (v,S))) . x = v . x
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al st not x in dom (S `2) holds
(v . (Val_S (v,S))) . x = v . x
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( not x in dom (S `2) implies (v . (Val_S (v,S))) . x = v . x )
assume  not x in dom (S `2) ; ::_thesis: (v . (Val_S (v,S))) . x = v . x
then A1: not x in dom (@ (S `2)) by SUBSTUT1:def_2;
 dom ((@ (S `2)) * v) c= dom (@ (S `2)) by RELAT_1:25;
then  not x in dom (Val_S (v,S)) by A1;
hence  (v . (Val_S (v,S))) . x = v . x by FUNCT_4:11; ::_thesis: verum
end;

theorem Th12: :: SUBLEMMA:12
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st x in dom (S `2) holds
(v . (Val_S (v,S))) . x = (Val_S (v,S)) . x
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st x in dom (S `2) holds
(v . (Val_S (v,S))) . x = (Val_S (v,S)) . x
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st x in dom (S `2) holds
(v . (Val_S (v,S))) . x = (Val_S (v,S)) . x
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st x in dom (S `2) holds
(v . (Val_S (v,S))) . x = (Val_S (v,S)) . x
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al st x in dom (S `2) holds
(v . (Val_S (v,S))) . x = (Val_S (v,S)) . x
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( x in dom (S `2) implies (v . (Val_S (v,S))) . x = (Val_S (v,S)) . x )
assume  x in dom (S `2) ; ::_thesis: (v . (Val_S (v,S))) . x = (Val_S (v,S)) . x
then A1: x in dom (@ (S `2)) by SUBSTUT1:def_2;
 ( rng (@ (S `2)) c= bound_QC-variables Al & dom v = bound_QC-variables Al ) by FUNCT_2:def_1;
then  x in dom (Val_S (v,S)) by A1, RELAT_1:27;
hence  (v . (Val_S (v,S))) . x = (Val_S (v,S)) . x by FUNCT_4:13; ::_thesis: verum
end;

theorem Th13: :: SUBLEMMA:13
for Al being QC-alphabet
for k being Element of NAT
for A being non empty set
for v being Element of Valuations_in (Al,A)
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds (v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll = v *' (CQC_Subst (ll,Sub))
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for A being non empty set
for v being Element of Valuations_in (Al,A)
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds (v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll = v *' (CQC_Subst (ll,Sub))
let k be Element of NAT ; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds (v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll = v *' (CQC_Subst (ll,Sub))
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds (v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll = v *' (CQC_Subst (ll,Sub))
let v be Element of Valuations_in (Al,A); ::_thesis: for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds (v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll = v *' (CQC_Subst (ll,Sub))
let P be QC-pred_symbol of k,Al; ::_thesis: for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds (v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll = v *' (CQC_Subst (ll,Sub))
let ll be CQC-variable_list of k,Al; ::_thesis: for Sub being CQC_Substitution of Al holds (v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll = v *' (CQC_Subst (ll,Sub))
let Sub be CQC_Substitution of Al; ::_thesis: (v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll = v *' (CQC_Subst (ll,Sub))
set S9 = Sub_P (P,ll,Sub);
set ll9 = CQC_Subst (ll,Sub);
A1: len ll = k by CARD_1:def_7;
 Sub_P (P,ll,Sub) = [(P ! ll),Sub] by SUBSTUT1:9;
then A2: (Sub_P (P,ll,Sub)) `2 = Sub by MCART_1:7;
A3: len ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) = k by VALUAT_1:def_3;
then A4: dom ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) = Seg k by FINSEQ_1:def_3;
A5: for j being natural number st j in dom ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) holds
((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) . j = (v *' (CQC_Subst (ll,Sub))) . j
proof
let j be natural number ; ::_thesis: ( j in dom ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) implies ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) . j = (v *' (CQC_Subst (ll,Sub))) . j )
assume A6: j in dom ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) ; ::_thesis: ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) . j = (v *' (CQC_Subst (ll,Sub))) . j
A7: ( 1 <= j & j <= k ) by A4, A6, FINSEQ_1:1;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
 j in Seg (len ll) by A4, A6, CARD_1:def_7;
then  j in dom ll by FINSEQ_1:def_3;
then reconsider x = ll . j as bound_QC-variable of Al by Th5;
A8: now__::_thesis:_(_ll_._j_in_dom_Sub_implies_((v_._(Val_S_(v,(Sub_P_(P,ll,Sub)))))_*'_ll)_._j_=_(v_*'_(CQC_Subst_(ll,Sub)))_._j_)
assume A9: ll . j in dom Sub ; ::_thesis: ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) . j = (v *' (CQC_Subst (ll,Sub))) . j
then  ( (v . (Val_S (v,(Sub_P (P,ll,Sub))))) . (ll . j) = (Val_S (v,(Sub_P (P,ll,Sub)))) . x & ll . j in dom (@ ((Sub_P (P,ll,Sub)) `2)) ) by A2, Th12, SUBSTUT1:def_2;
then  (v . (Val_S (v,(Sub_P (P,ll,Sub))))) . (ll . j) = v . ((@ ((Sub_P (P,ll,Sub)) `2)) . (ll . j)) by FUNCT_1:13;
then A10: (v . (Val_S (v,(Sub_P (P,ll,Sub))))) . (ll . j) = v . (((Sub_P (P,ll,Sub)) `2) . (ll . j)) by SUBSTUT1:def_2;
A11: ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) . j = (v . (Val_S (v,(Sub_P (P,ll,Sub))))) . (ll . j) by A7, VALUAT_1:def_3;
 v . ((CQC_Subst (ll,Sub)) . j) = v . (((Sub_P (P,ll,Sub)) `2) . (ll . j)) by A2, A1, A7, A9, SUBSTUT1:def_3;
hence  ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) . j = (v *' (CQC_Subst (ll,Sub))) . j by A7, A10, A11, VALUAT_1:def_3; ::_thesis: verum
end;
now__::_thesis:_(_not_ll_._j_in_dom_Sub_implies_((v_._(Val_S_(v,(Sub_P_(P,ll,Sub)))))_*'_ll)_._j_=_(v_*'_(CQC_Subst_(ll,Sub)))_._j_)
assume  not ll . j in dom Sub ; ::_thesis: ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) . j = (v *' (CQC_Subst (ll,Sub))) . j
then A12: ( v . ((CQC_Subst (ll,Sub)) . j) = v . (ll . j) & (v . (Val_S (v,(Sub_P (P,ll,Sub))))) . (ll . j) = v . x ) by A2, A1, A7, Th11, SUBSTUT1:def_3;
 (v *' (CQC_Subst (ll,Sub))) . j = v . ((CQC_Subst (ll,Sub)) . j) by A7, VALUAT_1:def_3;
hence  ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) . j = (v *' (CQC_Subst (ll,Sub))) . j by A7, A12, VALUAT_1:def_3; ::_thesis: verum
end;
hence  ((v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll) . j = (v *' (CQC_Subst (ll,Sub))) . j by A8; ::_thesis: verum
end;
 len (v *' (CQC_Subst (ll,Sub))) = k by VALUAT_1:def_3;
hence  (v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll = v *' (CQC_Subst (ll,Sub)) by A3, A5, FINSEQ_2:9; ::_thesis: verum
end;

theorem Th14: :: SUBLEMMA:14
for Al being QC-alphabet
for k being Element of NAT
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds (Sub_P (P,ll,Sub)) `1 = P ! ll
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds (Sub_P (P,ll,Sub)) `1 = P ! ll
let k be Element of NAT ; ::_thesis: for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds (Sub_P (P,ll,Sub)) `1 = P ! ll
let P be QC-pred_symbol of k,Al; ::_thesis: for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al holds (Sub_P (P,ll,Sub)) `1 = P ! ll
let ll be CQC-variable_list of k,Al; ::_thesis: for Sub being CQC_Substitution of Al holds (Sub_P (P,ll,Sub)) `1 = P ! ll
let Sub be CQC_Substitution of Al; ::_thesis: (Sub_P (P,ll,Sub)) `1 = P ! ll
 Sub_P (P,ll,Sub) = [(P ! ll),Sub] by SUBSTUT1:9;
hence  (Sub_P (P,ll,Sub)) `1 = P ! ll by MCART_1:7; ::_thesis: verum
end;

theorem Th15: :: SUBLEMMA:15
for Al being QC-alphabet
for k being Element of NAT
for A being non empty set
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= Sub_P (P,ll,Sub) )
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for A being non empty set
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= Sub_P (P,ll,Sub) )
let k be Element of NAT ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= Sub_P (P,ll,Sub) )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= Sub_P (P,ll,Sub) )
let J be interpretation of Al,A; ::_thesis: for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= Sub_P (P,ll,Sub) )
let P be QC-pred_symbol of k,Al; ::_thesis: for ll being CQC-variable_list of k,Al
for Sub being CQC_Substitution of Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= Sub_P (P,ll,Sub) )
let ll be CQC-variable_list of k,Al; ::_thesis: for Sub being CQC_Substitution of Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= Sub_P (P,ll,Sub) )
let Sub be CQC_Substitution of Al; ::_thesis: for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= Sub_P (P,ll,Sub) )
set S9 = Sub_P (P,ll,Sub);
set ll9 = CQC_Subst (ll,Sub);
reconsider p = P ! (CQC_Subst (ll,Sub)) as Element of CQC-WFF Al ;
reconsider ll9 = CQC_Subst (ll,Sub) as CQC-variable_list of k,Al ;
let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= Sub_P (P,ll,Sub) )
A1: ( (Valid (p,J)) . v = TRUE iff v *' ll9 in J . P ) by VALUAT_1:7;
A2: ( (v . (Val_S (v,(Sub_P (P,ll,Sub))))) *' ll in J . P iff (Valid ((P ! ll),J)) . (v . (Val_S (v,(Sub_P (P,ll,Sub))))) = TRUE ) by VALUAT_1:7;
A3: ( J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= P ! ll iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= (Sub_P (P,ll,Sub)) `1 ) by Th14;
 ( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v |= p ) by Th8;
hence  ( J,v |= CQC_Sub (Sub_P (P,ll,Sub)) iff J,v . (Val_S (v,(Sub_P (P,ll,Sub)))) |= Sub_P (P,ll,Sub) ) by A1, A2, A3, Def2, Th13, VALUAT_1:def_7; ::_thesis: verum
end;

theorem Th16: :: SUBLEMMA:16
for Al being QC-alphabet
for S being Element of CQC-Sub-WFF Al holds
( (Sub_not S) `1 = 'not' (S `1) & (Sub_not S) `2 = S `2 )
proof
let Al be QC-alphabet ; ::_thesis: for S being Element of CQC-Sub-WFF Al holds
( (Sub_not S) `1 = 'not' (S `1) & (Sub_not S) `2 = S `2 )
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( (Sub_not S) `1 = 'not' (S `1) & (Sub_not S) `2 = S `2 )
 Sub_not S = [('not' (S `1)),(S `2)] by SUBSTUT1:def_20;
hence  ( (Sub_not S) `1 = 'not' (S `1) & (Sub_not S) `2 = S `2 ) by MCART_1:7; ::_thesis: verum
end;

definition
let Al be QC-alphabet ;
let S be Element of CQC-Sub-WFF Al;
:: original: Sub_not
redefine func Sub_not S ->  Element of CQC-Sub-WFF Al;
coherence
 Sub_not S is Element of CQC-Sub-WFF Al
proof
set X =  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  ;
  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  = CQC-Sub-WFF Al by SUBSTUT1:def_39;
then A1: for G being Element of QC-Sub-WFF Al st G `1 is Element of CQC-WFF Al holds
G in CQC-Sub-WFF Al ;
 'not' (S `1) in CQC-WFF Al ;
then  (Sub_not S) `1 in CQC-WFF Al by Th16;
hence  Sub_not S is Element of CQC-Sub-WFF Al by A1; ::_thesis: verum
end;
end;

theorem Th17: :: SUBLEMMA:17
for Al being QC-alphabet
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al holds
( not J,v . (Val_S (v,S)) |= S iff J,v . (Val_S (v,S)) |= Sub_not S )
proof
let Al be QC-alphabet ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al holds
( not J,v . (Val_S (v,S)) |= S iff J,v . (Val_S (v,S)) |= Sub_not S )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al holds
( not J,v . (Val_S (v,S)) |= S iff J,v . (Val_S (v,S)) |= Sub_not S )
let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al holds
( not J,v . (Val_S (v,S)) |= S iff J,v . (Val_S (v,S)) |= Sub_not S )
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al holds
( not J,v . (Val_S (v,S)) |= S iff J,v . (Val_S (v,S)) |= Sub_not S )
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( not J,v . (Val_S (v,S)) |= S iff J,v . (Val_S (v,S)) |= Sub_not S )
A1: ( not J,v . (Val_S (v,S)) |= S `1 iff J,v . (Val_S (v,S)) |= 'not' (S `1) ) by VALUAT_1:17;
 ( J,v . (Val_S (v,S)) |= 'not' (S `1) iff J,v . (Val_S (v,S)) |= (Sub_not S) `1 ) by Th16;
hence  ( not J,v . (Val_S (v,S)) |= S iff J,v . (Val_S (v,S)) |= Sub_not S ) by A1, Def2; ::_thesis: verum
end;

theorem Th18: :: SUBLEMMA:18
for Al being QC-alphabet
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al holds Val_S (v,S) = Val_S (v,(Sub_not S))
proof
let Al be QC-alphabet ; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al holds Val_S (v,S) = Val_S (v,(Sub_not S))
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al holds Val_S (v,S) = Val_S (v,(Sub_not S))
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al holds Val_S (v,S) = Val_S (v,(Sub_not S))
let S be Element of CQC-Sub-WFF Al; ::_thesis: Val_S (v,S) = Val_S (v,(Sub_not S))
 Sub_not S = [('not' (S `1)),(S `2)] by SUBSTUT1:def_20;
hence  Val_S (v,S) = Val_S (v,(Sub_not S)) by MCART_1:7; ::_thesis: verum
end;

theorem Th19: :: SUBLEMMA:19
for Al being QC-alphabet
for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al st ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S (v,(Sub_not S))) |= Sub_not S )
proof
let Al be QC-alphabet ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al st ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S (v,(Sub_not S))) |= Sub_not S )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al st ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S (v,(Sub_not S))) |= Sub_not S )
let J be interpretation of Al,A; ::_thesis: for S being Element of CQC-Sub-WFF Al st ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S (v,(Sub_not S))) |= Sub_not S )
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) implies for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S (v,(Sub_not S))) |= Sub_not S ) )
assume A1: for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ; ::_thesis: for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S (v,(Sub_not S))) |= Sub_not S )
let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S (v,(Sub_not S))) |= Sub_not S )
A2: ( J,v |= 'not' (CQC_Sub S) iff not J,v |= CQC_Sub S ) by VALUAT_1:17;
 ( not J,v . (Val_S (v,S)) |= S iff J,v . (Val_S (v,S)) |= Sub_not S ) by Th17;
hence  ( J,v |= CQC_Sub (Sub_not S) iff J,v . (Val_S (v,(Sub_not S))) |= Sub_not S ) by A1, A2, Th18, SUBSTUT1:29; ::_thesis: verum
end;

definition
let Al be QC-alphabet ;
let S1, S2 be Element of CQC-Sub-WFF Al;
assume A1: S1 `2 = S2 `2 ;
func CQCSub_& (S1,S2) ->  Element of CQC-Sub-WFF Al equals :Def3: :: SUBLEMMA:def 3
 Sub_& (S1,S2);
coherence
 Sub_& (S1,S2) is Element of CQC-Sub-WFF Al
proof
set X =  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  ;
  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  = CQC-Sub-WFF Al by SUBSTUT1:def_39;
then A2: for G being Element of QC-Sub-WFF Al st G `1 is Element of CQC-WFF Al holds
G in CQC-Sub-WFF Al ;
 Sub_& (S1,S2) = [((S1 `1) '&' (S2 `1)),(S1 `2)] by A1, SUBSTUT1:def_21;
then  (Sub_& (S1,S2)) `1 = (S1 `1) '&' (S2 `1) by MCART_1:7;
hence  Sub_& (S1,S2) is Element of CQC-Sub-WFF Al by A2; ::_thesis: verum
end;
end;

:: deftheorem Def3 defines CQCSub_& SUBLEMMA:def_3_:_
 for Al being QC-alphabet
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
CQCSub_& (S1,S2) = Sub_& (S1,S2);

theorem Th20: :: SUBLEMMA:20
for Al being QC-alphabet
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
( (CQCSub_& (S1,S2)) `1 = (S1 `1) '&' (S2 `1) & (CQCSub_& (S1,S2)) `2 = S1 `2 )
proof
let Al be QC-alphabet ; ::_thesis: for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
( (CQCSub_& (S1,S2)) `1 = (S1 `1) '&' (S2 `1) & (CQCSub_& (S1,S2)) `2 = S1 `2 )
let S1, S2 be Element of CQC-Sub-WFF Al; ::_thesis: ( S1 `2 = S2 `2 implies ( (CQCSub_& (S1,S2)) `1 = (S1 `1) '&' (S2 `1) & (CQCSub_& (S1,S2)) `2 = S1 `2 ) )
assume A1: S1 `2 = S2 `2 ; ::_thesis: ( (CQCSub_& (S1,S2)) `1 = (S1 `1) '&' (S2 `1) & (CQCSub_& (S1,S2)) `2 = S1 `2 )
then  Sub_& (S1,S2) = [((S1 `1) '&' (S2 `1)),(S1 `2)] by SUBSTUT1:def_21;
then  CQCSub_& (S1,S2) = [((S1 `1) '&' (S2 `1)),(S1 `2)] by A1, Def3;
hence  ( (CQCSub_& (S1,S2)) `1 = (S1 `1) '&' (S2 `1) & (CQCSub_& (S1,S2)) `2 = S1 `2 ) by MCART_1:7; ::_thesis: verum
end;

theorem Th21: :: SUBLEMMA:21
for Al being QC-alphabet
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
(CQCSub_& (S1,S2)) `2 = S1 `2
proof
let Al be QC-alphabet ; ::_thesis: for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
(CQCSub_& (S1,S2)) `2 = S1 `2
let S1, S2 be Element of CQC-Sub-WFF Al; ::_thesis: ( S1 `2 = S2 `2 implies (CQCSub_& (S1,S2)) `2 = S1 `2 )
assume A1: S1 `2 = S2 `2 ; ::_thesis: (CQCSub_& (S1,S2)) `2 = S1 `2
then  CQCSub_& (S1,S2) = Sub_& (S1,S2) by Def3;
then  CQCSub_& (S1,S2) = [((S1 `1) '&' (S2 `1)),(S1 `2)] by A1, SUBSTUT1:def_21;
hence  (CQCSub_& (S1,S2)) `2 = S1 `2 by MCART_1:7; ::_thesis: verum
end;

theorem :: SUBLEMMA:22
for Al being QC-alphabet
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
( Val_S (v,S1) = Val_S (v,(CQCSub_& (S1,S2))) & Val_S (v,S2) = Val_S (v,(CQCSub_& (S1,S2))) ) by Th21;

theorem Th23: :: SUBLEMMA:23
for Al being QC-alphabet
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
CQC_Sub (CQCSub_& (S1,S2)) = (CQC_Sub S1) '&' (CQC_Sub S2)
proof
let Al be QC-alphabet ; ::_thesis: for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
CQC_Sub (CQCSub_& (S1,S2)) = (CQC_Sub S1) '&' (CQC_Sub S2)
let S1, S2 be Element of CQC-Sub-WFF Al; ::_thesis: ( S1 `2 = S2 `2 implies CQC_Sub (CQCSub_& (S1,S2)) = (CQC_Sub S1) '&' (CQC_Sub S2) )
assume A1: S1 `2 = S2 `2 ; ::_thesis: CQC_Sub (CQCSub_& (S1,S2)) = (CQC_Sub S1) '&' (CQC_Sub S2)
then  CQCSub_& (S1,S2) = Sub_& (S1,S2) by Def3;
hence  CQC_Sub (CQCSub_& (S1,S2)) = (CQC_Sub S1) '&' (CQC_Sub S2) by A1, SUBSTUT1:31; ::_thesis: verum
end;

theorem Th24: :: SUBLEMMA:24
for Al being QC-alphabet
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
( ( J,v . (Val_S (v,S1)) |= S1 & J,v . (Val_S (v,S2)) |= S2 ) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
proof
let Al be QC-alphabet ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
( ( J,v . (Val_S (v,S1)) |= S1 & J,v . (Val_S (v,S2)) |= S2 ) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
( ( J,v . (Val_S (v,S1)) |= S1 & J,v . (Val_S (v,S2)) |= S2 ) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A)
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
( ( J,v . (Val_S (v,S1)) |= S1 & J,v . (Val_S (v,S2)) |= S2 ) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
let v be Element of Valuations_in (Al,A); ::_thesis: for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 holds
( ( J,v . (Val_S (v,S1)) |= S1 & J,v . (Val_S (v,S2)) |= S2 ) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
let S1, S2 be Element of CQC-Sub-WFF Al; ::_thesis: ( S1 `2 = S2 `2 implies ( ( J,v . (Val_S (v,S1)) |= S1 & J,v . (Val_S (v,S2)) |= S2 ) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) ) )
assume A1: S1 `2 = S2 `2 ; ::_thesis: ( ( J,v . (Val_S (v,S1)) |= S1 & J,v . (Val_S (v,S2)) |= S2 ) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
then  Val_S (v,S1) = Val_S (v,(CQCSub_& (S1,S2))) by Th21;
then A2: ( ( J,v . (Val_S (v,S1)) |= S1 `1 & J,v . (Val_S (v,S1)) |= S2 `1 ) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= (S1 `1) '&' (S2 `1) ) by VALUAT_1:18;
 ( J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= (S1 `1) '&' (S2 `1) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= (CQCSub_& (S1,S2)) `1 ) by A1, Th20;
hence  ( ( J,v . (Val_S (v,S1)) |= S1 & J,v . (Val_S (v,S2)) |= S2 ) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) ) by A1, A2, Def2; ::_thesis: verum
end;

theorem Th25: :: SUBLEMMA:25
for Al being QC-alphabet
for A being non empty set
for J being interpretation of Al,A
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S1 iff J,v . (Val_S (v,S1)) |= S1 ) ) & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S2 iff J,v . (Val_S (v,S2)) |= S2 ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
proof
let Al be QC-alphabet ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S1 iff J,v . (Val_S (v,S1)) |= S1 ) ) & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S2 iff J,v . (Val_S (v,S2)) |= S2 ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S1 iff J,v . (Val_S (v,S1)) |= S1 ) ) & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S2 iff J,v . (Val_S (v,S2)) |= S2 ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
let J be interpretation of Al,A; ::_thesis: for S1, S2 being Element of CQC-Sub-WFF Al st S1 `2 = S2 `2 & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S1 iff J,v . (Val_S (v,S1)) |= S1 ) ) & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S2 iff J,v . (Val_S (v,S2)) |= S2 ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
let S1, S2 be Element of CQC-Sub-WFF Al; ::_thesis: ( S1 `2 = S2 `2 & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S1 iff J,v . (Val_S (v,S1)) |= S1 ) ) & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S2 iff J,v . (Val_S (v,S2)) |= S2 ) ) implies for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) ) )
assume that
A1: S1 `2 = S2 `2 and
A2: ( ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S1 iff J,v . (Val_S (v,S1)) |= S1 ) ) & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S2 iff J,v . (Val_S (v,S2)) |= S2 ) ) ) ; ::_thesis: for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) )
A3: ( J,v |= CQC_Sub S1 & J,v |= CQC_Sub S2 iff ( J,v . (Val_S (v,S1)) |= S1 & J,v . (Val_S (v,S2)) |= S2 ) ) by A2;
 ( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v |= (CQC_Sub S1) '&' (CQC_Sub S2) ) by A1, Th23;
hence  ( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) ) by A1, A3, Th24, VALUAT_1:18; ::_thesis: verum
end;


theorem Th26: :: SUBLEMMA:26
for Al being QC-alphabet
for B being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B st B is quantifiable holds
( (Sub_All (B,SQ)) `1 = All ((B `2),((B `1) `1)) & (Sub_All (B,SQ)) `2 = SQ )
proof
let Al be QC-alphabet ; ::_thesis: for B being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B st B is quantifiable holds
( (Sub_All (B,SQ)) `1 = All ((B `2),((B `1) `1)) & (Sub_All (B,SQ)) `2 = SQ )
let B be Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]; ::_thesis: for SQ being second_Q_comp of B st B is quantifiable holds
( (Sub_All (B,SQ)) `1 = All ((B `2),((B `1) `1)) & (Sub_All (B,SQ)) `2 = SQ )
let SQ be second_Q_comp of B; ::_thesis: ( B is quantifiable implies ( (Sub_All (B,SQ)) `1 = All ((B `2),((B `1) `1)) & (Sub_All (B,SQ)) `2 = SQ ) )
assume  B is quantifiable ; ::_thesis: ( (Sub_All (B,SQ)) `1 = All ((B `2),((B `1) `1)) & (Sub_All (B,SQ)) `2 = SQ )
then  Sub_All (B,SQ) = [(All ((B `2),((B `1) `1))),SQ] by SUBSTUT1:def_24;
hence  ( (Sub_All (B,SQ)) `1 = All ((B `2),((B `1) `1)) & (Sub_All (B,SQ)) `2 = SQ ) by MCART_1:7; ::_thesis: verum
end;

definition
let Al be QC-alphabet ;
let B be Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):];
attrB is CQC-WFF-like  means :Def4: :: SUBLEMMA:def 4
B `1 in CQC-Sub-WFF Al;
end;

:: deftheorem Def4 defines CQC-WFF-like SUBLEMMA:def_4_:_
 for Al being QC-alphabet
for B being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):] holds
( B is CQC-WFF-like iff B `1 in CQC-Sub-WFF Al );

registration
let Al be QC-alphabet ;
cluster CQC-WFF-like for Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):];
existence
 ex b1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):] st b1 is CQC-WFF-like
proof
set Sub = the CQC_Substitution of Al;
set x = the bound_QC-variable of Al;
set B = [[(VERUM Al), the CQC_Substitution of Al], the bound_QC-variable of Al];
A1: VERUM Al = <*[0,0]*> by QC_LANG1:def_14;
A2: [<*[0,0]*>, the CQC_Substitution of Al] in QC-Sub-WFF Al by SUBSTUT1:def_16;
reconsider S = [(VERUM Al), the CQC_Substitution of Al] as Element of QC-Sub-WFF Al by A1, SUBSTUT1:def_16;
 [(VERUM Al), the CQC_Substitution of Al] in QC-Sub-WFF Al by A2, QC_LANG1:def_14;
then reconsider B = [[(VERUM Al), the CQC_Substitution of Al], the bound_QC-variable of Al] as Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):] by ZFMISC_1:87;
take  B ; ::_thesis: B is CQC-WFF-like
set X =  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  ;
A3:  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  = CQC-Sub-WFF Al by SUBSTUT1:def_39;
 S `1 is Element of CQC-WFF Al by MCART_1:7;
then  S in CQC-Sub-WFF Al by A3;
then  B `1 in CQC-Sub-WFF Al by MCART_1:7;
hence  B is CQC-WFF-like by Def4; ::_thesis: verum
end;
end;

definition
let Al be QC-alphabet ;
let S be Element of CQC-Sub-WFF Al;
let x be bound_QC-variable of Al;
:: original: [
redefine func[S,x] ->  CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):];
coherence
 [S,x] is CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
proof
 [S,x] `1 = S ;
hence  [S,x] is CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):] by Def4; ::_thesis: verum
end;
end;


definition
let Al be QC-alphabet ;
let B be CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):];
:: original: `1
redefine funcB `1  ->  Element of CQC-Sub-WFF Al;
coherence
 B `1 is Element of CQC-Sub-WFF Al by Def4;
end;

definition
let Al be QC-alphabet ;
let B be CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):];
let SQ be second_Q_comp of B;
assume A1: B is quantifiable ;
func CQCSub_All (B,SQ) ->  Element of CQC-Sub-WFF Al equals :Def5: :: SUBLEMMA:def 5
 Sub_All (B,SQ);
coherence
 Sub_All (B,SQ) is Element of CQC-Sub-WFF Al
proof
set X =  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  ;
  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  = CQC-Sub-WFF Al by SUBSTUT1:def_39;
then A2: for G being Element of QC-Sub-WFF Al st G `1 is Element of CQC-WFF Al holds
G in CQC-Sub-WFF Al ;
 All ((B `2),((B `1) `1)) in CQC-WFF Al ;
then  (Sub_All (B,SQ)) `1 in CQC-WFF Al by A1, Th26;
hence  Sub_All (B,SQ) is Element of CQC-Sub-WFF Al by A2; ::_thesis: verum
end;
end;

:: deftheorem Def5 defines CQCSub_All SUBLEMMA:def_5_:_
 for Al being QC-alphabet
for B being CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B st B is quantifiable holds
CQCSub_All (B,SQ) = Sub_All (B,SQ);

theorem Th27: :: SUBLEMMA:27
for Al being QC-alphabet
for B being CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B st B is quantifiable holds
CQCSub_All (B,SQ) is Sub_universal
proof
let Al be QC-alphabet ; ::_thesis: for B being CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B st B is quantifiable holds
CQCSub_All (B,SQ) is Sub_universal
let B be CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]; ::_thesis: for SQ being second_Q_comp of B st B is quantifiable holds
CQCSub_All (B,SQ) is Sub_universal
let SQ be second_Q_comp of B; ::_thesis: ( B is quantifiable implies CQCSub_All (B,SQ) is Sub_universal )
assume A1: B is quantifiable ; ::_thesis: CQCSub_All (B,SQ) is Sub_universal
then  Sub_All (B,SQ) is Sub_universal by SUBSTUT1:14;
hence  CQCSub_All (B,SQ) is Sub_universal by A1, Def5; ::_thesis: verum
end;

definition
let Al be QC-alphabet ;
let S be Element of CQC-Sub-WFF Al;
assume X1: S is Sub_universal ;
func CQCSub_the_scope_of S ->  Element of CQC-Sub-WFF Al equals :Def6: :: SUBLEMMA:def 6
 Sub_the_scope_of S;
coherence
 Sub_the_scope_of S is Element of CQC-Sub-WFF Al
proof
consider B being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):], SQ being second_Q_comp of B such that
A1: S = Sub_All (B,SQ) and
A2: B `1 = Sub_the_scope_of S and
A3: B is quantifiable by X1, SUBSTUT1:def_34;
set X =  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  ;
  {  G where G is Element of QC-Sub-WFF Al : G `1 is Element of CQC-WFF Al  }  = CQC-Sub-WFF Al by SUBSTUT1:def_39;
then A4: for G being Element of QC-Sub-WFF Al st G `1 is Element of CQC-WFF Al holds
G in CQC-Sub-WFF Al ;
 S `1 = All ((B `2),((B `1) `1)) by A1, A3, Th26;
then  (B `1) `1 is Element of CQC-WFF Al by CQC_LANG:13;
hence  Sub_the_scope_of S is Element of CQC-Sub-WFF Al by A4, A2; ::_thesis: verum
end;
end;

:: deftheorem Def6 defines CQCSub_the_scope_of SUBLEMMA:def_6_:_
 for Al being QC-alphabet
for S being Element of CQC-Sub-WFF Al st S is Sub_universal holds
CQCSub_the_scope_of S = Sub_the_scope_of S;

definition
let Al be QC-alphabet ;
let S1 be Element of CQC-Sub-WFF Al;
let p be Element of CQC-WFF Al;
assume that
A1: S1 is Sub_universal and
A2: p = CQC_Sub (CQCSub_the_scope_of S1) ;
func CQCQuant (S1,p) ->  Element of CQC-WFF Al equals :Def7: :: SUBLEMMA:def 7
 Quant (S1,p);
coherence
 Quant (S1,p) is Element of CQC-WFF Al
proof
 CQCSub_the_scope_of S1 = Sub_the_scope_of S1 by A1, Def6;
then  CQC_Sub S1 = Quant (S1,p) by A1, A2, SUBSTUT1:32;
hence  Quant (S1,p) is Element of CQC-WFF Al ; ::_thesis: verum
end;
end;

:: deftheorem Def7 defines CQCQuant SUBLEMMA:def_7_:_
 for Al being QC-alphabet
for S1 being Element of CQC-Sub-WFF Al
for p being Element of CQC-WFF Al st S1 is Sub_universal & p = CQC_Sub (CQCSub_the_scope_of S1) holds
CQCQuant (S1,p) = Quant (S1,p);

theorem Th28: :: SUBLEMMA:28
for Al being QC-alphabet
for S being Element of CQC-Sub-WFF Al st S is Sub_universal holds
CQC_Sub S = CQCQuant (S,(CQC_Sub (CQCSub_the_scope_of S)))
proof
let Al be QC-alphabet ; ::_thesis: for S being Element of CQC-Sub-WFF Al st S is Sub_universal holds
CQC_Sub S = CQCQuant (S,(CQC_Sub (CQCSub_the_scope_of S)))
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( S is Sub_universal implies CQC_Sub S = CQCQuant (S,(CQC_Sub (CQCSub_the_scope_of S))) )
assume A1: S is Sub_universal ; ::_thesis: CQC_Sub S = CQCQuant (S,(CQC_Sub (CQCSub_the_scope_of S)))
then  CQCSub_the_scope_of S = Sub_the_scope_of S by Def6;
then  CQCQuant (S,(CQC_Sub (CQCSub_the_scope_of S))) = Quant (S,(CQC_Sub (Sub_the_scope_of S))) by A1, Def7;
hence  CQC_Sub S = CQCQuant (S,(CQC_Sub (CQCSub_the_scope_of S))) by A1, SUBSTUT1:32; ::_thesis: verum
end;

theorem Th29: :: SUBLEMMA:29
for Al being QC-alphabet
for B being CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B st B is quantifiable holds
CQCSub_the_scope_of (CQCSub_All (B,SQ)) = B `1
proof
let Al be QC-alphabet ; ::_thesis: for B being CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B st B is quantifiable holds
CQCSub_the_scope_of (CQCSub_All (B,SQ)) = B `1
let B be CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]; ::_thesis: for SQ being second_Q_comp of B st B is quantifiable holds
CQCSub_the_scope_of (CQCSub_All (B,SQ)) = B `1
let SQ be second_Q_comp of B; ::_thesis: ( B is quantifiable implies CQCSub_the_scope_of (CQCSub_All (B,SQ)) = B `1 )
assume A1: B is quantifiable ; ::_thesis: CQCSub_the_scope_of (CQCSub_All (B,SQ)) = B `1
then A2: CQCSub_All (B,SQ) = Sub_All (B,SQ) by Def5;
then  CQCSub_All (B,SQ) is Sub_universal by A1, SUBSTUT1:14;
then  CQCSub_the_scope_of (CQCSub_All (B,SQ)) = Sub_the_scope_of (Sub_All (B,SQ)) by A2, Def6;
hence  CQCSub_the_scope_of (CQCSub_All (B,SQ)) = B `1 by A1, SUBSTUT1:21; ::_thesis: verum
end;

begin

theorem Th30: :: SUBLEMMA:30
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) )
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies ( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) ) )
assume  [S,x] is quantifiable ; ::_thesis: ( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) )
then  CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = [S,x] `1 by Th29;
hence  ( CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)) = S & CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) ) ; ::_thesis: verum
end;

theorem Th31: :: SUBLEMMA:31
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) = All ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))),(CQC_Sub S))
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) = All ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))),(CQC_Sub S))
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) = All ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))),(CQC_Sub S))
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) = All ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))),(CQC_Sub S))
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) = All ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))),(CQC_Sub S)) )
set S1 = CQCSub_All ([S,x],xSQ);
set p = CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)));
A1: Quant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = All ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) by SUBSTUT1:def_37;
assume A2: [S,x] is quantifiable ; ::_thesis: CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) = All ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))),(CQC_Sub S))
then  CQCSub_All ([S,x],xSQ) = Sub_All ([S,x],xSQ) by Def5;
then  CQCSub_All ([S,x],xSQ) is Sub_universal by A2, SUBSTUT1:14;
then A3: CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) = Quant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) by Def7;
 CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) = CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) by A2, Th30;
hence  CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) = All ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))),(CQC_Sub S)) by A2, A3, A1, Th30; ::_thesis: verum
end;

theorem :: SUBLEMMA:32
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st x in dom (S `2) holds
v . ((@ (S `2)) . x) = (v . (Val_S (v,S))) . x
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st x in dom (S `2) holds
v . ((@ (S `2)) . x) = (v . (Val_S (v,S))) . x
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st x in dom (S `2) holds
v . ((@ (S `2)) . x) = (v . (Val_S (v,S))) . x
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al st x in dom (S `2) holds
v . ((@ (S `2)) . x) = (v . (Val_S (v,S))) . x
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al st x in dom (S `2) holds
v . ((@ (S `2)) . x) = (v . (Val_S (v,S))) . x
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( x in dom (S `2) implies v . ((@ (S `2)) . x) = (v . (Val_S (v,S))) . x )
assume  x in dom (S `2) ; ::_thesis: v . ((@ (S `2)) . x) = (v . (Val_S (v,S))) . x
then  ( (v . (Val_S (v,S))) . x = (Val_S (v,S)) . x & x in dom (@ (S `2)) ) by Th12, SUBSTUT1:def_2;
hence  v . ((@ (S `2)) . x) = (v . (Val_S (v,S))) . x by FUNCT_1:13; ::_thesis: verum
end;

theorem :: SUBLEMMA:33
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al st x in dom (@ (S `2)) holds
(@ (S `2)) . x is bound_QC-variable of Al
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al st x in dom (@ (S `2)) holds
(@ (S `2)) . x is bound_QC-variable of Al
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al st x in dom (@ (S `2)) holds
(@ (S `2)) . x is bound_QC-variable of Al
let S be Element of CQC-Sub-WFF Al; ::_thesis: ( x in dom (@ (S `2)) implies (@ (S `2)) . x is bound_QC-variable of Al )
assume  x in dom (@ (S `2)) ; ::_thesis: (@ (S `2)) . x is bound_QC-variable of Al
then  (@ (S `2)) . x in rng (@ (S `2)) by FUNCT_1:3;
hence  (@ (S `2)) . x is bound_QC-variable of Al ; ::_thesis: verum
end;

theorem Th34: :: SUBLEMMA:34
for Al being QC-alphabet holds [:(QC-WFF Al),(vSUB Al):] c= dom (QSub Al)
proof
let Al be QC-alphabet ; ::_thesis: [:(QC-WFF Al),(vSUB Al):] c= dom (QSub Al)
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in [:(QC-WFF Al),(vSUB Al):] or a in dom (QSub Al) )
assume  a in [:(QC-WFF Al),(vSUB Al):] ; ::_thesis: a in dom (QSub Al)
then consider b, c being set  such that
A1: b in QC-WFF Al and
A2: c in vSUB Al and
A3: a = [b,c] by ZFMISC_1:def_2;
reconsider Sub = c as CQC_Substitution of Al by A2;
reconsider p = b as Element of QC-WFF Al by A1;
A4: now__::_thesis:_(_not_p_is_universal_implies_a_in_dom_(QSub_Al)_)
set b = {} ;
set a = [[p,Sub],{}];
assume  not p is universal ; ::_thesis: a in dom (QSub Al)
then  p,Sub PQSub {} by SUBSTUT1:def_14;
then  [[p,Sub],{}] in QSub Al by SUBSTUT1:def_15;
hence  a in dom (QSub Al) by A3, FUNCT_1:1; ::_thesis: verum
end;
now__::_thesis:_(_p_is_universal_implies_a_in_dom_(QSub_Al)_)
set b = ExpandSub ((bound_in p),(the_scope_of p),(RestrictSub ((bound_in p),p,Sub)));
set a = [[p,Sub],(ExpandSub ((bound_in p),(the_scope_of p),(RestrictSub ((bound_in p),p,Sub))))];
assume  p is universal ; ::_thesis: a in dom (QSub Al)
then  p,Sub PQSub ExpandSub ((bound_in p),(the_scope_of p),(RestrictSub ((bound_in p),p,Sub))) by SUBSTUT1:def_14;
then  [[p,Sub],(ExpandSub ((bound_in p),(the_scope_of p),(RestrictSub ((bound_in p),p,Sub))))] in QSub Al by SUBSTUT1:def_15;
hence  a in dom (QSub Al) by A3, FUNCT_1:1; ::_thesis: verum
end;
hence  a in dom (QSub Al) by A4; ::_thesis: verum
end;


theorem Th35: :: SUBLEMMA:35
for Al being QC-alphabet
for B being CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B
for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
proof
let Al be QC-alphabet ; ::_thesis: for B being CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B
for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let B be CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]; ::_thesis: for SQ being second_Q_comp of B
for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let SQ be second_Q_comp of B; ::_thesis: for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let B1 be Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]; ::_thesis: for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let SQ1 be second_Q_comp of B1; ::_thesis: ( B is quantifiable & B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) implies ( B `2 = B1 `2 & SQ = SQ1 ) )
assume that
A1: B is quantifiable and
A2: ( B1 is quantifiable & Sub_All (B,SQ) = Sub_All (B1,SQ1) ) ; ::_thesis: ( B `2 = B1 `2 & SQ = SQ1 )
 Sub_All (B,SQ) = [(All ((B `2),((B `1) `1))),SQ] by A1, SUBSTUT1:def_24;
then A3: [(All ((B `2),((B `1) `1))),SQ] = [(All ((B1 `2),((B1 `1) `1))),SQ1] by A2, SUBSTUT1:def_24;
then  All ((B `2),((B `1) `1)) = All ((B1 `2),((B1 `1) `1)) by XTUPLE_0:1;
hence  ( B `2 = B1 `2 & SQ = SQ1 ) by A3, QC_LANG2:5, XTUPLE_0:1; ::_thesis: verum
end;

theorem Th36: :: SUBLEMMA:36
for Al being QC-alphabet
for B being CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B
for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & CQCSub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
proof
let Al be QC-alphabet ; ::_thesis: for B being CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ being second_Q_comp of B
for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & CQCSub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let B be CQC-WFF-like Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]; ::_thesis: for SQ being second_Q_comp of B
for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & CQCSub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let SQ be second_Q_comp of B; ::_thesis: for B1 being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]
for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & CQCSub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let B1 be Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):]; ::_thesis: for SQ1 being second_Q_comp of B1 st B is quantifiable & B1 is quantifiable & CQCSub_All (B,SQ) = Sub_All (B1,SQ1) holds
( B `2 = B1 `2 & SQ = SQ1 )
let SQ1 be second_Q_comp of B1; ::_thesis: ( B is quantifiable & B1 is quantifiable & CQCSub_All (B,SQ) = Sub_All (B1,SQ1) implies ( B `2 = B1 `2 & SQ = SQ1 ) )
assume that
A1: B is quantifiable and
A2: B1 is quantifiable and
A3: CQCSub_All (B,SQ) = Sub_All (B1,SQ1) ; ::_thesis: ( B `2 = B1 `2 & SQ = SQ1 )
 Sub_All (B,SQ) = Sub_All (B1,SQ1) by A1, A3, Def5;
hence  ( B `2 = B1 `2 & SQ = SQ1 ) by A1, A2, Th35; ::_thesis: verum
end;

theorem :: SUBLEMMA:37
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
Sub_the_bound_of (CQCSub_All ([S,x],xSQ)) = x
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
Sub_the_bound_of (CQCSub_All ([S,x],xSQ)) = x
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
Sub_the_bound_of (CQCSub_All ([S,x],xSQ)) = x
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
Sub_the_bound_of (CQCSub_All ([S,x],xSQ)) = x
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies Sub_the_bound_of (CQCSub_All ([S,x],xSQ)) = x )
set S1 = CQCSub_All ([S,x],xSQ);
assume A1: [S,x] is quantifiable ; ::_thesis: Sub_the_bound_of (CQCSub_All ([S,x],xSQ)) = x
then  CQCSub_All ([S,x],xSQ) = Sub_All ([S,x],xSQ) by Def5;
then  CQCSub_All ([S,x],xSQ) is Sub_universal by A1, SUBSTUT1:14;
then consider B being Element of [:(QC-Sub-WFF Al),(bound_QC-variables Al):], SQ being second_Q_comp of B such that
A2: CQCSub_All ([S,x],xSQ) = Sub_All (B,SQ) and
A3: B `2 = Sub_the_bound_of (CQCSub_All ([S,x],xSQ)) and
A4: B is quantifiable by SUBSTUT1:def_33;
 [S,x] `2 = B `2 by A1, A2, A4, Th36;
hence  Sub_the_bound_of (CQCSub_All ([S,x],xSQ)) = x by A3; ::_thesis: verum
end;

theorem Th38: :: SUBLEMMA:38
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
( not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) & not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in Bound_Vars (S `1) )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
( not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) & not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in Bound_Vars (S `1) )
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
( not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) & not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in Bound_Vars (S `1) )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
( not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) & not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in Bound_Vars (S `1) )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable & x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) implies ( not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) & not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in Bound_Vars (S `1) ) )
set S1 = CQCSub_All ([S,x],xSQ);
assume that
A1: [S,x] is quantifiable and
A2: x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) ; ::_thesis: ( not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) & not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in Bound_Vars (S `1) )
A3: CQCSub_All ([S,x],xSQ) = Sub_All ([S,x],xSQ) by A1, Def5;
then  (CQCSub_All ([S,x],xSQ)) `1 = All (([S,x] `2),(([S,x] `1) `1)) by A1, Th26;
then A4: (CQCSub_All ([S,x],xSQ)) `1 = All (x,(([S,x] `1) `1)) ;
then A5: bound_in ((CQCSub_All ([S,x],xSQ)) `1) = x by QC_LANG2:7;
set finSub = RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2));
A6: Dom_Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)) =  {  s where s is QC-symbol of Al : x. s in Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))  }  by SUBSTUT1:def_9;
 (CQCSub_All ([S,x],xSQ)) `2 = xSQ by A1, A3, Th26;
then A7: RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2)) = RestrictSub (x,(All (x,(S `1))),xSQ) by A4, A5;
set Y = (Dom_Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) \/ (Sub_Var (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))));
set n = upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)));
 NSub ((the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)),(RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2)))) = NAT \ ((Dom_Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) \/ (Sub_Var (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))))) by SUBSTUT1:def_11;
then reconsider X = NAT \ ((Dom_Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) \/ (Sub_Var (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))))) as non empty Subset of (QC-symbols Al) ;
A8: upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) in NSub ((the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)),(RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))))
proof
 upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) = the Element of NSub ((the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)),(RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2)))) by SUBSTUT1:def_12;
hence  upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) in NSub ((the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)),(RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2)))) ; ::_thesis: verum
end;
 ( Dom_Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)) c= (Dom_Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) \/ (Sub_Var (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2)))) & upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) in NAT \ ((Dom_Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) \/ (Sub_Var (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))))) ) by A8, SUBSTUT1:def_11, XBOOLE_1:7;
then  not upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) in Dom_Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)) by XBOOLE_0:def_5;
then A9: not x. (upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)))) in Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)) by A6;
 (CQCSub_All ([S,x],xSQ)) `1 = All (x,(S `1)) by A4;
then A10: not x. (upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)))) in Bound_Vars (S `1) by A9, QC_LANG2:7;
 ( Sub_Var (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))) c= (Dom_Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) \/ (Sub_Var (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2)))) & upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) in NAT \ ((Dom_Bound_Vars (the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) \/ (Sub_Var (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))))) ) by A8, SUBSTUT1:def_11, XBOOLE_1:7;
then A11: not upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1))) in Sub_Var (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))) by XBOOLE_0:def_5;
 Sub_Var (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))) =  {  s where s is QC-symbol of Al : x. s in rng (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2)))  }  by SUBSTUT1:def_10;
then A12: not x. (upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)))) in rng (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))) by A11;
 CQCSub_All ([S,x],xSQ) = @ (CQCSub_All ([S,x],xSQ)) by SUBSTUT1:def_35;
hence  ( not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) & not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in Bound_Vars (S `1) ) by A2, A5, A7, A12, A10, SUBSTUT1:def_36; ::_thesis: verum
end;

theorem Th39: :: SUBLEMMA:39
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ))
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ))
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ))
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ))
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable & not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) implies not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) )
set S1 = CQCSub_All ([S,x],xSQ);
assume that
A1: [S,x] is quantifiable and
A2: not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) ; ::_thesis: not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ))
A3: CQCSub_All ([S,x],xSQ) = Sub_All ([S,x],xSQ) by A1, Def5;
then A4: (CQCSub_All ([S,x],xSQ)) `1 = All (([S,x] `2),(([S,x] `1) `1)) by A1, Th26;
then A5: (CQCSub_All ([S,x],xSQ)) `1 = All (x,(([S,x] `1) `1)) ;
set finSub = RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2));
A6: CQCSub_All ([S,x],xSQ) = @ (CQCSub_All ([S,x],xSQ)) by SUBSTUT1:def_35;
 (CQCSub_All ([S,x],xSQ)) `1 = All (x,(([S,x] `1) `1)) by A4;
then A7: bound_in ((CQCSub_All ([S,x],xSQ)) `1) = x by QC_LANG2:7;
 (CQCSub_All ([S,x],xSQ)) `2 = xSQ by A1, A3, Th26;
then  not bound_in ((CQCSub_All ([S,x],xSQ)) `1) in rng (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))) by A2, A7, A5;
hence  not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) by A2, A7, A6, SUBSTUT1:def_36; ::_thesis: verum
end;

theorem Th40: :: SUBLEMMA:40
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ))
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ))
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ))
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ))
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) )
assume A1: [S,x] is quantifiable ; ::_thesis: not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ))
then  ( x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) implies not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) ) by Th38;
hence  not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) by A1, Th39; ::_thesis: verum
end;

theorem Th41: :: SUBLEMMA:41
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
S `2 = ExpandSub (x,(S `1),(RestrictSub (x,(All (x,(S `1))),xSQ)))
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
S `2 = ExpandSub (x,(S `1),(RestrictSub (x,(All (x,(S `1))),xSQ)))
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
S `2 = ExpandSub (x,(S `1),(RestrictSub (x,(All (x,(S `1))),xSQ)))
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
S `2 = ExpandSub (x,(S `1),(RestrictSub (x,(All (x,(S `1))),xSQ)))
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies S `2 = ExpandSub (x,(S `1),(RestrictSub (x,(All (x,(S `1))),xSQ))) )
set Z = [(All (x,(S `1))),xSQ];
set q = All (x,(S `1));
assume  [S,x] is quantifiable ; ::_thesis: S `2 = ExpandSub (x,(S `1),(RestrictSub (x,(All (x,(S `1))),xSQ)))
then A1: ([S,x] `1) `2 = (QSub Al) . [(All (([S,x] `2),(([S,x] `1) `1))),xSQ] by SUBSTUT1:def_23;
A2: [(All (x,(S `1))),xSQ] in [:(QC-WFF Al),(vSUB Al):] by ZFMISC_1:def_2;
 [:(QC-WFF Al),(vSUB Al):] c= dom (QSub Al) by Th34;
then  [[(All (x,(S `1))),xSQ],(([S,x] `1) `2)] in QSub Al by A2, A1, FUNCT_1:1;
then  [[(All (x,(S `1))),xSQ],(S `2)] in QSub Al ;
then consider p being QC-formula of Al, Sub1 being CQC_Substitution of Al, b being set  such that
A3: [[(All (x,(S `1))),xSQ],(S `2)] = [[p,Sub1],b] and
A4: p,Sub1 PQSub b by SUBSTUT1:def_15;
 [(All (x,(S `1))),xSQ] = [p,Sub1] by A3, XTUPLE_0:1;
then A5: ( All (x,(S `1)) = p & xSQ = Sub1 ) by XTUPLE_0:1;
A6: All (x,(S `1)) is universal by QC_LANG1:def_21;
then A7: bound_in (All (x,(S `1))) = x by QC_LANG1:def_27;
 S `2 = b by A3, XTUPLE_0:1;
then  S `2 = ExpandSub ((bound_in (All (x,(S `1)))),(the_scope_of (All (x,(S `1)))),(RestrictSub ((bound_in (All (x,(S `1)))),(All (x,(S `1))),xSQ))) by A4, A5, A6, SUBSTUT1:def_14;
hence  S `2 = ExpandSub (x,(S `1),(RestrictSub (x,(All (x,(S `1))),xSQ))) by A6, A7, QC_LANG1:def_28; ::_thesis: verum
end;

theorem :: SUBLEMMA:42
for Al being QC-alphabet holds still_not-bound_in (VERUM Al) c= Bound_Vars (VERUM Al)
proof
let Al be QC-alphabet ; ::_thesis: still_not-bound_in (VERUM Al) c= Bound_Vars (VERUM Al)
 Bound_Vars (VERUM Al) = {} by SUBSTUT1:2;
hence  still_not-bound_in (VERUM Al) c= Bound_Vars (VERUM Al) by QC_LANG3:3; ::_thesis: verum
end;

theorem Th43: :: SUBLEMMA:43
for Al being QC-alphabet
for k being Element of NAT
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al holds still_not-bound_in (P ! ll) = Bound_Vars (P ! ll)
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al holds still_not-bound_in (P ! ll) = Bound_Vars (P ! ll)
let k be Element of NAT ; ::_thesis: for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al holds still_not-bound_in (P ! ll) = Bound_Vars (P ! ll)
let P be QC-pred_symbol of k,Al; ::_thesis: for ll being CQC-variable_list of k,Al holds still_not-bound_in (P ! ll) = Bound_Vars (P ! ll)
let ll be CQC-variable_list of k,Al; ::_thesis: still_not-bound_in (P ! ll) = Bound_Vars (P ! ll)
set l1 = the_arguments_of (P ! ll);
A1: P ! ll is atomic by QC_LANG1:def_18;
then consider n being Element of NAT , P9 being QC-pred_symbol of n,Al, ll9 being QC-variable_list of n,Al such that
A2: the_arguments_of (P ! ll) = ll9 and
A3: P ! ll = P9 ! ll9 by QC_LANG1:def_23;
 Bound_Vars (P ! ll) = Bound_Vars (the_arguments_of (P ! ll)) by A1, SUBSTUT1:3;
then A4: Bound_Vars (P ! ll) =  {  ((the_arguments_of (P ! ll)) . i) where i is Element of NAT : ( 1 <= i & i <= len (the_arguments_of (P ! ll)) & (the_arguments_of (P ! ll)) . i in bound_QC-variables Al )  }  by SUBSTUT1:def_7;
 still_not-bound_in (P ! ll) = still_not-bound_in ll by QC_LANG3:5;
then A5: still_not-bound_in (P ! ll) = variables_in (ll,(bound_QC-variables Al)) by QC_LANG3:2;
A6: ( (<*P9*> ^ ll9) . 1 = P9 & (<*P*> ^ ll) . 1 = P ) by FINSEQ_1:41;
 ( P ! ll = <*P*> ^ ll & P9 ! ll9 = <*P9*> ^ ll9 ) by QC_LANG1:8;
then  ll9 = ll by A3, A6, FINSEQ_1:33;
hence  still_not-bound_in (P ! ll) = Bound_Vars (P ! ll) by A4, A5, A2, QC_LANG3:def_1; ::_thesis: verum
end;

theorem Th44: :: SUBLEMMA:44
for Al being QC-alphabet
for p being Element of CQC-WFF Al st still_not-bound_in p c= Bound_Vars p holds
still_not-bound_in ('not' p) c= Bound_Vars ('not' p)
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al st still_not-bound_in p c= Bound_Vars p holds
still_not-bound_in ('not' p) c= Bound_Vars ('not' p)
let p be Element of CQC-WFF Al; ::_thesis: ( still_not-bound_in p c= Bound_Vars p implies still_not-bound_in ('not' p) c= Bound_Vars ('not' p) )
 'not' p is negative by QC_LANG1:def_19;
then  Bound_Vars ('not' p) = Bound_Vars (the_argument_of ('not' p)) by SUBSTUT1:4;
then A1: Bound_Vars ('not' p) = Bound_Vars p by QC_LANG2:1;
assume  still_not-bound_in p c= Bound_Vars p ; ::_thesis: still_not-bound_in ('not' p) c= Bound_Vars ('not' p)
hence  still_not-bound_in ('not' p) c= Bound_Vars ('not' p) by A1, QC_LANG3:7; ::_thesis: verum
end;

theorem Th45: :: SUBLEMMA:45
for Al being QC-alphabet
for p, q being Element of CQC-WFF Al st still_not-bound_in p c= Bound_Vars p & still_not-bound_in q c= Bound_Vars q holds
still_not-bound_in (p '&' q) c= Bound_Vars (p '&' q)
proof
let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al st still_not-bound_in p c= Bound_Vars p & still_not-bound_in q c= Bound_Vars q holds
still_not-bound_in (p '&' q) c= Bound_Vars (p '&' q)
let p, q be Element of CQC-WFF Al; ::_thesis: ( still_not-bound_in p c= Bound_Vars p & still_not-bound_in q c= Bound_Vars q implies still_not-bound_in (p '&' q) c= Bound_Vars (p '&' q) )
A1: still_not-bound_in (p '&' q) = (still_not-bound_in p) \/ (still_not-bound_in q) by QC_LANG3:10;
 p '&' q is conjunctive by QC_LANG1:def_20;
then  Bound_Vars (p '&' q) = (Bound_Vars (the_left_argument_of (p '&' q))) \/ (Bound_Vars (the_right_argument_of (p '&' q))) by SUBSTUT1:5;
then  Bound_Vars (p '&' q) = (Bound_Vars p) \/ (Bound_Vars (the_right_argument_of (p '&' q))) by QC_LANG2:4;
then A2: Bound_Vars (p '&' q) = (Bound_Vars p) \/ (Bound_Vars q) by QC_LANG2:4;
assume  ( still_not-bound_in p c= Bound_Vars p & still_not-bound_in q c= Bound_Vars q ) ; ::_thesis: still_not-bound_in (p '&' q) c= Bound_Vars (p '&' q)
hence  still_not-bound_in (p '&' q) c= Bound_Vars (p '&' q) by A2, A1, XBOOLE_1:13; ::_thesis: verum
end;

theorem Th46: :: SUBLEMMA:46
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al st still_not-bound_in p c= Bound_Vars p holds
still_not-bound_in (All (x,p)) c= Bound_Vars (All (x,p))
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al st still_not-bound_in p c= Bound_Vars p holds
still_not-bound_in (All (x,p)) c= Bound_Vars (All (x,p))
let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al st still_not-bound_in p c= Bound_Vars p holds
still_not-bound_in (All (x,p)) c= Bound_Vars (All (x,p))
let x be bound_QC-variable of Al; ::_thesis: ( still_not-bound_in p c= Bound_Vars p implies still_not-bound_in (All (x,p)) c= Bound_Vars (All (x,p)) )
A1: still_not-bound_in (All (x,p)) = (still_not-bound_in p) \ {x} by QC_LANG3:12;
 All (x,p) is universal by QC_LANG1:def_21;
then  Bound_Vars (All (x,p)) = (Bound_Vars (the_scope_of (All (x,p)))) \/ {(bound_in (All (x,p)))} by SUBSTUT1:6;
then  Bound_Vars (All (x,p)) = (Bound_Vars p) \/ {(bound_in (All (x,p)))} by QC_LANG2:7;
then  ( (Bound_Vars p) \ {x} c= Bound_Vars p & Bound_Vars p c= Bound_Vars (All (x,p)) ) by XBOOLE_1:7, XBOOLE_1:36;
then A2: (Bound_Vars p) \ {x} c= Bound_Vars (All (x,p)) by XBOOLE_1:1;
assume  still_not-bound_in p c= Bound_Vars p ; ::_thesis: still_not-bound_in (All (x,p)) c= Bound_Vars (All (x,p))
then  still_not-bound_in (All (x,p)) c= (Bound_Vars p) \ {x} by A1, XBOOLE_1:33;
hence  still_not-bound_in (All (x,p)) c= Bound_Vars (All (x,p)) by A2, XBOOLE_1:1; ::_thesis: verum
end;

theorem Th47: :: SUBLEMMA:47
for Al being QC-alphabet
for p being Element of CQC-WFF Al holds still_not-bound_in p c= Bound_Vars p
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al holds still_not-bound_in p c= Bound_Vars p
defpred S1[ Element of QC-WFF Al] means  still_not-bound_in $1 c= Bound_Vars $1;
 Bound_Vars (VERUM Al) = {} by SUBSTUT1:2;
then A1: for p, q being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for k being Element of NAT
for l being CQC-variable_list of k,Al
for P being QC-pred_symbol of k,Al holds
( S1[ VERUM Al] & S1[P ! l] & ( S1[p] implies S1[ 'not' p] ) & ( S1[p] & S1[q] implies S1[p '&' q] ) & ( S1[p] implies S1[ All (x,p)] ) ) by Th43, Th44, Th45, Th46, QC_LANG3:3;
thus  for p being Element of CQC-WFF Al holds S1[p] from CQC_LANG:sch_1(A1); ::_thesis: verum
end;

notation
let Al be QC-alphabet ;
let A be non empty set ;
let x be bound_QC-variable of Al;
let a be Element of A;
synonym x | a for Al .--> A;
end;

definition
let Al be QC-alphabet ;
let A be non empty set ;
let x be bound_QC-variable of Al;
let a be Element of A;
:: original: |
redefine funcx | a ->  Val_Sub of A,Al;
coherence
 | is Val_Sub of A,Al
proof
 ( dom (x .--> a) = {x} & rng (x .--> a) = {a} ) by FUNCOP_1:8, FUNCOP_1:13;
hence  | is Val_Sub of A,Al by RELSET_1:4; ::_thesis: verum
end;
end;

theorem Th48: :: SUBLEMMA:48
for Al being QC-alphabet
for b being set
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A st x <> b holds
(v . (x | a)) . b = v . b
proof
let Al be QC-alphabet ; ::_thesis: for b being set
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A st x <> b holds
(v . (x | a)) . b = v . b
let b be set ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A st x <> b holds
(v . (x | a)) . b = v . b
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A st x <> b holds
(v . (x | a)) . b = v . b
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for a being Element of A st x <> b holds
(v . (x | a)) . b = v . b
let v be Element of Valuations_in (Al,A); ::_thesis: for a being Element of A st x <> b holds
(v . (x | a)) . b = v . b
let a be Element of A; ::_thesis: ( x <> b implies (v . (x | a)) . b = v . b )
assume  x <> b ; ::_thesis: (v . (x | a)) . b = v . b
then  not b in dom ({x} --> a) by TARSKI:def_1;
hence  (v . (x | a)) . b = v . b by FUNCT_4:11; ::_thesis: verum
end;

theorem Th49: :: SUBLEMMA:49
for Al being QC-alphabet
for x, y being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A st x = y holds
(v . (x | a)) . y = a
proof
let Al be QC-alphabet ; ::_thesis: for x, y being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A st x = y holds
(v . (x | a)) . y = a
let x, y be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A st x = y holds
(v . (x | a)) . y = a
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for a being Element of A st x = y holds
(v . (x | a)) . y = a
let v be Element of Valuations_in (Al,A); ::_thesis: for a being Element of A st x = y holds
(v . (x | a)) . y = a
let a be Element of A; ::_thesis: ( x = y implies (v . (x | a)) . y = a )
assume A1: x = y ; ::_thesis: (v . (x | a)) . y = a
then  y in {x} by TARSKI:def_1;
then A2: y in dom (x .--> a) by FUNCOP_1:13;
 (x .--> a) . y = a by A1, FUNCOP_1:72;
hence  (v . (x | a)) . y = a by A2, FUNCT_4:13; ::_thesis: verum
end;

theorem Th50: :: SUBLEMMA:50
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A) holds
( J,v |= All (x,p) iff for a being Element of A holds J,v . (x | a) |= p )
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A) holds
( J,v |= All (x,p) iff for a being Element of A holds J,v . (x | a) |= p )
let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A) holds
( J,v |= All (x,p) iff for a being Element of A holds J,v . (x | a) |= p )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A) holds
( J,v |= All (x,p) iff for a being Element of A holds J,v . (x | a) |= p )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A) holds
( J,v |= All (x,p) iff for a being Element of A holds J,v . (x | a) |= p )
let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A) holds
( J,v |= All (x,p) iff for a being Element of A holds J,v . (x | a) |= p )
let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= All (x,p) iff for a being Element of A holds J,v . (x | a) |= p )
thus  ( J,v |= All (x,p) implies for a being Element of A holds J,v . (x | a) |= p ) ::_thesis: ( ( for a being Element of A holds J,v . (x | a) |= p ) implies J,v |= All (x,p) )
proof
assume A1: J,v |= All (x,p) ; ::_thesis: for a being Element of A holds J,v . (x | a) |= p
let a be Element of A; ::_thesis: J,v . (x | a) |= p
 for y being bound_QC-variable of Al st x <> y holds
(v . (x | a)) . y = v . y by Th48;
hence  J,v . (x | a) |= p by A1, VALUAT_1:29; ::_thesis: verum
end;
thus  ( ( for a being Element of A holds J,v . (x | a) |= p ) implies J,v |= All (x,p) ) ::_thesis: verum
proof
assume A2: for a being Element of A holds J,v . (x | a) |= p ; ::_thesis: J,v |= All (x,p)
 for w being Element of Valuations_in (Al,A) st ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) holds
J,w |= p
proof
let w be Element of Valuations_in (Al,A); ::_thesis: ( ( for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ) implies J,w |= p )
assume A3: for y being bound_QC-variable of Al st x <> y holds
w . y = v . y ; ::_thesis: J,w |= p
set c = w . x;
A4: for b being set st b in dom w holds
w . b = (v . (x | (w . x))) . b
proof
let b be set ; ::_thesis: ( b in dom w implies w . b = (v . (x | (w . x))) . b )
assume  b in dom w ; ::_thesis: w . b = (v . (x | (w . x))) . b
then reconsider y = b as bound_QC-variable of Al ;
now__::_thesis:_(_x_<>_y_implies_w_._b_=_(v_._(x_|_(w_._x)))_._b_)
assume A5: x <> y ; ::_thesis: w . b = (v . (x | (w . x))) . b
then  w . y = v . y by A3;
hence  w . b = (v . (x | (w . x))) . b by A5, Th48; ::_thesis: verum
end;
hence  w . b = (v . (x | (w . x))) . b by Th49; ::_thesis: verum
end;
 v . (x | (w . x)) is Element of Funcs ((bound_QC-variables Al),A) by VALUAT_1:def_1;
then A6: ex f being Function st
( v . (x | (w . x)) = f & dom f = bound_QC-variables Al & rng f c= A ) by FUNCT_2:def_2;
 w is Element of Funcs ((bound_QC-variables Al),A) by VALUAT_1:def_1;
then  ex f being Function st
( w = f & dom f = bound_QC-variables Al & rng f c= A ) by FUNCT_2:def_2;
then  v . (x | (w . x)) = w by A4, A6, FUNCT_1:2;
hence  J,w |= p by A2; ::_thesis: verum
end;
hence  J,v |= All (x,p) by VALUAT_1:29; ::_thesis: verum
end;
end;

definition
let Al be QC-alphabet ;
let S be Element of CQC-Sub-WFF Al;
let x be bound_QC-variable of Al;
let xSQ be second_Q_comp of [S,x];
let A be non empty set ;
let v be Element of Valuations_in (Al,A);
func NEx_Val (v,S,x,xSQ) ->  Val_Sub of A,Al equals :: SUBLEMMA:def 8
(@ (RestrictSub (x,(All (x,(S `1))),xSQ))) * v;
coherence
 (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) * v is Val_Sub of A,Al ;
end;

:: deftheorem  defines NEx_Val SUBLEMMA:def_8_:_
 for Al being QC-alphabet
for S being Element of CQC-Sub-WFF Al
for x being bound_QC-variable of Al
for xSQ being second_Q_comp of [S,x]
for A being non empty set
for v being Element of Valuations_in (Al,A) holds NEx_Val (v,S,x,xSQ) = (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) * v;

definition
let Al be QC-alphabet ;
let A be non empty set ;
let v, w be Val_Sub of A,Al;
:: original: .
redefine funcv +* w ->  Val_Sub of A,Al;
coherence
 . is Val_Sub of A,Al
proof
 ( dom (v +* w) = (dom v) \/ (dom w) & rng (v +* w) c= A ) by FUNCT_4:def_1;
hence  . is Val_Sub of A,Al ; ::_thesis: verum
end;
end;

theorem Th51: :: SUBLEMMA:51
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1)))
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1)))
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1)))
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1)))
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable & x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) implies S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1))) )
set S1 = CQCSub_All ([S,x],xSQ);
assume that
A1: [S,x] is quantifiable and
A2: x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) ; ::_thesis: S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1)))
A3: CQCSub_All ([S,x],xSQ) = Sub_All ([S,x],xSQ) by A1, Def5;
then A4: (CQCSub_All ([S,x],xSQ)) `2 = xSQ by A1, Th26;
A5: (CQCSub_All ([S,x],xSQ)) `1 = All (([S,x] `2),(([S,x] `1) `1)) by A1, A3, Th26;
then A6: (CQCSub_All ([S,x],xSQ)) `1 = All (x,(([S,x] `1) `1)) ;
then A7: ( (CQCSub_All ([S,x],xSQ)) `1 = All (x,(S `1)) & x = bound_in ((CQCSub_All ([S,x],xSQ)) `1) ) by QC_LANG2:7;
set finSub = RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2));
A8: @ (CQCSub_All ([S,x],xSQ)) = CQCSub_All ([S,x],xSQ) by SUBSTUT1:def_35;
 (CQCSub_All ([S,x],xSQ)) `1 = All (x,(([S,x] `1) `1)) by A5;
then  bound_in ((CQCSub_All ([S,x],xSQ)) `1) = x by QC_LANG2:7;
then  bound_in ((CQCSub_All ([S,x],xSQ)) `1) in rng (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))) by A2, A4, A6;
then  S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x. (upVar ((RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))),(the_scope_of ((CQCSub_All ([S,x],xSQ)) `1)))) by A8, SUBSTUT1:def_36;
hence  S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1))) by A4, A7, QC_LANG2:7; ::_thesis: verum
end;

theorem Th52: :: SUBLEMMA:52
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) holds
S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable & not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) implies S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x )
set S1 = CQCSub_All ([S,x],xSQ);
assume that
A1: [S,x] is quantifiable and
A2: not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) ; ::_thesis: S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x
A3: CQCSub_All ([S,x],xSQ) = Sub_All ([S,x],xSQ) by A1, Def5;
then A4: (CQCSub_All ([S,x],xSQ)) `1 = All (([S,x] `2),(([S,x] `1) `1)) by A1, Th26;
then A5: (CQCSub_All ([S,x],xSQ)) `1 = All (x,(([S,x] `1) `1)) ;
set finSub = RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2));
A6: @ (CQCSub_All ([S,x],xSQ)) = CQCSub_All ([S,x],xSQ) by SUBSTUT1:def_35;
 (CQCSub_All ([S,x],xSQ)) `1 = All (x,(([S,x] `1) `1)) by A4;
then A7: bound_in ((CQCSub_All ([S,x],xSQ)) `1) = x by QC_LANG2:7;
 (CQCSub_All ([S,x],xSQ)) `2 = xSQ by A1, A3, Th26;
then  not bound_in ((CQCSub_All ([S,x],xSQ)) `1) in rng (RestrictSub ((bound_in ((CQCSub_All ([S,x],xSQ)) `1)),((CQCSub_All ([S,x],xSQ)) `1),((CQCSub_All ([S,x],xSQ)) `2))) by A2, A7, A5;
hence  S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x by A7, A6, SUBSTUT1:def_36; ::_thesis: verum
end;

theorem Th53: :: SUBLEMMA:53
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds
( Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) & dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds
( Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) & dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds
( Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) & dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} )
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds
( Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) & dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} )
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds
( Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) & dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds
( Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) & dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies for a being Element of A holds
( Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) & dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} ) )
assume A1: [S,x] is quantifiable ; ::_thesis: for a being Element of A holds
( Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) & dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} )
set finSub = RestrictSub (x,(All (x,(S `1))),xSQ);
let a be Element of A; ::_thesis: ( Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) & dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} )
set S1 = CQCSub_All ([S,x],xSQ);
set z = S_Bound (@ (CQCSub_All ([S,x],xSQ)));
A2: S `2 = ExpandSub (x,(S `1),(RestrictSub (x,(All (x,(S `1))),xSQ))) by A1, Th41;
A3: now__::_thesis:_(_not_x_in_rng_(RestrictSub_(x,(All_(x,(S_`1))),xSQ))_implies_(_dom_(RestrictSub_(x,(All_(x,(S_`1))),xSQ))_misses_{x}_&_Val_S_((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a)),S)_=_(NEx_Val_((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a)),S,x,xSQ))_+*_(x_|_a)_)_)
reconsider F = {[x,x]} as Function ;
assume A4: not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) ; ::_thesis: ( dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} & Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) )
then  S `2 = (RestrictSub (x,(All (x,(S `1))),xSQ)) \/ F by A2, SUBSTUT1:def_13;
then A5: @ (S `2) = (RestrictSub (x,(All (x,(S `1))),xSQ)) \/ F by SUBSTUT1:def_2;
A6: now__::_thesis:_dom_(RestrictSub_(x,(All_(x,(S_`1))),xSQ))_misses_{x}
set q = All (x,(S `1));
set X =  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  ;
assume  not dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} ; ::_thesis: contradiction
then  (dom (RestrictSub (x,(All (x,(S `1))),xSQ))) /\ {x} <> {} by XBOOLE_0:def_7;
then consider b being set  such that
A7: b in (dom (RestrictSub (x,(All (x,(S `1))),xSQ))) /\ {x} by XBOOLE_0:def_1;
 RestrictSub (x,(All (x,(S `1))),xSQ) = xSQ |  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by SUBSTUT1:def_6;
then  RestrictSub (x,(All (x,(S `1))),xSQ) = (@ xSQ) |  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by SUBSTUT1:def_2;
then  @ (RestrictSub (x,(All (x,(S `1))),xSQ)) = (@ xSQ) |  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by SUBSTUT1:def_2;
then  dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) = (dom (@ xSQ)) /\  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by RELAT_1:61;
then A8: dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) c=  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by XBOOLE_1:17;
 b in dom (RestrictSub (x,(All (x,(S `1))),xSQ)) by A7, XBOOLE_0:def_4;
then  b in dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) by SUBSTUT1:def_2;
then  b in  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by A8;
then A9: ex y being bound_QC-variable of Al st
( y = b & y in still_not-bound_in (All (x,(S `1))) & y is Element of dom xSQ & y <> x & y <> xSQ . y ) ;
 b in {x} by A7, XBOOLE_0:def_4;
hence  contradiction by A9, TARSKI:def_1; ::_thesis: verum
end;
hence  dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} ; ::_thesis: Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)
 dom {[x,x]} = {x} by RELAT_1:9;
then  dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) misses dom F by A6, SUBSTUT1:def_2;
then A10: (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) \/ F = (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* F by FUNCT_4:31;
 v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) is Element of Funcs ((bound_QC-variables Al),A) by VALUAT_1:def_1;
then A11: ex f being Function st
( v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) = f & dom f = bound_QC-variables Al & rng f c= A ) by FUNCT_2:def_2;
A12: rng F = {x} by RELAT_1:9;
then  dom (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) = dom F by A11, RELAT_1:27;
then A13: dom (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) = {x} by RELAT_1:9;
A14: {[x,x]} = x .--> x by FUNCT_4:82;
 for b being set st b in dom (x | a) holds
(x | a) . b = (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) . b
proof
let b be set ; ::_thesis: ( b in dom (x | a) implies (x | a) . b = (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) . b )
assume A15: b in dom (x | a) ; ::_thesis: (x | a) . b = (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) . b
A16: (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) . b = (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (F . b) by A13, A15, FUNCT_1:12;
 b = x by A15, TARSKI:def_1;
then  ( (x | a) . b = a & F . b = x ) by A14, FUNCOP_1:72;
hence  (x | a) . b = (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) . b by A1, A4, A16, Th49, Th52; ::_thesis: verum
end;
then A17: x | a = F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) by A13, FUNCOP_1:13, FUNCT_1:2;
 ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* F) * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) = ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) +* (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) by A11, A12, FUNCT_7:9;
hence  Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) by A10, A5, A17, SUBSTUT1:def_2; ::_thesis: verum
end;
now__::_thesis:_(_x_in_rng_(RestrictSub_(x,(All_(x,(S_`1))),xSQ))_implies_(_dom_(RestrictSub_(x,(All_(x,(S_`1))),xSQ))_misses_{x}_&_Val_S_((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a)),S)_=_(NEx_Val_((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a)),S,x,xSQ))_+*_(x_|_a)_)_)
reconsider F = {[x,(x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1))))]} as Function ;
assume A18: x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) ; ::_thesis: ( dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} & Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) )
A19: now__::_thesis:_not_dom_(RestrictSub_(x,(All_(x,(S_`1))),xSQ))_meets_{x}
set q = All (x,(S `1));
set X =  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  ;
assume  dom (RestrictSub (x,(All (x,(S `1))),xSQ)) meets {x} ; ::_thesis: contradiction
then consider b being set  such that
A20: b in dom (RestrictSub (x,(All (x,(S `1))),xSQ)) and
A21: b in {x} by XBOOLE_0:3;
 RestrictSub (x,(All (x,(S `1))),xSQ) = xSQ |  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by SUBSTUT1:def_6;
then  RestrictSub (x,(All (x,(S `1))),xSQ) = (@ xSQ) |  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by SUBSTUT1:def_2;
then  @ (RestrictSub (x,(All (x,(S `1))),xSQ)) = (@ xSQ) |  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by SUBSTUT1:def_2;
then  dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) = (dom (@ xSQ)) /\  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by RELAT_1:61;
then A22: dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) c=  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by XBOOLE_1:17;
 b in dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) by A20, SUBSTUT1:def_2;
then  b in  {  y1 where y1 is bound_QC-variable of Al : ( y1 in still_not-bound_in (All (x,(S `1))) & y1 is Element of dom xSQ & y1 <> x & y1 <> xSQ . y1 )  }  by A22;
then  ex y being bound_QC-variable of Al st
( y = b & y in still_not-bound_in (All (x,(S `1))) & y is Element of dom xSQ & y <> x & y <> xSQ . y ) ;
hence  contradiction by A21, TARSKI:def_1; ::_thesis: verum
end;
hence  dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} ; ::_thesis: Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)
 dom {[x,(x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1))))]} = {x} by RELAT_1:9;
then  dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) misses dom F by A19, SUBSTUT1:def_2;
then A23: (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) \/ F = (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* F by FUNCT_4:31;
 v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) is Element of Funcs ((bound_QC-variables Al),A) by VALUAT_1:def_1;
then A24: ex f being Function st
( v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) = f & dom f = bound_QC-variables Al & rng f c= A ) by FUNCT_2:def_2;
 rng F = {(x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1))))} by RELAT_1:9;
then  dom (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) = dom F by A24, RELAT_1:27;
then A25: dom (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) = {x} by RELAT_1:9;
A26: {[x,(x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1))))]} = x .--> (x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1)))) by FUNCT_4:82;
 for b being set st b in dom (x | a) holds
(x | a) . b = (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) . b
proof
let b be set ; ::_thesis: ( b in dom (x | a) implies (x | a) . b = (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) . b )
assume A27: b in dom (x | a) ; ::_thesis: (x | a) . b = (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) . b
A28: (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) . b = (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (F . b) by A25, A27, FUNCT_1:12;
 b = x by A27, TARSKI:def_1;
then  ( (x | a) . b = a & F . b = x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1))) ) by A26, FUNCOP_1:72;
hence  (x | a) . b = (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) . b by A1, A18, A28, Th49, Th51; ::_thesis: verum
end;
then A29: x | a = F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) by A25, FUNCOP_1:13, FUNCT_1:2;
 rng F = {(x. (upVar ((RestrictSub (x,(All (x,(S `1))),xSQ)),(S `1))))} by RELAT_1:9;
then A30: ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* F) * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) = ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) +* (F * (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) by A24, FUNCT_7:9;
 S `2 = (RestrictSub (x,(All (x,(S `1))),xSQ)) \/ F by A2, A18, SUBSTUT1:def_13;
then  @ (S `2) = (RestrictSub (x,(All (x,(S `1))),xSQ)) \/ F by SUBSTUT1:def_2;
hence  Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) by A23, A30, A29, SUBSTUT1:def_2; ::_thesis: verum
end;
hence  ( Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) & dom (RestrictSub (x,(All (x,(S `1))),xSQ)) misses {x} ) by A3; ::_thesis: verum
end;

theorem Th54: :: SUBLEMMA:54
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S )
let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S )
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) )
set S1 = CQCSub_All ([S,x],xSQ);
set z = S_Bound (@ (CQCSub_All ([S,x],xSQ)));
assume A1: [S,x] is quantifiable ; ::_thesis: ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S )
thus  ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) ::_thesis: ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S )
proof
assume A2: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ; ::_thesis: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S
let a be Element of A; ::_thesis: J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S
 Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) by A1, Th53;
hence  J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S by A2; ::_thesis: verum
end;
thus  ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) ::_thesis: verum
proof
assume A3: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ; ::_thesis: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S
let a be Element of A; ::_thesis: J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S
 Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S) = (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a) by A1, Th53;
hence  J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S by A3; ::_thesis: verum
end;
end;

theorem Th55: :: SUBLEMMA:55
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ)
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ)
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ)
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ)
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ)
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for a being Element of A holds NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ)
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies for a being Element of A holds NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ) )
assume A1: [S,x] is quantifiable ; ::_thesis: for a being Element of A holds NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ)
set finSub = RestrictSub (x,(All (x,(S `1))),xSQ);
set NF1 = NEx_Val (v,S,x,xSQ);
set S1 = CQCSub_All ([S,x],xSQ);
let a be Element of A; ::_thesis: NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ)
set z = S_Bound (@ (CQCSub_All ([S,x],xSQ)));
set NF = NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ);
 v is Element of Funcs ((bound_QC-variables Al),A) by VALUAT_1:def_1;
then  ex f being Function st
( v = f & dom f = bound_QC-variables Al & rng f c= A ) by FUNCT_2:def_2;
then  rng (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) c= dom v ;
then A2: dom (NEx_Val (v,S,x,xSQ)) = dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) by RELAT_1:27;
 v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) is Element of Funcs ((bound_QC-variables Al),A) by VALUAT_1:def_1;
then  ex f being Function st
( v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) = f & dom f = bound_QC-variables Al & rng f c= A ) by FUNCT_2:def_2;
then A3: rng (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) c= dom (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) ;
then A4: dom (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) = dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) by RELAT_1:27;
 for b being set st b in dom (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) holds
(NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) . b = (NEx_Val (v,S,x,xSQ)) . b
proof
let b be set ; ::_thesis: ( b in dom (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) implies (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) . b = (NEx_Val (v,S,x,xSQ)) . b )
assume A5: b in dom (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) ; ::_thesis: (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) . b = (NEx_Val (v,S,x,xSQ)) . b
A6: (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) . b in rng (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) by A4, A5, FUNCT_1:3;
then reconsider x = (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) . b as bound_QC-variable of Al ;
 not S_Bound (@ (CQCSub_All ([S,x],xSQ))) in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) by A1, Th40;
then  S_Bound (@ (CQCSub_All ([S,x],xSQ))) <> x by A6, SUBSTUT1:def_2;
then A7: (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . x = v . x by Th48;
 (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) . b = (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . x by A5, FUNCT_1:12;
hence  (NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) . b = (NEx_Val (v,S,x,xSQ)) . b by A4, A2, A5, A7, FUNCT_1:12; ::_thesis: verum
end;
hence  NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ) by A3, A2, FUNCT_1:2, RELAT_1:27; ::_thesis: verum
end;

theorem Th56: :: SUBLEMMA:56
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) )
set S1 = CQCSub_All ([S,x],xSQ);
set z = S_Bound (@ (CQCSub_All ([S,x],xSQ)));
assume A1: [S,x] is quantifiable ; ::_thesis: ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
thus  ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) ::_thesis: ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S )
proof
assume A2: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ; ::_thesis: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
let a be Element of A; ::_thesis: J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
 NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ) by A1, Th55;
hence  J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S by A2; ::_thesis: verum
end;
thus  ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) ::_thesis: verum
proof
assume A3: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ; ::_thesis: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S
let a be Element of A; ::_thesis: J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S
 NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ) = NEx_Val (v,S,x,xSQ) by A1, Th55;
hence  J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S by A3; ::_thesis: verum
end;
end;

begin

theorem Th57: :: SUBLEMMA:57
for Al being QC-alphabet
for l1 being FinSequence of QC-variables Al st rng l1 c= bound_QC-variables Al holds
still_not-bound_in l1 = rng l1
proof
let Al be QC-alphabet ; ::_thesis: for l1 being FinSequence of QC-variables Al st rng l1 c= bound_QC-variables Al holds
still_not-bound_in l1 = rng l1
let l1 be FinSequence of QC-variables Al; ::_thesis: ( rng l1 c= bound_QC-variables Al implies still_not-bound_in l1 = rng l1 )
A1: variables_in (l1,(bound_QC-variables Al)) =  {  (l1 . k) where k is Element of NAT : ( 1 <= k & k <= len l1 & l1 . k in bound_QC-variables Al )  }  by QC_LANG3:def_1;
assume A2: rng l1 c= bound_QC-variables Al ; ::_thesis: still_not-bound_in l1 = rng l1
A3: rng l1 c= variables_in (l1,(bound_QC-variables Al))
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng l1 or b in variables_in (l1,(bound_QC-variables Al)) )
assume A4: b in rng l1 ; ::_thesis: b in variables_in (l1,(bound_QC-variables Al))
then consider k being natural number  such that
A5: k in dom l1 and
A6: l1 . k = b by FINSEQ_2:10;
 k in Seg (len l1) by A5, FINSEQ_1:def_3;
then  ( 1 <= k & k <= len l1 ) by FINSEQ_1:1;
hence  b in variables_in (l1,(bound_QC-variables Al)) by A2, A1, A4, A5, A6; ::_thesis: verum
end;
 variables_in (l1,(bound_QC-variables Al)) c= rng l1
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in variables_in (l1,(bound_QC-variables Al)) or b in rng l1 )
assume  b in variables_in (l1,(bound_QC-variables Al)) ; ::_thesis: b in rng l1
then consider k being Element of NAT  such that
A7: b = l1 . k and
A8: ( 1 <= k & k <= len l1 ) and
 l1 . k in bound_QC-variables Al by A1;
 k in Seg (len l1) by A8, FINSEQ_1:1;
then  k in dom l1 by FINSEQ_1:def_3;
hence  b in rng l1 by A7, FUNCT_1:3; ::_thesis: verum
end;
then  variables_in (l1,(bound_QC-variables Al)) = rng l1 by A3, XBOOLE_0:def_10;
hence  still_not-bound_in l1 = rng l1 by QC_LANG3:2; ::_thesis: verum
end;

theorem Th58: :: SUBLEMMA:58
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A holds
( dom v = bound_QC-variables Al & dom (x | a) = {x} )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A holds
( dom v = bound_QC-variables Al & dom (x | a) = {x} )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A holds
( dom v = bound_QC-variables Al & dom (x | a) = {x} )
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for a being Element of A holds
( dom v = bound_QC-variables Al & dom (x | a) = {x} )
let v be Element of Valuations_in (Al,A); ::_thesis: for a being Element of A holds
( dom v = bound_QC-variables Al & dom (x | a) = {x} )
let a be Element of A; ::_thesis: ( dom v = bound_QC-variables Al & dom (x | a) = {x} )
 v is Element of Funcs ((bound_QC-variables Al),A) by VALUAT_1:def_1;
then  ex f being Function st
( v = f & dom f = bound_QC-variables Al & rng f c= A ) by FUNCT_2:def_2;
hence  dom v = bound_QC-variables Al ; ::_thesis: dom (x | a) = {x}
thus  dom (x | a) = {x} by FUNCOP_1:13; ::_thesis: verum
end;

theorem Th59: :: SUBLEMMA:59
for Al being QC-alphabet
for k being Element of NAT
for A being non empty set
for v being Element of Valuations_in (Al,A)
for ll being CQC-variable_list of k,Al holds v *' ll = ll * (v | (still_not-bound_in ll))
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for A being non empty set
for v being Element of Valuations_in (Al,A)
for ll being CQC-variable_list of k,Al holds v *' ll = ll * (v | (still_not-bound_in ll))
let k be Element of NAT ; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for ll being CQC-variable_list of k,Al holds v *' ll = ll * (v | (still_not-bound_in ll))
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for ll being CQC-variable_list of k,Al holds v *' ll = ll * (v | (still_not-bound_in ll))
let v be Element of Valuations_in (Al,A); ::_thesis: for ll being CQC-variable_list of k,Al holds v *' ll = ll * (v | (still_not-bound_in ll))
let ll be CQC-variable_list of k,Al; ::_thesis: v *' ll = ll * (v | (still_not-bound_in ll))
 rng ll c= bound_QC-variables Al by RELAT_1:def_19;
then A1: rng ll = still_not-bound_in ll by Th57;
 dom (v | (still_not-bound_in ll)) = (dom v) /\ (still_not-bound_in ll) by RELAT_1:61;
then  dom (v | (still_not-bound_in ll)) = (bound_QC-variables Al) /\ (still_not-bound_in ll) by Th58;
then  rng ll = dom (v | (still_not-bound_in ll)) by A1, XBOOLE_1:28;
then A2: dom (ll * (v | (still_not-bound_in ll))) = dom ll by RELAT_1:27;
then A3: dom (ll * (v | (still_not-bound_in ll))) = Seg (len ll) by FINSEQ_1:def_3;
then reconsider f = ll * (v | (still_not-bound_in ll)) as FinSequence by FINSEQ_1:def_2;
 len f = len ll by A3, FINSEQ_1:def_3;
then A4: len f = k by SUBSTUT1:34;
then A5: dom f = Seg k by FINSEQ_1:def_3;
A6: for j being natural number st j in dom f holds
f . j = (v *' ll) . j
proof
A7: rng ll c= bound_QC-variables Al by RELAT_1:def_19;
let j be natural number ; ::_thesis: ( j in dom f implies f . j = (v *' ll) . j )
assume A8: j in dom f ; ::_thesis: f . j = (v *' ll) . j
reconsider j = j as Element of NAT by ORDINAL1:def_12;
 ll . j in rng ll by A2, A8, FUNCT_1:3;
then  ll . j in bound_QC-variables Al by A7;
then A9: ll . j in dom v by Th58;
 ll . j in still_not-bound_in ll by A1, A2, A8, FUNCT_1:3;
then  ll . j in (dom v) /\ (still_not-bound_in ll) by A9, XBOOLE_0:def_4;
then A10: (v | (still_not-bound_in ll)) . (ll . j) = v . (ll . j) by FUNCT_1:48;
 ( 1 <= j & j <= k ) by A5, A8, FINSEQ_1:1;
then  (v | (still_not-bound_in ll)) . (ll . j) = (v *' ll) . j by A10, VALUAT_1:def_3;
hence  f . j = (v *' ll) . j by A2, A8, FUNCT_1:13; ::_thesis: verum
end;
 len (v *' ll) = k by VALUAT_1:def_3;
hence  v *' ll = ll * (v | (still_not-bound_in ll)) by A4, A6, FINSEQ_2:9; ::_thesis: verum
end;

theorem Th60: :: SUBLEMMA:60
for Al being QC-alphabet
for k being Element of NAT
for A being non empty set
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (P ! ll)) = w | (still_not-bound_in (P ! ll)) holds
( J,v |= P ! ll iff J,w |= P ! ll )
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for A being non empty set
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (P ! ll)) = w | (still_not-bound_in (P ! ll)) holds
( J,v |= P ! ll iff J,w |= P ! ll )
let k be Element of NAT ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (P ! ll)) = w | (still_not-bound_in (P ! ll)) holds
( J,v |= P ! ll iff J,w |= P ! ll )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (P ! ll)) = w | (still_not-bound_in (P ! ll)) holds
( J,v |= P ! ll iff J,w |= P ! ll )
let J be interpretation of Al,A; ::_thesis: for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (P ! ll)) = w | (still_not-bound_in (P ! ll)) holds
( J,v |= P ! ll iff J,w |= P ! ll )
let P be QC-pred_symbol of k,Al; ::_thesis: for ll being CQC-variable_list of k,Al
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (P ! ll)) = w | (still_not-bound_in (P ! ll)) holds
( J,v |= P ! ll iff J,w |= P ! ll )
let ll be CQC-variable_list of k,Al; ::_thesis: for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (P ! ll)) = w | (still_not-bound_in (P ! ll)) holds
( J,v |= P ! ll iff J,w |= P ! ll )
let v, w be Element of Valuations_in (Al,A); ::_thesis: ( v | (still_not-bound_in (P ! ll)) = w | (still_not-bound_in (P ! ll)) implies ( J,v |= P ! ll iff J,w |= P ! ll ) )
assume A1: v | (still_not-bound_in (P ! ll)) = w | (still_not-bound_in (P ! ll)) ; ::_thesis: ( J,v |= P ! ll iff J,w |= P ! ll )
A2: still_not-bound_in (P ! ll) = still_not-bound_in ll by QC_LANG3:5;
A3: ( w *' ll in J . P iff (Valid ((P ! ll),J)) . w = TRUE ) by VALUAT_1:7;
A4: ( (Valid ((P ! ll),J)) . v = TRUE iff v *' ll in J . P ) by VALUAT_1:7;
 ( ll * (w | (still_not-bound_in ll)) in J . P iff w *' ll in J . P ) by Th59;
hence  ( J,v |= P ! ll iff J,w |= P ! ll ) by A1, A2, A4, A3, Th59, VALUAT_1:def_7; ::_thesis: verum
end;

theorem Th61: :: SUBLEMMA:61
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for A being non empty set
for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in ('not' p)) = w | (still_not-bound_in ('not' p)) holds
( J,v |= 'not' p iff J,w |= 'not' p )
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for A being non empty set
for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in ('not' p)) = w | (still_not-bound_in ('not' p)) holds
( J,v |= 'not' p iff J,w |= 'not' p )
let p be Element of CQC-WFF Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in ('not' p)) = w | (still_not-bound_in ('not' p)) holds
( J,v |= 'not' p iff J,w |= 'not' p )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in ('not' p)) = w | (still_not-bound_in ('not' p)) holds
( J,v |= 'not' p iff J,w |= 'not' p )
let J be interpretation of Al,A; ::_thesis: ( ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) implies for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in ('not' p)) = w | (still_not-bound_in ('not' p)) holds
( J,v |= 'not' p iff J,w |= 'not' p ) )
assume A1: for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ; ::_thesis: for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in ('not' p)) = w | (still_not-bound_in ('not' p)) holds
( J,v |= 'not' p iff J,w |= 'not' p )
let v, w be Element of Valuations_in (Al,A); ::_thesis: ( v | (still_not-bound_in ('not' p)) = w | (still_not-bound_in ('not' p)) implies ( J,v |= 'not' p iff J,w |= 'not' p ) )
A2: still_not-bound_in ('not' p) = still_not-bound_in p by QC_LANG3:7;
assume  v | (still_not-bound_in ('not' p)) = w | (still_not-bound_in ('not' p)) ; ::_thesis: ( J,v |= 'not' p iff J,w |= 'not' p )
then  ( not J,v |= p iff not J,w |= p ) by A1, A2;
hence  ( J,v |= 'not' p iff J,w |= 'not' p ) by VALUAT_1:17; ::_thesis: verum
end;

theorem Th62: :: SUBLEMMA:62
for Al being QC-alphabet
for p, q being Element of CQC-WFF Al
for A being non empty set
for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) & ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in q) = w | (still_not-bound_in q) holds
( J,v |= q iff J,w |= q ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (p '&' q)) = w | (still_not-bound_in (p '&' q)) holds
( J,v |= p '&' q iff J,w |= p '&' q )
proof
let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al
for A being non empty set
for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) & ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in q) = w | (still_not-bound_in q) holds
( J,v |= q iff J,w |= q ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (p '&' q)) = w | (still_not-bound_in (p '&' q)) holds
( J,v |= p '&' q iff J,w |= p '&' q )
let p, q be Element of CQC-WFF Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) & ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in q) = w | (still_not-bound_in q) holds
( J,v |= q iff J,w |= q ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (p '&' q)) = w | (still_not-bound_in (p '&' q)) holds
( J,v |= p '&' q iff J,w |= p '&' q )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) & ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in q) = w | (still_not-bound_in q) holds
( J,v |= q iff J,w |= q ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (p '&' q)) = w | (still_not-bound_in (p '&' q)) holds
( J,v |= p '&' q iff J,w |= p '&' q )
let J be interpretation of Al,A; ::_thesis: ( ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) & ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in q) = w | (still_not-bound_in q) holds
( J,v |= q iff J,w |= q ) ) implies for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (p '&' q)) = w | (still_not-bound_in (p '&' q)) holds
( J,v |= p '&' q iff J,w |= p '&' q ) )
assume A1: ( ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) & ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in q) = w | (still_not-bound_in q) holds
( J,v |= q iff J,w |= q ) ) ) ; ::_thesis: for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (p '&' q)) = w | (still_not-bound_in (p '&' q)) holds
( J,v |= p '&' q iff J,w |= p '&' q )
set X = (still_not-bound_in p) \/ (still_not-bound_in q);
let v, w be Element of Valuations_in (Al,A); ::_thesis: ( v | (still_not-bound_in (p '&' q)) = w | (still_not-bound_in (p '&' q)) implies ( J,v |= p '&' q iff J,w |= p '&' q ) )
A2: still_not-bound_in (p '&' q) = (still_not-bound_in p) \/ (still_not-bound_in q) by QC_LANG3:10;
assume  v | (still_not-bound_in (p '&' q)) = w | (still_not-bound_in (p '&' q)) ; ::_thesis: ( J,v |= p '&' q iff J,w |= p '&' q )
then  ( v | (still_not-bound_in p) = w | (still_not-bound_in p) & v | (still_not-bound_in q) = w | (still_not-bound_in q) ) by A2, RELAT_1:153, XBOOLE_1:7;
then  ( J,v |= p & J,v |= q iff ( J,w |= p & J,w |= q ) ) by A1;
hence  ( J,v |= p '&' q iff J,w |= p '&' q ) by VALUAT_1:18; ::_thesis: verum
end;

theorem Th63: :: SUBLEMMA:63
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A
for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A
for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A
for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for a being Element of A
for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
let v be Element of Valuations_in (Al,A); ::_thesis: for a being Element of A
for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
let a be Element of A; ::_thesis: for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
let X be set ; ::_thesis: ( X c= bound_QC-variables Al implies ( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X ) )
A1: dom ((v . (x | a)) | X) = (dom (v . (x | a))) /\ X by RELAT_1:61;
 dom (v | X) = (dom v) /\ X by RELAT_1:61;
then A2: dom (v | X) = (bound_QC-variables Al) /\ X by Th58;
assume  X c= bound_QC-variables Al ; ::_thesis: ( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
hence  ( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X ) by A2, A1, Th58, XBOOLE_1:28; ::_thesis: verum
end;

theorem :: SUBLEMMA:64
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for v, w being Element of Valuations_in (Al,A)
for a being Element of A st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for v, w being Element of Valuations_in (Al,A)
for a being Element of A st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v, w being Element of Valuations_in (Al,A)
for a being Element of A st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v, w being Element of Valuations_in (Al,A)
for a being Element of A st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
let A be non empty set ; ::_thesis: for v, w being Element of Valuations_in (Al,A)
for a being Element of A st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
let v, w be Element of Valuations_in (Al,A); ::_thesis: for a being Element of A st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
let a be Element of A; ::_thesis: ( v | (still_not-bound_in p) = w | (still_not-bound_in p) implies (v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p) )
assume A1: v | (still_not-bound_in p) = w | (still_not-bound_in p) ; ::_thesis: (v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
 dom (v | (still_not-bound_in p)) = dom ((v . (x | a)) | (still_not-bound_in p)) by Th63;
then A2: dom ((v . (x | a)) | (still_not-bound_in p)) = dom ((w . (x | a)) | (still_not-bound_in p)) by A1, Th63;
 for b being set st b in dom ((v . (x | a)) | (still_not-bound_in p)) holds
((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b
proof
let b be set ; ::_thesis: ( b in dom ((v . (x | a)) | (still_not-bound_in p)) implies ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b )
assume A3: b in dom ((v . (x | a)) | (still_not-bound_in p)) ; ::_thesis: ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b
A4: ( ((v . (x | a)) | (still_not-bound_in p)) . b = (v . (x | a)) . b & ((w . (x | a)) | (still_not-bound_in p)) . b = (w . (x | a)) . b ) by A2, A3, FUNCT_1:47;
 b in dom (v | (still_not-bound_in p)) by A3, Th63;
then A5: (v | (still_not-bound_in p)) . b = v . b by FUNCT_1:47;
 b in dom (w | (still_not-bound_in p)) by A1, A3, Th63;
then A6: v . b = w . b by A1, A5, FUNCT_1:47;
A7: now__::_thesis:_(_b_<>_x_implies_((v_._(x_|_a))_|_(still_not-bound_in_p))_._b_=_((w_._(x_|_a))_|_(still_not-bound_in_p))_._b_)
assume A8: b <> x ; ::_thesis: ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b
then  (v . (x | a)) . b = v . b by Th48;
hence  ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b by A4, A6, A8, Th48; ::_thesis: verum
end;
now__::_thesis:_(_b_=_x_implies_((v_._(x_|_a))_|_(still_not-bound_in_p))_._b_=_((w_._(x_|_a))_|_(still_not-bound_in_p))_._b_)
assume A9: b = x ; ::_thesis: ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b
then  (v . (x | a)) . b = a by Th49;
hence  ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b by A4, A9, Th49; ::_thesis: verum
end;
hence  ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b by A7; ::_thesis: verum
end;
hence  (v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p) by A2, FUNCT_1:2; ::_thesis: verum
end;

theorem :: SUBLEMMA:65
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al holds still_not-bound_in p c= (still_not-bound_in (All (x,p))) \/ {x}
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al holds still_not-bound_in p c= (still_not-bound_in (All (x,p))) \/ {x}
let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al holds still_not-bound_in p c= (still_not-bound_in (All (x,p))) \/ {x}
let x be bound_QC-variable of Al; ::_thesis: still_not-bound_in p c= (still_not-bound_in (All (x,p))) \/ {x}
set X = (still_not-bound_in p) \ {x};
 ( still_not-bound_in (All (x,p)) = (still_not-bound_in p) \ {x} & {x} \/ ((still_not-bound_in p) \ {x}) = {x} \/ (still_not-bound_in p) ) by QC_LANG3:12, XBOOLE_1:39;
hence  still_not-bound_in p c= (still_not-bound_in (All (x,p))) \/ {x} by XBOOLE_1:7; ::_thesis: verum
end;

theorem Th66: :: SUBLEMMA:66
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for v, w being Element of Valuations_in (Al,A)
for a being Element of A st v | ((still_not-bound_in p) \ {x}) = w | ((still_not-bound_in p) \ {x}) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for v, w being Element of Valuations_in (Al,A)
for a being Element of A st v | ((still_not-bound_in p) \ {x}) = w | ((still_not-bound_in p) \ {x}) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v, w being Element of Valuations_in (Al,A)
for a being Element of A st v | ((still_not-bound_in p) \ {x}) = w | ((still_not-bound_in p) \ {x}) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v, w being Element of Valuations_in (Al,A)
for a being Element of A st v | ((still_not-bound_in p) \ {x}) = w | ((still_not-bound_in p) \ {x}) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
let A be non empty set ; ::_thesis: for v, w being Element of Valuations_in (Al,A)
for a being Element of A st v | ((still_not-bound_in p) \ {x}) = w | ((still_not-bound_in p) \ {x}) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
let v, w be Element of Valuations_in (Al,A); ::_thesis: for a being Element of A st v | ((still_not-bound_in p) \ {x}) = w | ((still_not-bound_in p) \ {x}) holds
(v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
let a be Element of A; ::_thesis: ( v | ((still_not-bound_in p) \ {x}) = w | ((still_not-bound_in p) \ {x}) implies (v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p) )
A1: dom ((w . (x | a)) | (still_not-bound_in p)) = still_not-bound_in p by Th63;
then A2: dom ((v . (x | a)) | (still_not-bound_in p)) = dom ((w . (x | a)) | (still_not-bound_in p)) by Th63;
assume A3: v | ((still_not-bound_in p) \ {x}) = w | ((still_not-bound_in p) \ {x}) ; ::_thesis: (v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p)
 for b being set st b in dom ((v . (x | a)) | (still_not-bound_in p)) holds
((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b
proof
let b be set ; ::_thesis: ( b in dom ((v . (x | a)) | (still_not-bound_in p)) implies ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b )
assume A4: b in dom ((v . (x | a)) | (still_not-bound_in p)) ; ::_thesis: ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b
A5: ( ((v . (x | a)) | (still_not-bound_in p)) . b = (v . (x | a)) . b & ((w . (x | a)) | (still_not-bound_in p)) . b = (w . (x | a)) . b ) by A2, A4, FUNCT_1:47;
A6: now__::_thesis:_(_b_<>_x_implies_((v_._(x_|_a))_|_(still_not-bound_in_p))_._b_=_((w_._(x_|_a))_|_(still_not-bound_in_p))_._b_)
assume A7: b <> x ; ::_thesis: ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b
then A8: not b in {x} by TARSKI:def_1;
 b in still_not-bound_in p by A4, Th63;
then A9: b in (still_not-bound_in p) \ {x} by A8, XBOOLE_0:def_5;
then  b in dom (w | ((still_not-bound_in p) \ {x})) by Th63;
then A10: (w | ((still_not-bound_in p) \ {x})) . b = w . b by FUNCT_1:47;
A11: ( (v . (x | a)) . b = v . b & (w . (x | a)) . b = w . b ) by A7, Th48;
 b in dom (v | ((still_not-bound_in p) \ {x})) by A9, Th63;
hence  ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b by A3, A5, A10, A11, FUNCT_1:47; ::_thesis: verum
end;
now__::_thesis:_(_b_=_x_implies_((v_._(x_|_a))_|_(still_not-bound_in_p))_._b_=_((w_._(x_|_a))_|_(still_not-bound_in_p))_._b_)
A12: ((w . (x | a)) | (still_not-bound_in p)) . b = (w . (x | a)) . b by A2, A4, FUNCT_1:47;
assume A13: b = x ; ::_thesis: ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b
 ((v . (x | a)) | (still_not-bound_in p)) . b = (v . (x | a)) . b by A4, FUNCT_1:47;
then  ((v . (x | a)) | (still_not-bound_in p)) . b = a by A13, Th49;
hence  ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b by A13, A12, Th49; ::_thesis: verum
end;
hence  ((v . (x | a)) | (still_not-bound_in p)) . b = ((w . (x | a)) | (still_not-bound_in p)) . b by A6; ::_thesis: verum
end;
hence  (v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p) by A1, Th63, FUNCT_1:2; ::_thesis: verum
end;

theorem Th67: :: SUBLEMMA:67
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (All (x,p))) = w | (still_not-bound_in (All (x,p))) holds
( J,v |= All (x,p) iff J,w |= All (x,p) )
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (All (x,p))) = w | (still_not-bound_in (All (x,p))) holds
( J,v |= All (x,p) iff J,w |= All (x,p) )
let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (All (x,p))) = w | (still_not-bound_in (All (x,p))) holds
( J,v |= All (x,p) iff J,w |= All (x,p) )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (All (x,p))) = w | (still_not-bound_in (All (x,p))) holds
( J,v |= All (x,p) iff J,w |= All (x,p) )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A st ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) holds
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (All (x,p))) = w | (still_not-bound_in (All (x,p))) holds
( J,v |= All (x,p) iff J,w |= All (x,p) )
let J be interpretation of Al,A; ::_thesis: ( ( for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ) implies for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (All (x,p))) = w | (still_not-bound_in (All (x,p))) holds
( J,v |= All (x,p) iff J,w |= All (x,p) ) )
assume A1: for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p ) ; ::_thesis: for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in (All (x,p))) = w | (still_not-bound_in (All (x,p))) holds
( J,v |= All (x,p) iff J,w |= All (x,p) )
set X = (still_not-bound_in p) \ {x};
let v, w be Element of Valuations_in (Al,A); ::_thesis: ( v | (still_not-bound_in (All (x,p))) = w | (still_not-bound_in (All (x,p))) implies ( J,v |= All (x,p) iff J,w |= All (x,p) ) )
A2: v | (still_not-bound_in (All (x,p))) = v | ((still_not-bound_in p) \ {x}) by QC_LANG3:12;
assume  v | (still_not-bound_in (All (x,p))) = w | (still_not-bound_in (All (x,p))) ; ::_thesis: ( J,v |= All (x,p) iff J,w |= All (x,p) )
then A3: v | ((still_not-bound_in p) \ {x}) = w | ((still_not-bound_in p) \ {x}) by A2, QC_LANG3:12;
A4: ( ( for a being Element of A holds J,w . (x | a) |= p ) implies for a being Element of A holds J,v . (x | a) |= p )
proof
assume A5: for a being Element of A holds J,w . (x | a) |= p ; ::_thesis: for a being Element of A holds J,v . (x | a) |= p
let a be Element of A; ::_thesis: J,v . (x | a) |= p
 (v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p) by A3, Th66;
then  ( J,v . (x | a) |= p iff J,w . (x | a) |= p ) by A1;
hence  J,v . (x | a) |= p by A5; ::_thesis: verum
end;
 ( ( for a being Element of A holds J,v . (x | a) |= p ) implies for a being Element of A holds J,w . (x | a) |= p )
proof
assume A6: for a being Element of A holds J,v . (x | a) |= p ; ::_thesis: for a being Element of A holds J,w . (x | a) |= p
let a be Element of A; ::_thesis: J,w . (x | a) |= p
 (v . (x | a)) | (still_not-bound_in p) = (w . (x | a)) | (still_not-bound_in p) by A3, Th66;
then  ( J,v . (x | a) |= p iff J,w . (x | a) |= p ) by A1;
hence  J,w . (x | a) |= p by A6; ::_thesis: verum
end;
hence  ( J,v |= All (x,p) iff J,w |= All (x,p) ) by A4, Th50; ::_thesis: verum
end;

theorem Th68: :: SUBLEMMA:68
for Al being QC-alphabet
for A being non empty set
for J being interpretation of Al,A
for p being Element of CQC-WFF Al
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p )
proof
let Al be QC-alphabet ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for p being Element of CQC-WFF Al
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for p being Element of CQC-WFF Al
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p )
let J be interpretation of Al,A; ::_thesis: for p being Element of CQC-WFF Al
for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in p) = w | (still_not-bound_in p) holds
( J,v |= p iff J,w |= p )
defpred S1[ Element of CQC-WFF Al] means  for v, w being Element of Valuations_in (Al,A) st v | (still_not-bound_in $1) = w | (still_not-bound_in $1) holds
( J,v |= $1 iff J,w |= $1 );
A1: for p, q being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for k being Element of NAT
for l being CQC-variable_list of k,Al
for P being QC-pred_symbol of k,Al holds
( S1[ VERUM Al] & S1[P ! l] & ( S1[p] implies S1[ 'not' p] ) & ( S1[p] & S1[q] implies S1[p '&' q] ) & ( S1[p] implies S1[ All (x,p)] ) ) by Th60, Th61, Th62, Th67, VALUAT_1:32;
thus  for p being Element of CQC-WFF Al holds S1[p] from CQC_LANG:sch_1(A1); ::_thesis: verum
end;

theorem Th69: :: SUBLEMMA:69
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x]
for a being Element of A st [S,x] is quantifiable holds
((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x]
for a being Element of A st [S,x] is quantifiable holds
((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x]
for a being Element of A st [S,x] is quantifiable holds
((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x]
for a being Element of A st [S,x] is quantifiable holds
((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x]
for a being Element of A st [S,x] is quantifiable holds
((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x]
for a being Element of A st [S,x] is quantifiable holds
((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))
let xSQ be second_Q_comp of [S,x]; ::_thesis: for a being Element of A st [S,x] is quantifiable holds
((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))
let a be Element of A; ::_thesis: ( [S,x] is quantifiable implies ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) )
set S1 = CQCSub_All ([S,x],xSQ);
set z = S_Bound (@ (CQCSub_All ([S,x],xSQ)));
set finSub = RestrictSub (x,(All (x,(S `1))),xSQ);
set V1 = (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a));
set V2 = v . ((NEx_Val (v,S,x,xSQ)) +* (x | a));
set X = still_not-bound_in (S `1);
A1: dom ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) = (dom (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) \/ (dom ((NEx_Val (v,S,x,xSQ)) +* (x | a))) by FUNCT_4:def_1;
 dom (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) = bound_QC-variables Al by Th58;
then  dom (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) = dom v by Th58;
then A2: dom ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) = dom (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) by A1, FUNCT_4:def_1;
assume A3: [S,x] is quantifiable ; ::_thesis: ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))
A4: now__::_thesis:_(_not_x_in_rng_(RestrictSub_(x,(All_(x,(S_`1))),xSQ))_implies_((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a))_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_|_(still_not-bound_in_(S_`1))_=_(v_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_|_(still_not-bound_in_(S_`1))_)
assume  not x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) ; ::_thesis: ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))
then A5: S_Bound (@ (CQCSub_All ([S,x],xSQ))) = x by A3, Th52;
 for b being set st b in dom ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) holds
((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b
proof
let b be set ; ::_thesis: ( b in dom ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) implies ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b )
assume A6: b in dom ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) ; ::_thesis: ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b
A7: now__::_thesis:_(_b_<>_S_Bound_(@_(CQCSub_All_([S,x],xSQ)))_implies_((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a))_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_._b_=_(v_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_._b_)
assume A8: b <> S_Bound (@ (CQCSub_All ([S,x],xSQ))) ; ::_thesis: ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b
A9: now__::_thesis:_(_not_b_in_dom_(NEx_Val_(v,S,x,xSQ))_implies_((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a))_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_._b_=_(v_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_._b_)
A10: not b in dom (x | a) by A5, A8, TARSKI:def_1;
assume  not b in dom (NEx_Val (v,S,x,xSQ)) ; ::_thesis: ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b
then  not b in (dom (NEx_Val (v,S,x,xSQ))) \/ (dom (x | a)) by A10, XBOOLE_0:def_3;
then A11: not b in dom ((NEx_Val (v,S,x,xSQ)) +* (x | a)) by FUNCT_4:def_1;
reconsider x = b as bound_QC-variable of Al by A6;
 ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . b by A11, FUNCT_4:11;
then  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = v . x by A8, Th48;
hence  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b by A11, FUNCT_4:11; ::_thesis: verum
end;
now__::_thesis:_(_b_in_dom_(NEx_Val_(v,S,x,xSQ))_implies_((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a))_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_._b_=_(v_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_._b_)
 dom ((NEx_Val (v,S,x,xSQ)) +* (x | a)) = (dom (NEx_Val (v,S,x,xSQ))) \/ (dom (x | a)) by FUNCT_4:def_1;
then A12: dom (NEx_Val (v,S,x,xSQ)) c= dom ((NEx_Val (v,S,x,xSQ)) +* (x | a)) by XBOOLE_1:7;
assume A13: b in dom (NEx_Val (v,S,x,xSQ)) ; ::_thesis: ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b
then  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = ((NEx_Val (v,S,x,xSQ)) +* (x | a)) . b by A12, FUNCT_4:13;
hence  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b by A13, A12, FUNCT_4:13; ::_thesis: verum
end;
hence  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b by A9; ::_thesis: verum
end;
now__::_thesis:_(_b_=_S_Bound_(@_(CQCSub_All_([S,x],xSQ)))_implies_((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a))_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_._b_=_(v_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_._b_)
assume  b = S_Bound (@ (CQCSub_All ([S,x],xSQ))) ; ::_thesis: ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b
then  b in {x} by A5, TARSKI:def_1;
then A14: b in dom (x | a) by Th58;
 dom ((NEx_Val (v,S,x,xSQ)) +* (x | a)) = (dom (NEx_Val (v,S,x,xSQ))) \/ (dom (x | a)) by FUNCT_4:def_1;
then A15: dom (x | a) c= dom ((NEx_Val (v,S,x,xSQ)) +* (x | a)) by XBOOLE_1:7;
then  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = ((NEx_Val (v,S,x,xSQ)) +* (x | a)) . b by A14, FUNCT_4:13;
hence  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b by A14, A15, FUNCT_4:13; ::_thesis: verum
end;
hence  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) . b by A7; ::_thesis: verum
end;
hence  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) by A2, FUNCT_1:2; ::_thesis: verum
end;
now__::_thesis:_(_x_in_rng_(RestrictSub_(x,(All_(x,(S_`1))),xSQ))_implies_((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a))_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_|_(still_not-bound_in_(S_`1))_=_(v_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_|_(still_not-bound_in_(S_`1))_)
assume A16: x in rng (RestrictSub (x,(All (x,(S `1))),xSQ)) ; ::_thesis: ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))
A17: dom (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) = ((dom (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) \/ (dom ((NEx_Val (v,S,x,xSQ)) +* (x | a)))) /\ (still_not-bound_in (S `1)) by A1, RELAT_1:61;
A18: for b being set st b in dom (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) holds
(((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = ((v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b
proof
A19: still_not-bound_in (S `1) c= Bound_Vars (S `1) by Th47;
let b be set ; ::_thesis: ( b in dom (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) implies (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = ((v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b )
assume A20: b in dom (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) ; ::_thesis: (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = ((v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b
 b in still_not-bound_in (S `1) by A17, A20, XBOOLE_0:def_4;
then A21: b <> S_Bound (@ (CQCSub_All ([S,x],xSQ))) by A3, A16, A19, Th38;
A22: (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v | (still_not-bound_in (S `1))) +* (((NEx_Val (v,S,x,xSQ)) +* (x | a)) | (still_not-bound_in (S `1))) by FUNCT_4:71;
A23: ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) | (still_not-bound_in (S `1))) +* (((NEx_Val (v,S,x,xSQ)) +* (x | a)) | (still_not-bound_in (S `1))) by FUNCT_4:71;
then A24: dom (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) = (dom ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) | (still_not-bound_in (S `1)))) \/ (dom (((NEx_Val (v,S,x,xSQ)) +* (x | a)) | (still_not-bound_in (S `1)))) by FUNCT_4:def_1;
A25: now__::_thesis:_(_not_b_in_dom_(((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a))_|_(still_not-bound_in_(S_`1)))_implies_(((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a))_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_|_(still_not-bound_in_(S_`1)))_._b_=_((v_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_|_(still_not-bound_in_(S_`1)))_._b_)
assume A26: not b in dom (((NEx_Val (v,S,x,xSQ)) +* (x | a)) | (still_not-bound_in (S `1))) ; ::_thesis: (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = ((v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b
then A27: b in dom ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) | (still_not-bound_in (S `1))) by A20, A24, XBOOLE_0:def_3;
then  b in (dom (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a))) /\ (still_not-bound_in (S `1)) by RELAT_1:61;
then A28: b in still_not-bound_in (S `1) by XBOOLE_0:def_4;
 b in bound_QC-variables Al by A20;
then  b in dom v by Th58;
then A29: b in (dom v) /\ (still_not-bound_in (S `1)) by A28, XBOOLE_0:def_4;
 (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) | (still_not-bound_in (S `1))) . b by A23, A26, FUNCT_4:11;
then  (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . b by A27, FUNCT_1:47;
then A30: (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = v . b by A21, Th48;
 ((v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = (v | (still_not-bound_in (S `1))) . b by A22, A26, FUNCT_4:11;
hence  (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = ((v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b by A30, A29, FUNCT_1:48; ::_thesis: verum
end;
now__::_thesis:_(_b_in_dom_(((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a))_|_(still_not-bound_in_(S_`1)))_implies_(((v_._((S_Bound_(@_(CQCSub_All_([S,x],xSQ))))_|_a))_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_|_(still_not-bound_in_(S_`1)))_._b_=_((v_._((NEx_Val_(v,S,x,xSQ))_+*_(x_|_a)))_|_(still_not-bound_in_(S_`1)))_._b_)
assume A31: b in dom (((NEx_Val (v,S,x,xSQ)) +* (x | a)) | (still_not-bound_in (S `1))) ; ::_thesis: (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = ((v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b
then  (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = (((NEx_Val (v,S,x,xSQ)) +* (x | a)) | (still_not-bound_in (S `1))) . b by A23, FUNCT_4:13;
hence  (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = ((v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b by A22, A31, FUNCT_4:13; ::_thesis: verum
end;
hence  (((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b = ((v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) . b by A25; ::_thesis: verum
end;
 dom ((v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1))) = (dom ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)))) /\ (still_not-bound_in (S `1)) by A2, RELAT_1:61;
hence  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) by A18, FUNCT_1:2, RELAT_1:61; ::_thesis: verum
end;
hence  ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) by A4; ::_thesis: verum
end;

theorem Th70: :: SUBLEMMA:70
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) )
set S1 = CQCSub_All ([S,x],xSQ);
set z = S_Bound (@ (CQCSub_All ([S,x],xSQ)));
assume A1: [S,x] is quantifiable ; ::_thesis: ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
thus  ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) implies for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) ::_thesis: ( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
proof
set X = still_not-bound_in (S `1);
assume A2: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ; ::_thesis: for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
let a be Element of A; ::_thesis: J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
set V1 = (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a));
set V2 = v . ((NEx_Val (v,S,x,xSQ)) +* (x | a));
 ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) by A1, Th69;
then A3: ( J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S `1 iff J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S `1 ) by Th68;
 J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S by A2;
hence  J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S by A3, Def2; ::_thesis: verum
end;
thus  ( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) ::_thesis: verum
proof
set X = still_not-bound_in (S `1);
assume A4: for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ; ::_thesis: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
let a be Element of A; ::_thesis: J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
set V1 = (v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a));
set V2 = v . ((NEx_Val (v,S,x,xSQ)) +* (x | a));
 ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) = (v . ((NEx_Val (v,S,x,xSQ)) +* (x | a))) | (still_not-bound_in (S `1)) by A1, Th69;
then A5: ( J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S `1 iff J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S `1 ) by Th68;
 J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S by A4;
hence  J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S by A5, Def2; ::_thesis: verum
end;
end;

theorem Th71: :: SUBLEMMA:71
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds dom (NEx_Val (v,S,x,xSQ)) = dom (RestrictSub (x,(All (x,(S `1))),xSQ))
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds dom (NEx_Val (v,S,x,xSQ)) = dom (RestrictSub (x,(All (x,(S `1))),xSQ))
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds dom (NEx_Val (v,S,x,xSQ)) = dom (RestrictSub (x,(All (x,(S `1))),xSQ))
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds dom (NEx_Val (v,S,x,xSQ)) = dom (RestrictSub (x,(All (x,(S `1))),xSQ))
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds dom (NEx_Val (v,S,x,xSQ)) = dom (RestrictSub (x,(All (x,(S `1))),xSQ))
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] holds dom (NEx_Val (v,S,x,xSQ)) = dom (RestrictSub (x,(All (x,(S `1))),xSQ))
let xSQ be second_Q_comp of [S,x]; ::_thesis: dom (NEx_Val (v,S,x,xSQ)) = dom (RestrictSub (x,(All (x,(S `1))),xSQ))
 rng (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) c= bound_QC-variables Al ;
then  rng (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) c= dom v by Th58;
then  dom (NEx_Val (v,S,x,xSQ)) = dom (@ (RestrictSub (x,(All (x,(S `1))),xSQ))) by RELAT_1:27;
hence  dom (NEx_Val (v,S,x,xSQ)) = dom (RestrictSub (x,(All (x,(S `1))),xSQ)) by SUBSTUT1:def_2; ::_thesis: verum
end;

theorem Th72: :: SUBLEMMA:72
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
thus  ( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) ::_thesis: ( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) implies for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
proof
assume A1: for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ; ::_thesis: for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S
let a be Element of A; ::_thesis: J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S
 v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) = (v . (NEx_Val (v,S,x,xSQ))) . (x | a) by FUNCT_4:14;
hence  J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S by A1; ::_thesis: verum
end;
thus  ( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) implies for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) ::_thesis: verum
proof
assume A2: for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ; ::_thesis: for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
let a be Element of A; ::_thesis: J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
 v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) = (v . (NEx_Val (v,S,x,xSQ))) . (x | a) by FUNCT_4:14;
hence  J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S by A2; ::_thesis: verum
end;
end;

theorem Th73: :: SUBLEMMA:73
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 )
let J be interpretation of Al,A; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 )
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 )
thus  ( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) implies for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 ) ::_thesis: ( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 ) implies for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
proof
assume A1: for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ; ::_thesis: for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1
let a be Element of A; ::_thesis: J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1
 ( J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S iff J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 ) by Def2;
hence  J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 by A1; ::_thesis: verum
end;
thus  ( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 ) implies for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) ::_thesis: verum
proof
assume A2: for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 ; ::_thesis: for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S
let a be Element of A; ::_thesis: J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S
 ( J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S iff J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 ) by Def2;
hence  J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S by A2; ::_thesis: verum
end;
end;

theorem Th74: :: SUBLEMMA:74
for Al being QC-alphabet
for k being Element of NAT
for A being non empty set
for ll being CQC-variable_list of k,Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ll ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
(v . vS) *' ll = (v . ((vS +* vS1) +* vS2)) *' ll
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for A being non empty set
for ll being CQC-variable_list of k,Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ll ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
(v . vS) *' ll = (v . ((vS +* vS1) +* vS2)) *' ll
let k be Element of NAT ; ::_thesis: for A being non empty set
for ll being CQC-variable_list of k,Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ll ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
(v . vS) *' ll = (v . ((vS +* vS1) +* vS2)) *' ll
let A be non empty set ; ::_thesis: for ll being CQC-variable_list of k,Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ll ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
(v . vS) *' ll = (v . ((vS +* vS1) +* vS2)) *' ll
let ll be CQC-variable_list of k,Al; ::_thesis: for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ll ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
(v . vS) *' ll = (v . ((vS +* vS1) +* vS2)) *' ll
let v be Element of Valuations_in (Al,A); ::_thesis: for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ll ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
(v . vS) *' ll = (v . ((vS +* vS1) +* vS2)) *' ll
let vS, vS1, vS2 be Val_Sub of A,Al; ::_thesis: ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ll ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 implies (v . vS) *' ll = (v . ((vS +* vS1) +* vS2)) *' ll )
assume that
A1: for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ll and
A2: for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y and
A3: dom vS misses dom vS2 ; ::_thesis: (v . vS) *' ll = (v . ((vS +* vS1) +* vS2)) *' ll
set ll2 = (v . ((vS +* vS1) +* vS2)) *' ll;
set ll1 = (v . vS) *' ll;
A4: len ((v . vS) *' ll) = k by VALUAT_1:def_3;
then A5: dom ((v . vS) *' ll) = Seg k by FINSEQ_1:def_3;
A6: len ((v . ((vS +* vS1) +* vS2)) *' ll) = k by VALUAT_1:def_3;
 for i being natural number st i in dom ((v . vS) *' ll) holds
((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i
proof
let i be natural number ; ::_thesis: ( i in dom ((v . vS) *' ll) implies ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i )
assume A7: i in dom ((v . vS) *' ll) ; ::_thesis: ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i
A8: i in dom ((v . ((vS +* vS1) +* vS2)) *' ll) by A6, A5, A7, FINSEQ_1:def_3;
reconsider i = i as Element of NAT by ORDINAL1:def_12;
A9: dom ((v . ((vS +* vS1) +* vS2)) *' ll) c= dom ll by RELAT_1:25;
A10: now__::_thesis:_(_ll_._i_in_dom_((vS_+*_vS1)_+*_vS2)_implies_((v_._vS)_*'_ll)_._i_=_((v_._((vS_+*_vS1)_+*_vS2))_*'_ll)_._i_)
reconsider x = ll . i as bound_QC-variable of Al by A8, A9, Th5;
assume A11: ll . i in dom ((vS +* vS1) +* vS2) ; ::_thesis: ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i
A12: now__::_thesis:_(_not_x_in_dom_vS2_implies_((v_._vS)_*'_ll)_._i_=_((v_._((vS_+*_vS1)_+*_vS2))_*'_ll)_._i_)
A13: now__::_thesis:_not_x_in_dom_vS1
 len ll = k by SUBSTUT1:34;
then A14: i <= len ll by A5, A7, FINSEQ_1:1;
 1 <= i by A5, A7, FINSEQ_1:1;
then  x in  {  (ll . n) where n is Element of NAT : ( 1 <= n & n <= len ll & ll . n in bound_QC-variables Al )  }  by A14;
then A15: x in variables_in (ll,(bound_QC-variables Al)) by QC_LANG3:def_1;
assume  x in dom vS1 ; ::_thesis: contradiction
then  not x in still_not-bound_in ll by A1;
hence  contradiction by A15, QC_LANG3:2; ::_thesis: verum
end;
assume A16: not x in dom vS2 ; ::_thesis: ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i
then A17: ((vS +* vS1) +* vS2) . x = (vS +* vS1) . x by FUNCT_4:11;
A18: x in dom (vS +* vS1) by A11, A16, FUNCT_4:12;
now__::_thesis:_(_not_x_in_dom_vS1_implies_((v_._vS)_*'_ll)_._i_=_((v_._((vS_+*_vS1)_+*_vS2))_*'_ll)_._i_)
assume A19: not x in dom vS1 ; ::_thesis: ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i
then  ((vS +* vS1) +* vS2) . x = vS . x by A17, FUNCT_4:11;
then A20: (v +* ((vS +* vS1) +* vS2)) . x = vS . x by A11, FUNCT_4:13;
 x in dom vS by A18, A19, FUNCT_4:12;
then  (v . ((vS +* vS1) +* vS2)) . x = (v +* vS) . x by A20, FUNCT_4:13;
then  ((v . ((vS +* vS1) +* vS2)) *' ll) . i = (v . vS) . x by A8, FUNCT_1:12;
hence  ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i by A7, FUNCT_1:12; ::_thesis: verum
end;
hence  ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i by A13; ::_thesis: verum
end;
now__::_thesis:_(_x_in_dom_vS2_implies_((v_._vS)_*'_ll)_._i_=_((v_._((vS_+*_vS1)_+*_vS2))_*'_ll)_._i_)
assume A21: x in dom vS2 ; ::_thesis: ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i
then  ((vS +* vS1) +* vS2) . x = vS2 . x by FUNCT_4:13;
then  ((vS +* vS1) +* vS2) . x = v . x by A2, A21;
then  (v +* ((vS +* vS1) +* vS2)) . x = v . x by A11, FUNCT_4:13;
then A22: ((v . ((vS +* vS1) +* vS2)) *' ll) . i = v . x by A8, FUNCT_1:12;
 not x in dom vS by A3, A21, XBOOLE_0:3;
then  (v +* vS) . x = v . x by FUNCT_4:11;
hence  ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i by A7, A22, FUNCT_1:12; ::_thesis: verum
end;
hence  ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i by A12; ::_thesis: verum
end;
now__::_thesis:_(_not_ll_._i_in_dom_((vS_+*_vS1)_+*_vS2)_implies_((v_._vS)_*'_ll)_._i_=_((v_._((vS_+*_vS1)_+*_vS2))_*'_ll)_._i_)
assume A23: not ll . i in dom ((vS +* vS1) +* vS2) ; ::_thesis: ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i
then  not ll . i in dom (vS +* vS1) by FUNCT_4:12;
then  not ll . i in dom vS by FUNCT_4:12;
then  (v +* vS) . (ll . i) = v . (ll . i) by FUNCT_4:11;
then A24: ((v . vS) *' ll) . i = v . (ll . i) by A7, FUNCT_1:12;
 (v +* ((vS +* vS1) +* vS2)) . (ll . i) = v . (ll . i) by A23, FUNCT_4:11;
hence  ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i by A8, A24, FUNCT_1:12; ::_thesis: verum
end;
hence  ((v . vS) *' ll) . i = ((v . ((vS +* vS1) +* vS2)) *' ll) . i by A10; ::_thesis: verum
end;
hence  (v . vS) *' ll = (v . ((vS +* vS1) +* vS2)) *' ll by A4, A6, FINSEQ_2:9; ::_thesis: verum
end;

theorem Th75: :: SUBLEMMA:75
for Al being QC-alphabet
for k being Element of NAT
for A being non empty set
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
proof
let Al be QC-alphabet ; ::_thesis: for k being Element of NAT
for A being non empty set
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let k be Element of NAT ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let J be interpretation of Al,A; ::_thesis: for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let P be QC-pred_symbol of k,Al; ::_thesis: for ll being CQC-variable_list of k,Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let ll be CQC-variable_list of k,Al; ::_thesis: for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let v be Element of Valuations_in (Al,A); ::_thesis: for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
let vS, vS1, vS2 be Val_Sub of A,Al; ::_thesis: ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 implies ( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll ) )
assume that
A1: for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (P ! ll) and
A2: ( ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 ) ; ::_thesis: ( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll )
A3: for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ll
proof
let y be bound_QC-variable of Al; ::_thesis: ( y in dom vS1 implies not y in still_not-bound_in ll )
assume  y in dom vS1 ; ::_thesis: not y in still_not-bound_in ll
then  not y in still_not-bound_in (P ! ll) by A1;
hence  not y in still_not-bound_in ll by QC_LANG3:5; ::_thesis: verum
end;
A4: ( (v . ((vS +* vS1) +* vS2)) *' ll in J . P iff (Valid ((P ! ll),J)) . (v . ((vS +* vS1) +* vS2)) = TRUE ) by VALUAT_1:7;
 ( (Valid ((P ! ll),J)) . (v . vS) = TRUE iff (v . vS) *' ll in J . P ) by VALUAT_1:7;
hence  ( J,v . vS |= P ! ll iff J,v . ((vS +* vS1) +* vS2) |= P ! ll ) by A2, A3, A4, Th74, VALUAT_1:def_7; ::_thesis: verum
end;

theorem Th76: :: SUBLEMMA:76
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for A being non empty set
for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ('not' p) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= 'not' p iff J,v . ((vS +* vS1) +* vS2) |= 'not' p )
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for A being non empty set
for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ('not' p) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= 'not' p iff J,v . ((vS +* vS1) +* vS2) |= 'not' p )
let p be Element of CQC-WFF Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ('not' p) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= 'not' p iff J,v . ((vS +* vS1) +* vS2) |= 'not' p )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ('not' p) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= 'not' p iff J,v . ((vS +* vS1) +* vS2) |= 'not' p )
let J be interpretation of Al,A; ::_thesis: ( ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) implies for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ('not' p) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= 'not' p iff J,v . ((vS +* vS1) +* vS2) |= 'not' p ) )
assume A1: for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ; ::_thesis: for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ('not' p) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= 'not' p iff J,v . ((vS +* vS1) +* vS2) |= 'not' p )
let v be Element of Valuations_in (Al,A); ::_thesis: for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ('not' p) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= 'not' p iff J,v . ((vS +* vS1) +* vS2) |= 'not' p )
let vS, vS1, vS2 be Val_Sub of A,Al; ::_thesis: ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ('not' p) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 implies ( J,v . vS |= 'not' p iff J,v . ((vS +* vS1) +* vS2) |= 'not' p ) )
assume that
A2: for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in ('not' p) and
A3: ( ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 ) ; ::_thesis: ( J,v . vS |= 'not' p iff J,v . ((vS +* vS1) +* vS2) |= 'not' p )
 for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p
proof
let y be bound_QC-variable of Al; ::_thesis: ( y in dom vS1 implies not y in still_not-bound_in p )
assume  y in dom vS1 ; ::_thesis: not y in still_not-bound_in p
then  not y in still_not-bound_in ('not' p) by A2;
hence  not y in still_not-bound_in p by QC_LANG3:7; ::_thesis: verum
end;
then  ( not J,v . vS |= p iff not J,v . ((vS +* vS1) +* vS2) |= p ) by A1, A3;
hence  ( J,v . vS |= 'not' p iff J,v . ((vS +* vS1) +* vS2) |= 'not' p ) by VALUAT_1:17; ::_thesis: verum
end;

theorem Th77: :: SUBLEMMA:77
for Al being QC-alphabet
for p, q being Element of CQC-WFF Al
for A being non empty set
for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) & ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in q ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= q iff J,v . ((vS +* vS1) +* vS2) |= q ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (p '&' q) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p '&' q iff J,v . ((vS +* vS1) +* vS2) |= p '&' q )
proof
let Al be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF Al
for A being non empty set
for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) & ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in q ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= q iff J,v . ((vS +* vS1) +* vS2) |= q ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (p '&' q) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p '&' q iff J,v . ((vS +* vS1) +* vS2) |= p '&' q )
let p, q be Element of CQC-WFF Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) & ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in q ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= q iff J,v . ((vS +* vS1) +* vS2) |= q ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (p '&' q) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p '&' q iff J,v . ((vS +* vS1) +* vS2) |= p '&' q )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) & ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in q ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= q iff J,v . ((vS +* vS1) +* vS2) |= q ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (p '&' q) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p '&' q iff J,v . ((vS +* vS1) +* vS2) |= p '&' q )
let J be interpretation of Al,A; ::_thesis: ( ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) & ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in q ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= q iff J,v . ((vS +* vS1) +* vS2) |= q ) ) implies for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (p '&' q) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p '&' q iff J,v . ((vS +* vS1) +* vS2) |= p '&' q ) )
assume A1: ( ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) & ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in q ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= q iff J,v . ((vS +* vS1) +* vS2) |= q ) ) ) ; ::_thesis: for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (p '&' q) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p '&' q iff J,v . ((vS +* vS1) +* vS2) |= p '&' q )
let v be Element of Valuations_in (Al,A); ::_thesis: for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (p '&' q) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p '&' q iff J,v . ((vS +* vS1) +* vS2) |= p '&' q )
let vS, vS1, vS2 be Val_Sub of A,Al; ::_thesis: ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (p '&' q) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 implies ( J,v . vS |= p '&' q iff J,v . ((vS +* vS1) +* vS2) |= p '&' q ) )
assume that
A2: for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (p '&' q) and
A3: ( ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 ) ; ::_thesis: ( J,v . vS |= p '&' q iff J,v . ((vS +* vS1) +* vS2) |= p '&' q )
A4: for y being bound_QC-variable of Al st y in dom vS1 holds
( not y in still_not-bound_in p & not y in still_not-bound_in q )
proof
let y be bound_QC-variable of Al; ::_thesis: ( y in dom vS1 implies ( not y in still_not-bound_in p & not y in still_not-bound_in q ) )
assume  y in dom vS1 ; ::_thesis: ( not y in still_not-bound_in p & not y in still_not-bound_in q )
then  not y in still_not-bound_in (p '&' q) by A2;
then  not y in (still_not-bound_in p) \/ (still_not-bound_in q) by QC_LANG3:10;
hence  ( not y in still_not-bound_in p & not y in still_not-bound_in q ) by XBOOLE_0:def_3; ::_thesis: verum
end;
A5: ( J,v . ((vS +* vS1) +* vS2) |= p & J,v . ((vS +* vS1) +* vS2) |= q implies ( J,v . vS |= p & J,v . vS |= q ) )
proof
assume A6: ( J,v . ((vS +* vS1) +* vS2) |= p & J,v . ((vS +* vS1) +* vS2) |= q ) ; ::_thesis: ( J,v . vS |= p & J,v . vS |= q )
 ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in q ) ) by A4;
hence  ( J,v . vS |= p & J,v . vS |= q ) by A1, A3, A6; ::_thesis: verum
end;
 ( J,v . vS |= p & J,v . vS |= q implies ( J,v . ((vS +* vS1) +* vS2) |= p & J,v . ((vS +* vS1) +* vS2) |= q ) )
proof
assume A7: ( J,v . vS |= p & J,v . vS |= q ) ; ::_thesis: ( J,v . ((vS +* vS1) +* vS2) |= p & J,v . ((vS +* vS1) +* vS2) |= q )
 ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in q ) ) by A4;
hence  ( J,v . ((vS +* vS1) +* vS2) |= p & J,v . ((vS +* vS1) +* vS2) |= q ) by A1, A3, A7; ::_thesis: verum
end;
hence  ( J,v . vS |= p '&' q iff J,v . ((vS +* vS1) +* vS2) |= p '&' q ) by A5, VALUAT_1:18; ::_thesis: verum
end;

theorem Th78: :: SUBLEMMA:78
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for vS1 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
not y in still_not-bound_in p
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for vS1 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
not y in still_not-bound_in p
let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for vS1 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
not y in still_not-bound_in p
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for vS1 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
not y in still_not-bound_in p
let A be non empty set ; ::_thesis: for vS1 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
not y in still_not-bound_in p
let vS1 be Val_Sub of A,Al; ::_thesis: ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) implies for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
not y in still_not-bound_in p )
assume A1: for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ; ::_thesis: for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
not y in still_not-bound_in p
let y be bound_QC-variable of Al; ::_thesis: ( y in (dom vS1) \ {x} implies not y in still_not-bound_in p )
assume A2: y in (dom vS1) \ {x} ; ::_thesis: not y in still_not-bound_in p
 (dom vS1) \ {x} c= dom vS1 by XBOOLE_1:36;
then  not y in still_not-bound_in (All (x,p)) by A1, A2;
then A3: not y in (still_not-bound_in p) \ {x} by QC_LANG3:12;
A4: {x} \/ ((still_not-bound_in p) \ {x}) = {x} \/ (still_not-bound_in p) by XBOOLE_1:39;
 not y in {x} by A2, XBOOLE_0:def_5;
then  not y in {x} \/ ((still_not-bound_in p) \ {x}) by A3, XBOOLE_0:def_3;
hence  not y in still_not-bound_in p by A4, XBOOLE_0:def_3; ::_thesis: verum
end;

theorem Th79: :: SUBLEMMA:79
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for vS being Val_Sub of A,Al
for vS1 being Function st ( for y being bound_QC-variable of Al st y in dom vS1 holds
vS1 . y = v . y ) & dom vS misses dom vS1 holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
(vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for vS being Val_Sub of A,Al
for vS1 being Function st ( for y being bound_QC-variable of Al st y in dom vS1 holds
vS1 . y = v . y ) & dom vS misses dom vS1 holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
(vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for vS being Val_Sub of A,Al
for vS1 being Function st ( for y being bound_QC-variable of Al st y in dom vS1 holds
vS1 . y = v . y ) & dom vS misses dom vS1 holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
(vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for vS being Val_Sub of A,Al
for vS1 being Function st ( for y being bound_QC-variable of Al st y in dom vS1 holds
vS1 . y = v . y ) & dom vS misses dom vS1 holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
(vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y
let v be Element of Valuations_in (Al,A); ::_thesis: for vS being Val_Sub of A,Al
for vS1 being Function st ( for y being bound_QC-variable of Al st y in dom vS1 holds
vS1 . y = v . y ) & dom vS misses dom vS1 holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
(vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y
let vS be Val_Sub of A,Al; ::_thesis: for vS1 being Function st ( for y being bound_QC-variable of Al st y in dom vS1 holds
vS1 . y = v . y ) & dom vS misses dom vS1 holds
for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
(vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y
let vS1 be Function; ::_thesis: ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
vS1 . y = v . y ) & dom vS misses dom vS1 implies for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
(vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y )
assume that
A1: for y being bound_QC-variable of Al st y in dom vS1 holds
vS1 . y = v . y and
A2: dom vS misses dom vS1 ; ::_thesis: for y being bound_QC-variable of Al st y in (dom vS1) \ {x} holds
(vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y
let y be bound_QC-variable of Al; ::_thesis: ( y in (dom vS1) \ {x} implies (vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y )
assume A3: y in (dom vS1) \ {x} ; ::_thesis: (vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y
 y in (dom vS1) /\ ((dom vS1) \ {x}) by A3, XBOOLE_0:def_4;
then  (vS1 | ((dom vS1) \ {x})) . y = vS1 . y by FUNCT_1:48;
then A4: (vS1 | ((dom vS1) \ {x})) . y = v . y by A1, A3;
 not y in dom vS by A2, A3, XBOOLE_0:3;
hence  (vS1 | ((dom vS1) \ {x})) . y = (v . vS) . y by A4, FUNCT_4:11; ::_thesis: verum
end;

theorem Th80: :: SUBLEMMA:80
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) )
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) )
let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A st ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) holds
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) )
let J be interpretation of Al,A; ::_thesis: ( ( for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ) implies for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) ) )
assume A1: for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p ) ; ::_thesis: for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) )
let v be Element of Valuations_in (Al,A); ::_thesis: for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) )
let vS, vS1, vS2 be Val_Sub of A,Al; ::_thesis: ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 implies ( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) ) )
assume that
A2: for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,p)) and
A3: ( ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 ) ; ::_thesis: ( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) )
set vS19 = vS1 | ((dom vS1) \ {x});
set vS29 = vS2 | ((dom vS2) \ {x});
A4: for y being bound_QC-variable of Al st y in dom (vS2 | ((dom vS2) \ {x})) holds
(vS2 | ((dom vS2) \ {x})) . y = (v . vS) . y by A3, Th79;
A5: dom (vS2 | ((dom vS2) \ {x})) misses {x} by XBOOLE_1:63, XBOOLE_1:79;
A6: for y being bound_QC-variable of Al st y in dom (vS1 | ((dom vS1) \ {x})) holds
not y in still_not-bound_in p by A2, Th78;
A7: ( ( for a being Element of A holds J,(v . vS) . (x | a) |= p ) implies for a being Element of A holds J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p )
proof
assume A8: for a being Element of A holds J,(v . vS) . (x | a) |= p ; ::_thesis: for a being Element of A holds J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p
let a be Element of A; ::_thesis: J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p
 dom (vS2 | ((dom vS2) \ {x})) misses dom (x | a) by A5, Th58;
then  ( J,(v . vS) . (x | a) |= p iff J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p ) by A1, A6, A4;
hence  J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p by A8; ::_thesis: verum
end;
A9: dom (vS1 | ((dom vS1) \ {x})) misses {x} by XBOOLE_1:63, XBOOLE_1:79;
A10: for a being Element of A holds (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v . ((vS +* vS1) +* vS2)) . (x | a)
proof
let a be Element of A; ::_thesis: (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v . ((vS +* vS1) +* vS2)) . (x | a)
 dom (vS1 | ((dom vS1) \ {x})) misses dom (x | a) by A9, Th58;
then  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* (((vS1 | ((dom vS1) \ {x})) +* (x | a)) +* (vS2 | ((dom vS2) \ {x}))) by FUNCT_4:35;
then A11: (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 | ((dom vS1) \ {x})) +* ((x | a) +* (vS2 | ((dom vS2) \ {x})))) by FUNCT_4:14;
 dom (vS2 | ((dom vS2) \ {x})) misses dom (x | a) by A5, Th58;
then  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 | ((dom vS1) \ {x})) +* ((vS2 | ((dom vS2) \ {x})) +* (x | a))) by A11, FUNCT_4:35;
then A12: (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* (((vS1 | ((dom vS1) \ {x})) +* (vS2 | ((dom vS2) \ {x}))) +* (x | a)) by FUNCT_4:14;
A13: now__::_thesis:_(_x_in_dom_vS1_implies_(v_._vS)_._(((x_|_a)_+*_(vS1_|_((dom_vS1)_\_{x})))_+*_(vS2_|_((dom_vS2)_\_{x})))_=_(v_+*_vS)_+*_((vS1_+*_vS2)_+*_(x_|_a))_)
assume  x in dom vS1 ; ::_thesis: (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a))
then A14: (vS1 | ((dom vS1) \ {x})) +* (x .--> (vS1 . x)) = vS1 by Th2;
A15: now__::_thesis:_(_not_x_in_dom_vS2_implies_(v_._vS)_._(((x_|_a)_+*_(vS1_|_((dom_vS1)_\_{x})))_+*_(vS2_|_((dom_vS2)_\_{x})))_=_(v_+*_vS)_+*_((vS1_+*_vS2)_+*_(x_|_a))_)
assume  not x in dom vS2 ; ::_thesis: (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a))
then  vS2 | ((dom vS2) \ {x}) = vS2 | (dom vS2) by ZFMISC_1:57;
then  vS2 | ((dom vS2) \ {x}) = vS2 ;
then A16: (vS2 | ((dom vS2) \ {x})) +* {} = vS2 ;
 dom (x .--> (vS1 . x)) = {x} by FUNCOP_1:13;
then  ( dom {} c= dom (x | a) & dom (x .--> (vS1 . x)) = dom (x | a) ) by Th58, XBOOLE_1:2;
hence  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a)) by A12, A14, A16, Th1; ::_thesis: verum
end;
now__::_thesis:_(_x_in_dom_vS2_implies_(v_._vS)_._(((x_|_a)_+*_(vS1_|_((dom_vS1)_\_{x})))_+*_(vS2_|_((dom_vS2)_\_{x})))_=_(v_+*_vS)_+*_((vS1_+*_vS2)_+*_(x_|_a))_)
 dom (x .--> (vS2 . x)) = {x} by FUNCOP_1:13;
then A17: dom (x .--> (vS2 . x)) = dom (x | a) by Th58;
 dom (x .--> (vS1 . x)) = {x} by FUNCOP_1:13;
then A18: dom (x .--> (vS1 . x)) = dom (x | a) by Th58;
assume  x in dom vS2 ; ::_thesis: (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a))
then  (vS2 | ((dom vS2) \ {x})) +* (x .--> (vS2 . x)) = vS2 by Th2;
hence  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a)) by A12, A14, A18, A17, Th1; ::_thesis: verum
end;
hence  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a)) by A15; ::_thesis: verum
end;
now__::_thesis:_(_not_x_in_dom_vS1_implies_(v_._vS)_._(((x_|_a)_+*_(vS1_|_((dom_vS1)_\_{x})))_+*_(vS2_|_((dom_vS2)_\_{x})))_=_(v_+*_vS)_+*_((vS1_+*_vS2)_+*_(x_|_a))_)
A19: dom {} c= dom (x | a) by XBOOLE_1:2;
assume  not x in dom vS1 ; ::_thesis: (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a))
then  vS1 | ((dom vS1) \ {x}) = vS1 | (dom vS1) by ZFMISC_1:57;
then A20: vS1 | ((dom vS1) \ {x}) = vS1 by RELAT_1:68;
then A21: (vS1 | ((dom vS1) \ {x})) +* {} = vS1 ;
A22: now__::_thesis:_(_x_in_dom_vS2_implies_(v_._vS)_._(((x_|_a)_+*_(vS1_|_((dom_vS1)_\_{x})))_+*_(vS2_|_((dom_vS2)_\_{x})))_=_(v_+*_vS)_+*_((vS1_+*_vS2)_+*_(x_|_a))_)
 dom (x .--> (vS2 . x)) = {x} by FUNCOP_1:13;
then A23: dom (x .--> (vS2 . x)) = dom (x | a) by Th58;
assume  x in dom vS2 ; ::_thesis: (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a))
then  (vS2 | ((dom vS2) \ {x})) +* (x .--> (vS2 . x)) = vS2 by Th2;
hence  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a)) by A12, A21, A19, A23, Th1; ::_thesis: verum
end;
now__::_thesis:_(_not_x_in_dom_vS2_implies_(v_._vS)_._(((x_|_a)_+*_(vS1_|_((dom_vS1)_\_{x})))_+*_(vS2_|_((dom_vS2)_\_{x})))_=_(v_+*_vS)_+*_((vS1_+*_vS2)_+*_(x_|_a))_)
assume  not x in dom vS2 ; ::_thesis: (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a))
then  vS2 | ((dom vS2) \ {x}) = vS2 | (dom vS2) by ZFMISC_1:57;
hence  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a)) by A12, A20, RELAT_1:68; ::_thesis: verum
end;
hence  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v +* vS) +* ((vS1 +* vS2) +* (x | a)) by A22; ::_thesis: verum
end;
then  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = ((v +* vS) +* (vS1 +* vS2)) +* (x | a) by A13, FUNCT_4:14;
then  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (((v +* vS) +* vS1) +* vS2) +* (x | a) by FUNCT_4:14;
then  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = ((v +* (vS +* vS1)) +* vS2) +* (x | a) by FUNCT_4:14;
hence  (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v . ((vS +* vS1) +* vS2)) . (x | a) by FUNCT_4:14; ::_thesis: verum
end;
A24: ( ( for a being Element of A holds J,(v . ((vS +* vS1) +* vS2)) . (x | a) |= p ) implies for a being Element of A holds J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p )
proof
assume A25: for a being Element of A holds J,(v . ((vS +* vS1) +* vS2)) . (x | a) |= p ; ::_thesis: for a being Element of A holds J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p
let a be Element of A; ::_thesis: J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p
 (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v . ((vS +* vS1) +* vS2)) . (x | a) by A10;
hence  J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p by A25; ::_thesis: verum
end;
A26: ( ( for a being Element of A holds J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p ) implies for a being Element of A holds J,(v . vS) . (x | a) |= p )
proof
assume A27: for a being Element of A holds J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p ; ::_thesis: for a being Element of A holds J,(v . vS) . (x | a) |= p
let a be Element of A; ::_thesis: J,(v . vS) . (x | a) |= p
 dom (vS2 | ((dom vS2) \ {x})) misses dom (x | a) by A5, Th58;
then  ( J,(v . vS) . (x | a) |= p iff J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p ) by A1, A6, A4;
hence  J,(v . vS) . (x | a) |= p by A27; ::_thesis: verum
end;
 ( ( for a being Element of A holds J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p ) implies for a being Element of A holds J,(v . ((vS +* vS1) +* vS2)) . (x | a) |= p )
proof
assume A28: for a being Element of A holds J,(v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) |= p ; ::_thesis: for a being Element of A holds J,(v . ((vS +* vS1) +* vS2)) . (x | a) |= p
let a be Element of A; ::_thesis: J,(v . ((vS +* vS1) +* vS2)) . (x | a) |= p
 (v . vS) . (((x | a) +* (vS1 | ((dom vS1) \ {x}))) +* (vS2 | ((dom vS2) \ {x}))) = (v . ((vS +* vS1) +* vS2)) . (x | a) by A10;
hence  J,(v . ((vS +* vS1) +* vS2)) . (x | a) |= p by A28; ::_thesis: verum
end;
hence  ( J,v . vS |= All (x,p) iff J,v . ((vS +* vS1) +* vS2) |= All (x,p) ) by A7, A26, A24, Th50; ::_thesis: verum
end;

theorem Th81: :: SUBLEMMA:81
for Al being QC-alphabet
for A being non empty set
for J being interpretation of Al,A
for p being Element of CQC-WFF Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p )
proof
let Al be QC-alphabet ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for p being Element of CQC-WFF Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for p being Element of CQC-WFF Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p )
let J be interpretation of Al,A; ::_thesis: for p being Element of CQC-WFF Al
for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in p ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= p iff J,v . ((vS +* vS1) +* vS2) |= p )
defpred S1[ Element of CQC-WFF Al] means  for v being Element of Valuations_in (Al,A)
for vS, vS1, vS2 being Val_Sub of A,Al st ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in $1 ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom vS misses dom vS2 holds
( J,v . vS |= $1 iff J,v . ((vS +* vS1) +* vS2) |= $1 );
A1: for p, q being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for k being Element of NAT
for l being CQC-variable_list of k,Al
for P being QC-pred_symbol of k,Al holds
( S1[ VERUM Al] & S1[P ! l] & ( S1[p] implies S1[ 'not' p] ) & ( S1[p] & S1[q] implies S1[p '&' q] ) & ( S1[p] implies S1[ All (x,p)] ) ) by Th75, Th76, Th77, Th80, VALUAT_1:32;
thus  for p being Element of CQC-WFF Al holds S1[p] from CQC_LANG:sch_1(A1); ::_thesis: verum
end;

definition
let Al be QC-alphabet ;
let p be Element of CQC-WFF Al;
func RSub1 p ->  set  means :Def9: :: SUBLEMMA:def 9
 for b being set holds
( b in it iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) );
existence
 ex b1 being set st
for b being set holds
( b in b1 iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) )
proof
defpred S1[ set ] means  ex x being bound_QC-variable of Al st
( x = $1 & not x in still_not-bound_in p );
consider X being set  such that
A1: for b being set holds
( b in X iff ( b in bound_QC-variables Al & S1[b] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for b being set holds
( b in X iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) )
thus  for b being set holds
( b in X iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) ) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for b being set holds
( b in b1 iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) ) ) & ( for b being set holds
( b in b2 iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) ) ) holds
b1 = b2
proof
let X1, X2 be set ; ::_thesis: ( ( for b being set holds
( b in X1 iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) ) ) & ( for b being set holds
( b in X2 iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) ) ) implies X1 = X2 )
assume that
A2: for b being set holds
( b in X1 iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) ) and
A3: for b being set holds
( b in X2 iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) ) ; ::_thesis: X1 = X2
now__::_thesis:_for_b_being_set_holds_
(_b_in_X1_iff_b_in_X2_)
let b be set ; ::_thesis: ( b in X1 iff b in X2 )
 ( b in X1 iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) ) by A2;
hence  ( b in X1 iff b in X2 ) by A3; ::_thesis: verum
end;
hence  X1 = X2 by TARSKI:1; ::_thesis: verum
end;
end;

:: deftheorem Def9 defines RSub1 SUBLEMMA:def_9_:_
 for Al being QC-alphabet
for p being Element of CQC-WFF Al
for b3 being set holds
( b3 = RSub1 p iff for b being set holds
( b in b3 iff ex x being bound_QC-variable of Al st
( x = b & not x in still_not-bound_in p ) ) );

definition
let Al be QC-alphabet ;
let p be Element of CQC-WFF Al;
let Sub be CQC_Substitution of Al;
func RSub2 (p,Sub) ->  set  means :Def10: :: SUBLEMMA:def 10
 for b being set holds
( b in it iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) );
existence
 ex b1 being set st
for b being set holds
( b in b1 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) )
proof
defpred S1[ set ] means  ex x being bound_QC-variable of Al st
( x = $1 & x in still_not-bound_in p & x = (@ Sub) . x );
consider X being set  such that
A1: for b being set holds
( b in X iff ( b in bound_QC-variables Al & S1[b] ) ) from XBOOLE_0:sch_1();
take  X ; ::_thesis: for b being set holds
( b in X iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) )
thus  for b being set holds
( b in X iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being set st ( for b being set holds
( b in b1 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) ) & ( for b being set holds
( b in b2 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) ) holds
b1 = b2
proof
let X1, X2 be set ; ::_thesis: ( ( for b being set holds
( b in X1 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) ) & ( for b being set holds
( b in X2 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) ) implies X1 = X2 )
assume that
A2: for b being set holds
( b in X1 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) and
A3: for b being set holds
( b in X2 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) ; ::_thesis: X1 = X2
now__::_thesis:_for_b_being_set_holds_
(_b_in_X1_iff_b_in_X2_)
let b be set ; ::_thesis: ( b in X1 iff b in X2 )
 ( b in X1 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) by A2;
hence  ( b in X1 iff b in X2 ) by A3; ::_thesis: verum
end;
hence  X1 = X2 by TARSKI:1; ::_thesis: verum
end;
end;

:: deftheorem Def10 defines RSub2 SUBLEMMA:def_10_:_
 for Al being QC-alphabet
for p being Element of CQC-WFF Al
for Sub being CQC_Substitution of Al
for b4 being set holds
( b4 = RSub2 (p,Sub) iff for b being set holds
( b in b4 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) );

theorem Th82: :: SUBLEMMA:82
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for Sub being CQC_Substitution of Al holds dom ((@ Sub) | (RSub1 p)) misses dom ((@ Sub) | (RSub2 (p,Sub)))
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for Sub being CQC_Substitution of Al holds dom ((@ Sub) | (RSub1 p)) misses dom ((@ Sub) | (RSub2 (p,Sub)))
let p be Element of CQC-WFF Al; ::_thesis: for Sub being CQC_Substitution of Al holds dom ((@ Sub) | (RSub1 p)) misses dom ((@ Sub) | (RSub2 (p,Sub)))
let Sub be CQC_Substitution of Al; ::_thesis: dom ((@ Sub) | (RSub1 p)) misses dom ((@ Sub) | (RSub2 (p,Sub)))
now__::_thesis:_not_dom_((@_Sub)_|_(RSub1_p))_meets_dom_((@_Sub)_|_(RSub2_(p,Sub)))
assume  dom ((@ Sub) | (RSub1 p)) meets dom ((@ Sub) | (RSub2 (p,Sub))) ; ::_thesis: contradiction
then consider a being set  such that
A1: a in (dom ((@ Sub) | (RSub1 p))) /\ (dom ((@ Sub) | (RSub2 (p,Sub)))) by XBOOLE_0:4;
 ( dom ((@ Sub) | (RSub1 p)) = (dom (@ Sub)) /\ (RSub1 p) & dom ((@ Sub) | (RSub2 (p,Sub))) = (dom (@ Sub)) /\ (RSub2 (p,Sub)) ) by RELAT_1:61;
then  a in ((dom (@ Sub)) /\ ((dom (@ Sub)) /\ (RSub1 p))) /\ (RSub2 (p,Sub)) by A1, XBOOLE_1:16;
then  a in (((dom (@ Sub)) /\ (dom (@ Sub))) /\ (RSub1 p)) /\ (RSub2 (p,Sub)) by XBOOLE_1:16;
then  a in (dom (@ Sub)) /\ ((RSub1 p) /\ (RSub2 (p,Sub))) by XBOOLE_1:16;
then A2: a in (RSub1 p) /\ (RSub2 (p,Sub)) by XBOOLE_0:def_4;
then  a in RSub2 (p,Sub) by XBOOLE_0:def_4;
then A3: ex b being bound_QC-variable of Al st
( b = a & b in still_not-bound_in p & b = (@ Sub) . b ) by Def10;
 a in RSub1 p by A2, XBOOLE_0:def_4;
then  ex b being bound_QC-variable of Al st
( b = a & not b in still_not-bound_in p ) by Def9;
hence  contradiction by A3; ::_thesis: verum
end;
hence  dom ((@ Sub) | (RSub1 p)) misses dom ((@ Sub) | (RSub2 (p,Sub))) ; ::_thesis: verum
end;

theorem Th83: :: SUBLEMMA:83
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for Sub being CQC_Substitution of Al holds @ (RestrictSub (x,(All (x,p)),Sub)) = (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub))))
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for Sub being CQC_Substitution of Al holds @ (RestrictSub (x,(All (x,p)),Sub)) = (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub))))
let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al
for Sub being CQC_Substitution of Al holds @ (RestrictSub (x,(All (x,p)),Sub)) = (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub))))
let x be bound_QC-variable of Al; ::_thesis: for Sub being CQC_Substitution of Al holds @ (RestrictSub (x,(All (x,p)),Sub)) = (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub))))
let Sub be CQC_Substitution of Al; ::_thesis: @ (RestrictSub (x,(All (x,p)),Sub)) = (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub))))
set X =  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  ;
thus  @ (RestrictSub (x,(All (x,p)),Sub)) c= (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub)))) :: according to XBOOLE_0:def_10 ::_thesis: (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub)))) c= @ (RestrictSub (x,(All (x,p)),Sub))
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in @ (RestrictSub (x,(All (x,p)),Sub)) or b in (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub)))) )
A1: dom ((@ Sub) | (RSub1 (All (x,p)))) misses dom ((@ Sub) | (RSub2 ((All (x,p)),Sub))) by Th82;
assume  b in @ (RestrictSub (x,(All (x,p)),Sub)) ; ::_thesis: b in (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub))))
then  b in RestrictSub (x,(All (x,p)),Sub) by SUBSTUT1:def_2;
then  b in Sub |  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  by SUBSTUT1:def_6;
then  b in (@ Sub) |  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  by SUBSTUT1:def_2;
then A2: b in (@ Sub) /\ [: {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  ,(rng (@ Sub)):] by RELAT_1:67;
then  b in [: {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  ,(rng (@ Sub)):] by XBOOLE_0:def_4;
then consider c, d being set  such that
A3: c in  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  and
 d in rng (@ Sub) and
A4: b = [c,d] by ZFMISC_1:def_2;
A5: ex y1 being bound_QC-variable of Al st
( y1 = c & y1 in still_not-bound_in (All (x,p)) & y1 is Element of dom Sub & y1 <> x & y1 <> Sub . y1 ) by A3;
now__::_thesis:_not_c_in_RSub2_((All_(x,p)),Sub)
assume  c in RSub2 ((All (x,p)),Sub) ; ::_thesis: contradiction
then  ex y being bound_QC-variable of Al st
( y = c & y in still_not-bound_in (All (x,p)) & y = (@ Sub) . y ) by Def10;
hence  contradiction by A5, SUBSTUT1:def_2; ::_thesis: verum
end;
then  not b in [:(RSub2 ((All (x,p)),Sub)),(rng (@ Sub)):] by A4, ZFMISC_1:87;
then  not b in (@ Sub) /\ [:(RSub2 ((All (x,p)),Sub)),(rng (@ Sub)):] by XBOOLE_0:def_4;
then A6: not b in (@ Sub) | (RSub2 ((All (x,p)),Sub)) by RELAT_1:67;
now__::_thesis:_not_c_in_RSub1_(All_(x,p))
assume  c in RSub1 (All (x,p)) ; ::_thesis: contradiction
then  ex y being bound_QC-variable of Al st
( y = c & not y in still_not-bound_in (All (x,p)) ) by Def9;
hence  contradiction by A5; ::_thesis: verum
end;
then  not b in [:(RSub1 (All (x,p))),(rng (@ Sub)):] by A4, ZFMISC_1:87;
then  not b in (@ Sub) /\ [:(RSub1 (All (x,p))),(rng (@ Sub)):] by XBOOLE_0:def_4;
then  not b in (@ Sub) | (RSub1 (All (x,p))) by RELAT_1:67;
then  not b in ((@ Sub) | (RSub1 (All (x,p)))) \/ ((@ Sub) | (RSub2 ((All (x,p)),Sub))) by A6, XBOOLE_0:def_3;
then A7: not b in ((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub))) by A1, FUNCT_4:31;
 b in @ Sub by A2, XBOOLE_0:def_4;
hence  b in (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub)))) by A7, XBOOLE_0:def_5; ::_thesis: verum
end;
thus  (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub)))) c= @ (RestrictSub (x,(All (x,p)),Sub)) ::_thesis: verum
proof
A8: dom ((@ Sub) | (RSub1 (All (x,p)))) misses dom ((@ Sub) | (RSub2 ((All (x,p)),Sub))) by Th82;
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub)))) or b in @ (RestrictSub (x,(All (x,p)),Sub)) )
assume A9: b in (@ Sub) \ (((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub)))) ; ::_thesis: b in @ (RestrictSub (x,(All (x,p)),Sub))
then A10: b in @ Sub by XBOOLE_0:def_5;
consider c, d being set  such that
A11: b = [c,d] by A9, RELAT_1:def_1;
A12: c in dom (@ Sub) by A10, A11, FUNCT_1:1;
then reconsider z = c as bound_QC-variable of Al ;
A13: d = (@ Sub) . c by A10, A11, FUNCT_1:1;
then A14: d in rng (@ Sub) by A12, FUNCT_1:3;
 not b in ((@ Sub) | (RSub1 (All (x,p)))) +* ((@ Sub) | (RSub2 ((All (x,p)),Sub))) by A9, XBOOLE_0:def_5;
then A15: not b in ((@ Sub) | (RSub1 (All (x,p)))) \/ ((@ Sub) | (RSub2 ((All (x,p)),Sub))) by A8, FUNCT_4:31;
then  not b in (@ Sub) | (RSub1 (All (x,p))) by XBOOLE_0:def_3;
then  not b in (@ Sub) /\ [:(RSub1 (All (x,p))),(rng (@ Sub)):] by RELAT_1:67;
then  not [z,d] in [:(RSub1 (All (x,p))),(rng (@ Sub)):] by A10, A11, XBOOLE_0:def_4;
then A16: not z in RSub1 (All (x,p)) by A14, ZFMISC_1:87;
then A17: z in still_not-bound_in (All (x,p)) by Def9;
then  z in (still_not-bound_in p) \ {x} by QC_LANG3:12;
then  not z in {x} by XBOOLE_0:def_5;
then A18: z <> x by TARSKI:def_1;
A19: d in rng (@ Sub) by A12, A13, FUNCT_1:3;
 not b in (@ Sub) | (RSub2 ((All (x,p)),Sub)) by A15, XBOOLE_0:def_3;
then  not b in (@ Sub) /\ [:(RSub2 ((All (x,p)),Sub)),(rng (@ Sub)):] by RELAT_1:67;
then  not [z,d] in [:(RSub2 ((All (x,p)),Sub)),(rng (@ Sub)):] by A10, A11, XBOOLE_0:def_4;
then  not z in RSub2 ((All (x,p)),Sub) by A19, ZFMISC_1:87;
then  ( not z in still_not-bound_in (All (x,p)) or z <> (@ Sub) . z ) by Def10;
then A20: z <> Sub . z by A16, Def9, SUBSTUT1:def_2;
A21: d in rng (@ Sub) by A12, A13, FUNCT_1:3;
 z is Element of dom Sub by A12, SUBSTUT1:def_2;
then  z in  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  by A17, A18, A20;
then  [z,d] in [: {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  ,(rng (@ Sub)):] by A21, ZFMISC_1:87;
then  b in (@ Sub) /\ [: {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  ,(rng (@ Sub)):] by A10, A11, XBOOLE_0:def_4;
then  b in (@ Sub) |  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  by RELAT_1:67;
then  b in Sub |  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,p)) & y is Element of dom Sub & y <> x & y <> Sub . y )  }  by SUBSTUT1:def_2;
then  b in RestrictSub (x,(All (x,p)),Sub) by SUBSTUT1:def_6;
hence  b in @ (RestrictSub (x,(All (x,p)),Sub)) by SUBSTUT1:def_2; ::_thesis: verum
end;
end;

theorem Th84: :: SUBLEMMA:84
for Al being QC-alphabet
for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for Sub being CQC_Substitution of Al holds dom (@ (RestrictSub (x,p,Sub))) misses (dom ((@ Sub) | (RSub1 p))) \/ (dom ((@ Sub) | (RSub2 (p,Sub))))
proof
let Al be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF Al
for x being bound_QC-variable of Al
for Sub being CQC_Substitution of Al holds dom (@ (RestrictSub (x,p,Sub))) misses (dom ((@ Sub) | (RSub1 p))) \/ (dom ((@ Sub) | (RSub2 (p,Sub))))
let p be Element of CQC-WFF Al; ::_thesis: for x being bound_QC-variable of Al
for Sub being CQC_Substitution of Al holds dom (@ (RestrictSub (x,p,Sub))) misses (dom ((@ Sub) | (RSub1 p))) \/ (dom ((@ Sub) | (RSub2 (p,Sub))))
let x be bound_QC-variable of Al; ::_thesis: for Sub being CQC_Substitution of Al holds dom (@ (RestrictSub (x,p,Sub))) misses (dom ((@ Sub) | (RSub1 p))) \/ (dom ((@ Sub) | (RSub2 (p,Sub))))
let Sub be CQC_Substitution of Al; ::_thesis: dom (@ (RestrictSub (x,p,Sub))) misses (dom ((@ Sub) | (RSub1 p))) \/ (dom ((@ Sub) | (RSub2 (p,Sub))))
set X =  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in p & y is Element of dom Sub & y <> x & y <> Sub . y )  }  ;
A1: dom ((@ Sub) | (RSub2 (p,Sub))) = (dom (@ Sub)) /\ (RSub2 (p,Sub)) by RELAT_1:61;
 RestrictSub (x,p,Sub) = Sub |  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in p & y is Element of dom Sub & y <> x & y <> Sub . y )  }  by SUBSTUT1:def_6;
then  RestrictSub (x,p,Sub) = (@ Sub) |  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in p & y is Element of dom Sub & y <> x & y <> Sub . y )  }  by SUBSTUT1:def_2;
then  dom (@ (RestrictSub (x,p,Sub))) = dom ((@ Sub) |  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in p & y is Element of dom Sub & y <> x & y <> Sub . y )  }  ) by SUBSTUT1:def_2;
then A2: dom (@ (RestrictSub (x,p,Sub))) = (dom (@ Sub)) /\  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in p & y is Element of dom Sub & y <> x & y <> Sub . y )  }  by RELAT_1:61;
A3: dom ((@ Sub) | (RSub1 p)) = (dom (@ Sub)) /\ (RSub1 p) by RELAT_1:61;
now__::_thesis:_not_dom_(@_(RestrictSub_(x,p,Sub)))_meets_(dom_((@_Sub)_|_(RSub1_p)))_\/_(dom_((@_Sub)_|_(RSub2_(p,Sub))))
assume  dom (@ (RestrictSub (x,p,Sub))) meets (dom ((@ Sub) | (RSub1 p))) \/ (dom ((@ Sub) | (RSub2 (p,Sub)))) ; ::_thesis: contradiction
then consider b being set  such that
A4: b in dom (@ (RestrictSub (x,p,Sub))) and
A5: b in (dom ((@ Sub) | (RSub1 p))) \/ (dom ((@ Sub) | (RSub2 (p,Sub)))) by XBOOLE_0:3;
 b in  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in p & y is Element of dom Sub & y <> x & y <> Sub . y )  }  by A2, A4, XBOOLE_0:def_4;
then A6: ex y being bound_QC-variable of Al st
( b = y & y in still_not-bound_in p & y is Element of dom Sub & y <> x & y <> Sub . y ) ;
A7: now__::_thesis:_not_b_in_dom_((@_Sub)_|_(RSub2_(p,Sub)))
assume  b in dom ((@ Sub) | (RSub2 (p,Sub))) ; ::_thesis: contradiction
then  b in RSub2 (p,Sub) by A1, XBOOLE_0:def_4;
then  ex y1 being bound_QC-variable of Al st
( y1 = b & y1 in still_not-bound_in p & y1 = (@ Sub) . y1 ) by Def10;
hence  contradiction by A6, SUBSTUT1:def_2; ::_thesis: verum
end;
now__::_thesis:_not_b_in_dom_((@_Sub)_|_(RSub1_p))
assume  b in dom ((@ Sub) | (RSub1 p)) ; ::_thesis: contradiction
then  b in RSub1 p by A3, XBOOLE_0:def_4;
then  ex y1 being bound_QC-variable of Al st
( y1 = b & not y1 in still_not-bound_in p ) by Def9;
hence  contradiction by A6; ::_thesis: verum
end;
hence  contradiction by A5, A7, XBOOLE_0:def_3; ::_thesis: verum
end;
hence  dom (@ (RestrictSub (x,p,Sub))) misses (dom ((@ Sub) | (RSub1 p))) \/ (dom ((@ Sub) | (RSub2 (p,Sub)))) ; ::_thesis: verum
end;

theorem Th85: :: SUBLEMMA:85
for Al being QC-alphabet
for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
@ ((CQCSub_All ([S,x],xSQ)) `2) = ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* ((@ xSQ) | (RSub1 (All (x,(S `1)))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
@ ((CQCSub_All ([S,x],xSQ)) `2) = ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* ((@ xSQ) | (RSub1 (All (x,(S `1)))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))
let x be bound_QC-variable of Al; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
@ ((CQCSub_All ([S,x],xSQ)) `2) = ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* ((@ xSQ) | (RSub1 (All (x,(S `1)))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
@ ((CQCSub_All ([S,x],xSQ)) `2) = ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* ((@ xSQ) | (RSub1 (All (x,(S `1)))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies @ ((CQCSub_All ([S,x],xSQ)) `2) = ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* ((@ xSQ) | (RSub1 (All (x,(S `1)))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) )
set S1 = CQCSub_All ([S,x],xSQ);
A1: (@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)) c= @ xSQ by RELAT_1:59;
 dom ((@ xSQ) | (RSub1 (All (x,(S `1))))) misses dom ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) by Th82;
then A2: ((@ xSQ) | (RSub1 (All (x,(S `1))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) = ((@ xSQ) | (RSub1 (All (x,(S `1))))) \/ ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) by FUNCT_4:31;
assume A3: [S,x] is quantifiable ; ::_thesis: @ ((CQCSub_All ([S,x],xSQ)) `2) = ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* ((@ xSQ) | (RSub1 (All (x,(S `1)))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))
then  CQCSub_All ([S,x],xSQ) = Sub_All ([S,x],xSQ) by Def5;
then A4: @ ((CQCSub_All ([S,x],xSQ)) `2) = @ xSQ by A3, Th26;
A5: @ (RestrictSub (x,(All (x,(S `1))),xSQ)) = (@ xSQ) \ (((@ xSQ) | (RSub1 (All (x,(S `1))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))) by Th83;
then reconsider F = (@ xSQ) \ (((@ xSQ) | (RSub1 (All (x,(S `1))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))) as PartFunc of (bound_QC-variables Al),(bound_QC-variables Al) ;
 dom F misses (dom ((@ xSQ) | (RSub1 (All (x,(S `1)))))) \/ (dom ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))) by A5, Th84;
then A6: dom F misses dom (((@ xSQ) | (RSub1 (All (x,(S `1))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))) by FUNCT_4:def_1;
 ( (((@ xSQ) | (RSub1 (All (x,(S `1))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))) \/ F = (((@ xSQ) | (RSub1 (All (x,(S `1))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))) \/ (@ xSQ) & (@ xSQ) | (RSub1 (All (x,(S `1)))) c= @ xSQ ) by RELAT_1:59, XBOOLE_1:39;
then  (((@ xSQ) | (RSub1 (All (x,(S `1))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))) \/ F = @ xSQ by A2, A1, XBOOLE_1:8, XBOOLE_1:12;
then  F +* (((@ xSQ) | (RSub1 (All (x,(S `1))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))) = @ xSQ by A6, FUNCT_4:31;
hence  @ ((CQCSub_All ([S,x],xSQ)) `2) = ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* ((@ xSQ) | (RSub1 (All (x,(S `1)))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) by A4, A5, FUNCT_4:14; ::_thesis: verum
end;

theorem Th86: :: SUBLEMMA:86
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
ex vS1, vS2 being Val_Sub of A,Al st
( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
ex vS1, vS2 being Val_Sub of A,Al st
( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
ex vS1, vS2 being Val_Sub of A,Al st
( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
let A be non empty set ; ::_thesis: for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
ex vS1, vS2 being Val_Sub of A,Al st
( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
let v be Element of Valuations_in (Al,A); ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
ex vS1, vS2 being Val_Sub of A,Al st
( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
ex vS1, vS2 being Val_Sub of A,Al st
( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies ex vS1, vS2 being Val_Sub of A,Al st
( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) ) )
set S1 = CQCSub_All ([S,x],xSQ);
assume  [S,x] is quantifiable ; ::_thesis: ex vS1, vS2 being Val_Sub of A,Al st
( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
then A1: Val_S (v,(CQCSub_All ([S,x],xSQ))) = (((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* ((@ xSQ) | (RSub1 (All (x,(S `1)))))) +* ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ)))) * v by Th85;
take vS1 = ((@ xSQ) | (RSub1 (All (x,(S `1))))) * v; ::_thesis: ex vS2 being Val_Sub of A,Al st
( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
take vS2 = ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) * v; ::_thesis: ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
 rng ((@ xSQ) | (RSub1 (All (x,(S `1))))) c= bound_QC-variables Al ;
then A2: rng ((@ xSQ) | (RSub1 (All (x,(S `1))))) c= dom v by Th58;
thus  for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ::_thesis: ( ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
proof
let y be bound_QC-variable of Al; ::_thesis: ( y in dom vS1 implies not y in still_not-bound_in (All (x,(S `1))) )
assume  y in dom vS1 ; ::_thesis: not y in still_not-bound_in (All (x,(S `1)))
then  y in dom ((@ xSQ) | (RSub1 (All (x,(S `1))))) by A2, RELAT_1:27;
then  y in (dom (@ xSQ)) /\ (RSub1 (All (x,(S `1)))) by RELAT_1:61;
then  y in RSub1 (All (x,(S `1))) by XBOOLE_0:def_4;
then  ex y1 being bound_QC-variable of Al st
( y1 = y & not y1 in still_not-bound_in (All (x,(S `1))) ) by Def9;
hence  not y in still_not-bound_in (All (x,(S `1))) ; ::_thesis: verum
end;
 rng ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) c= bound_QC-variables Al ;
then A3: rng ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) c= dom v by Th58;
thus  for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ::_thesis: ( dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 & v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) )
proof
let y be bound_QC-variable of Al; ::_thesis: ( y in dom vS2 implies vS2 . y = v . y )
assume  y in dom vS2 ; ::_thesis: vS2 . y = v . y
then A4: y in dom ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) by A3, RELAT_1:27;
then  y in (dom (@ xSQ)) /\ (RSub2 ((All (x,(S `1))),xSQ)) by RELAT_1:61;
then  y in RSub2 ((All (x,(S `1))),xSQ) by XBOOLE_0:def_4;
then  ex y1 being bound_QC-variable of Al st
( y1 = y & y1 in still_not-bound_in (All (x,(S `1))) & y1 = (@ xSQ) . y1 ) by Def10;
then  v . y = v . (((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) . y) by A4, FUNCT_1:47;
hence  vS2 . y = v . y by A4, FUNCT_1:13; ::_thesis: verum
end;
thus  dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 ::_thesis: v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2)
proof
set X =  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,(S `1))) & y is Element of dom xSQ & y <> x & y <> xSQ . y )  }  ;
 RestrictSub (x,(All (x,(S `1))),xSQ) = xSQ |  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,(S `1))) & y is Element of dom xSQ & y <> x & y <> xSQ . y )  }  by SUBSTUT1:def_6;
then  RestrictSub (x,(All (x,(S `1))),xSQ) = (@ xSQ) |  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,(S `1))) & y is Element of dom xSQ & y <> x & y <> xSQ . y )  }  by SUBSTUT1:def_2;
then  dom (NEx_Val (v,S,x,xSQ)) = dom ((@ xSQ) |  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,(S `1))) & y is Element of dom xSQ & y <> x & y <> xSQ . y )  }  ) by Th71;
then A5: dom (NEx_Val (v,S,x,xSQ)) = (dom (@ xSQ)) /\  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,(S `1))) & y is Element of dom xSQ & y <> x & y <> xSQ . y )  }  by RELAT_1:61;
 dom vS2 = dom ((@ xSQ) | (RSub2 ((All (x,(S `1))),xSQ))) by A3, RELAT_1:27;
then A6: dom vS2 = (dom (@ xSQ)) /\ (RSub2 ((All (x,(S `1))),xSQ)) by RELAT_1:61;
now__::_thesis:_not_dom_(NEx_Val_(v,S,x,xSQ))_meets_dom_vS2
assume  dom (NEx_Val (v,S,x,xSQ)) meets dom vS2 ; ::_thesis: contradiction
then consider b being set  such that
A7: b in dom (NEx_Val (v,S,x,xSQ)) and
A8: b in dom vS2 by XBOOLE_0:3;
 b in  {  y where y is bound_QC-variable of Al : ( y in still_not-bound_in (All (x,(S `1))) & y is Element of dom xSQ & y <> x & y <> xSQ . y )  }  by A5, A7, XBOOLE_0:def_4;
then A9: ex y being bound_QC-variable of Al st
( y = b & y in still_not-bound_in (All (x,(S `1))) & y is Element of dom xSQ & y <> x & y <> xSQ . y ) ;
 b in RSub2 ((All (x,(S `1))),xSQ) by A6, A8, XBOOLE_0:def_4;
then  ex y1 being bound_QC-variable of Al st
( y1 = b & y1 in still_not-bound_in (All (x,(S `1))) & y1 = (@ xSQ) . y1 ) by Def10;
hence  contradiction by A9, SUBSTUT1:def_2; ::_thesis: verum
end;
hence  dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 ; ::_thesis: verum
end;
 ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) +* ((@ xSQ) | (RSub1 (All (x,(S `1)))))) * v = ((@ (RestrictSub (x,(All (x,(S `1))),xSQ))) * v) +* (((@ xSQ) | (RSub1 (All (x,(S `1))))) * v) by A2, FUNCT_7:9;
hence  v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) by A1, A3, FUNCT_7:9; ::_thesis: verum
end;

theorem Th87: :: SUBLEMMA:87
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let J be interpretation of Al,A; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable implies for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) ) )
set S1 = CQCSub_All ([S,x],xSQ);
assume A1: [S,x] is quantifiable ; ::_thesis: for v being Element of Valuations_in (Al,A) holds
( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
then  CQCSub_All ([S,x],xSQ) = Sub_All ([S,x],xSQ) by Def5;
then  (CQCSub_All ([S,x],xSQ)) `1 = All (([S,x] `2),(([S,x] `1) `1)) by A1, Th26;
then  (CQCSub_All ([S,x],xSQ)) `1 = All (x,(([S,x] `1) `1)) by MCART_1:7;
then A2: (CQCSub_All ([S,x],xSQ)) `1 = All (x,(S `1)) by MCART_1:7;
let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
consider vS1, vS2 being Val_Sub of A,Al such that
A3: ( ( for y being bound_QC-variable of Al st y in dom vS1 holds
not y in still_not-bound_in (All (x,(S `1))) ) & ( for y being bound_QC-variable of Al st y in dom vS2 holds
vS2 . y = v . y ) & dom (NEx_Val (v,S,x,xSQ)) misses dom vS2 ) and
A4: v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) = v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) by A1, Th86;
 ( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (((NEx_Val (v,S,x,xSQ)) +* vS1) +* vS2) |= All (x,(S `1)) ) by A3, Th81;
hence  ( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) ) by A4, A2, Def2; ::_thesis: verum
end;

theorem Th88: :: SUBLEMMA:88
for Al being QC-alphabet
for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_All ([S,x],xSQ)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
proof
let Al be QC-alphabet ; ::_thesis: for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_All ([S,x],xSQ)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let x be bound_QC-variable of Al; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_All ([S,x],xSQ)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_All ([S,x],xSQ)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let J be interpretation of Al,A; ::_thesis: for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_All ([S,x],xSQ)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let S be Element of CQC-Sub-WFF Al; ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_All ([S,x],xSQ)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable & ( for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ) implies for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_All ([S,x],xSQ)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) ) )
assume that
A1: [S,x] is quantifiable and
A2: for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) ; ::_thesis: for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub (CQCSub_All ([S,x],xSQ)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
let v be Element of Valuations_in (Al,A); ::_thesis: ( J,v |= CQC_Sub (CQCSub_All ([S,x],xSQ)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) )
set S1 = CQCSub_All ([S,x],xSQ);
set z = S_Bound (@ (CQCSub_All ([S,x],xSQ)));
A3: ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) iff for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S,x,xSQ)) +* (x | a)) |= S ) by A1, Th54;
set q = CQC_Sub S;
A4: ( J,v |= All ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))),(CQC_Sub S)) iff for a being Element of A holds J,v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) |= CQC_Sub S ) by Th50;
A5: ( ( for a being Element of A holds J,v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) |= CQC_Sub S ) implies for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S )
proof
assume A6: for a being Element of A holds J,v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) |= CQC_Sub S ; ::_thesis: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S
let a be Element of A; ::_thesis: J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S
 J,v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) |= CQC_Sub S by A6;
hence  J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S by A2; ::_thesis: verum
end;
A7: ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ) implies for a being Element of A holds J,v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) |= CQC_Sub S )
proof
assume A8: for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S ; ::_thesis: for a being Element of A holds J,v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) |= CQC_Sub S
let a be Element of A; ::_thesis: J,v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) |= CQC_Sub S
 J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . (Val_S ((v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)),S)) |= S by A8;
hence  J,v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a) |= CQC_Sub S by A2; ::_thesis: verum
end;
set p = CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ)));
A9: ( J,v |= CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub (CQCSub_the_scope_of (CQCSub_All ([S,x],xSQ))))) iff J,v |= CQCQuant ((CQCSub_All ([S,x],xSQ)),(CQC_Sub S)) ) by A1, Th30;
A10: ( ( for a being Element of A holds J,(v . ((S_Bound (@ (CQCSub_All ([S,x],xSQ)))) | a)) . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) by A1, Th70;
A11: ( J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) implies for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
proof
assume  J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) ; ::_thesis: for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S
then  for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 by Th50;
hence  for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S by Th73; ::_thesis: verum
end;
A12: ( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) implies J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) )
proof
assume  for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ; ::_thesis: J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1))
then  for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S `1 by Th73;
hence  J,v . (NEx_Val (v,S,x,xSQ)) |= All (x,(S `1)) by Th50; ::_thesis: verum
end;
 CQCSub_All ([S,x],xSQ) is Sub_universal by A1, Th27;
hence  ( J,v |= CQC_Sub (CQCSub_All ([S,x],xSQ)) iff J,v . (Val_S (v,(CQCSub_All ([S,x],xSQ)))) |= CQCSub_All ([S,x],xSQ) ) by A1, A9, A4, A5, A7, A3, A10, A12, A11, Th28, Th31, Th56, Th72, Th87; ::_thesis: verum
end;

scheme :: SUBLEMMA:sch 1
SubCQCInd1{ F1() ->  QC-alphabet , P1[ set ] } :
 for S being Element of CQC-Sub-WFF F1() holds P1[S]
provided
A1: for S, S9 being Element of CQC-Sub-WFF F1()
for x being bound_QC-variable of F1()
for SQ being second_Q_comp of [S,x]
for k being Element of NAT
for ll being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1()
for e being Element of vSUB F1() holds
( P1[ Sub_P (P,ll,e)] & ( S is F1() -Sub_VERUM implies P1[S] ) & ( P1[S] implies P1[ Sub_not S] ) & ( S `2 = S9 `2 & P1[S] & P1[S9] implies P1[ CQCSub_& (S,S9)] ) & ( [S,x] is quantifiable & P1[S] implies P1[ CQCSub_All ([S,x],SQ)] ) )
proof
A2: for S, S9 being Element of CQC-Sub-WFF F1()
for x being bound_QC-variable of F1()
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & P1[S] holds
P1[ Sub_All ([S,x],xSQ)]
proof
let S, S9 be Element of CQC-Sub-WFF F1(); ::_thesis: for x being bound_QC-variable of F1()
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & P1[S] holds
P1[ Sub_All ([S,x],xSQ)]
let x be bound_QC-variable of F1(); ::_thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable & P1[S] holds
P1[ Sub_All ([S,x],xSQ)]
let xSQ be second_Q_comp of [S,x]; ::_thesis: ( [S,x] is quantifiable & P1[S] implies P1[ Sub_All ([S,x],xSQ)] )
assume that
A3: [S,x] is quantifiable and
A4: P1[S] ; ::_thesis: P1[ Sub_All ([S,x],xSQ)]
 P1[ CQCSub_All ([S,x],xSQ)] by A1, A3, A4;
hence  P1[ Sub_All ([S,x],xSQ)] by A3, Def5; ::_thesis: verum
end;
 for S, S9 being Element of CQC-Sub-WFF F1() st S `2 = S9 `2 & P1[S] & P1[S9] holds
P1[ Sub_& (S,S9)]
proof
let S, S9 be Element of CQC-Sub-WFF F1(); ::_thesis: ( S `2 = S9 `2 & P1[S] & P1[S9] implies P1[ Sub_& (S,S9)] )
assume that
A5: S `2 = S9 `2 and
A6: ( P1[S] & P1[S9] ) ; ::_thesis: P1[ Sub_& (S,S9)]
 CQCSub_& (S,S9) = Sub_& (S,S9) by A5, Def3;
hence  P1[ Sub_& (S,S9)] by A1, A5, A6; ::_thesis: verum
end;
then A7: for S, S9 being Element of CQC-Sub-WFF F1()
for x being bound_QC-variable of F1()
for SQ being second_Q_comp of [S,x]
for k being Element of NAT
for ll being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1()
for e being Element of vSUB F1() holds
( P1[ Sub_P (P,ll,e)] & ( S is F1() -Sub_VERUM implies P1[S] ) & ( P1[S] implies P1[ Sub_not S] ) & ( S `2 = S9 `2 & P1[S] & P1[S9] implies P1[ Sub_& (S,S9)] ) & ( [S,x] is quantifiable & P1[S] implies P1[ Sub_All ([S,x],SQ)] ) ) by A1, A2;
thus  for S being Element of CQC-Sub-WFF F1() holds P1[S] from SUBSTUT1:sch_5(A7); ::_thesis: verum
end;

theorem :: SUBLEMMA:89
for Al being QC-alphabet
for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
proof
let Al be QC-alphabet ; ::_thesis: for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
let A be non empty set ; ::_thesis: for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
let J be interpretation of Al,A; ::_thesis: for S being Element of CQC-Sub-WFF Al
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
defpred S1[ Element of CQC-Sub-WFF Al] means  for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub $1 iff J,v . (Val_S (v,$1)) |= $1 );
A1: for S, S9 being Element of CQC-Sub-WFF Al
for x being bound_QC-variable of Al
for SQ being second_Q_comp of [S,x]
for k being Element of NAT
for ll being CQC-variable_list of k,Al
for P being QC-pred_symbol of k,Al
for e being Element of vSUB Al holds
( S1[ Sub_P (P,ll,e)] & ( S is Al -Sub_VERUM implies S1[S] ) & ( S1[S] implies S1[ Sub_not S] ) & ( S `2 = S9 `2 & S1[S] & S1[S9] implies S1[ CQCSub_& (S,S9)] ) & ( [S,x] is quantifiable & S1[S] implies S1[ CQCSub_All ([S,x],SQ)] ) ) by Th4, Th15, Th19, Th25, Th88;
thus  for S being Element of CQC-Sub-WFF Al holds S1[S] from SUBLEMMA:sch_1(A1); ::_thesis: verum
end;