let n be non empty Nat; :: thesis: for J being non empty non void Signature

for T being non-empty MSAlgebra over J

for X being V3() GeneratorSet of T

for S1 being non empty non void b_{1} -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let J be non empty non void Signature; :: thesis: for T being non-empty MSAlgebra over J

for X being V3() GeneratorSet of T

for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let T be non-empty MSAlgebra over J; :: thesis: for X being V3() GeneratorSet of T

for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let X be V3() GeneratorSet of T; :: thesis: for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let S1 be non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X; :: thesis: for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis: for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let G be QC-theory of L; :: thesis: for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let A, B be Formula of L; :: thesis: for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let x be Element of Union X; :: thesis: ( L is subst-correct & L is vf-qc-correct implies (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G )

set Y = X extended_by ({}, the carrier of S1);

consider a being object such that

A1: ( a in dom X & x in X . a ) by CARD_5:2;

J is Subsignature of S1 by Def2;

then ( dom X = the carrier of J & the carrier of J c= the carrier of S1 & the carrier of S1 = dom (X extended_by ({}, the carrier of S1)) ) by INSTALG1:10, PARTFUN1:def 2;

then reconsider a = a as SortSymbol of S1 by A1;

assume A2: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

then (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) in G by Th107;

then A3: \for (x,((\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B))) in G by Def39;

A4: ( vf (\for (x,(A \imp B))) = (vf (A \imp B)) (\) (a -singleton x) & x in {x} ) by A1, A2, TARSKI:def 1;

then (vf (\for (x,(A \imp B)))) . a = ((vf (A \imp B)) . a) \ ((a -singleton x) . a) by PBOOLE:def 6

.= ((vf (A \imp B)) . a) \ {x} by AOFA_A00:6 ;

then x nin (vf (\for (x,(A \imp B)))) . a by A4, XBOOLE_0:def 5;

then A5: (\for (x,(A \imp B))) \imp (\for (x,((\for (x,A)) \imp B))) in G by A1, A3, Th108;

A6: ( vf (\for (x,A)) = (vf A) (\) (a -singleton x) & x in {x} ) by A1, A2, TARSKI:def 1;

then (vf (\for (x,A))) . a = ((vf A) . a) \ ((a -singleton x) . a) by PBOOLE:def 6

.= ((vf A) . a) \ {x} by AOFA_A00:6 ;

then x nin (vf (\for (x,A))) . a by A6, XBOOLE_0:def 5;

then (\for (x,((\for (x,A)) \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G by A1, Def39;

hence (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G by A5, Th45; :: thesis: verum

for T being non-empty MSAlgebra over J

for X being V3() GeneratorSet of T

for S1 being non empty non void b

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let J be non empty non void Signature; :: thesis: for T being non-empty MSAlgebra over J

for X being V3() GeneratorSet of T

for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let T be non-empty MSAlgebra over J; :: thesis: for X being V3() GeneratorSet of T

for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let X be V3() GeneratorSet of T; :: thesis: for S1 being non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let S1 be non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X; :: thesis: for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis: for G being QC-theory of L

for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let G be QC-theory of L; :: thesis: for A, B being Formula of L

for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let A, B be Formula of L; :: thesis: for x being Element of Union X st L is subst-correct & L is vf-qc-correct holds

(\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

let x be Element of Union X; :: thesis: ( L is subst-correct & L is vf-qc-correct implies (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G )

set Y = X extended_by ({}, the carrier of S1);

consider a being object such that

A1: ( a in dom X & x in X . a ) by CARD_5:2;

J is Subsignature of S1 by Def2;

then ( dom X = the carrier of J & the carrier of J c= the carrier of S1 & the carrier of S1 = dom (X extended_by ({}, the carrier of S1)) ) by INSTALG1:10, PARTFUN1:def 2;

then reconsider a = a as SortSymbol of S1 by A1;

assume A2: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G

then (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B) in G by Th107;

then A3: \for (x,((\for (x,(A \imp B))) \imp ((\for (x,A)) \imp B))) in G by Def39;

A4: ( vf (\for (x,(A \imp B))) = (vf (A \imp B)) (\) (a -singleton x) & x in {x} ) by A1, A2, TARSKI:def 1;

then (vf (\for (x,(A \imp B)))) . a = ((vf (A \imp B)) . a) \ ((a -singleton x) . a) by PBOOLE:def 6

.= ((vf (A \imp B)) . a) \ {x} by AOFA_A00:6 ;

then x nin (vf (\for (x,(A \imp B)))) . a by A4, XBOOLE_0:def 5;

then A5: (\for (x,(A \imp B))) \imp (\for (x,((\for (x,A)) \imp B))) in G by A1, A3, Th108;

A6: ( vf (\for (x,A)) = (vf A) (\) (a -singleton x) & x in {x} ) by A1, A2, TARSKI:def 1;

then (vf (\for (x,A))) . a = ((vf A) . a) \ ((a -singleton x) . a) by PBOOLE:def 6

.= ((vf A) . a) \ {x} by AOFA_A00:6 ;

then x nin (vf (\for (x,A))) . a by A6, XBOOLE_0:def 5;

then (\for (x,((\for (x,A)) \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G by A1, Def39;

hence (\for (x,(A \imp B))) \imp ((\for (x,A)) \imp (\for (x,B))) in G by A5, Th45; :: thesis: verum