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

assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: (\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B))) in G

(A \and B) \imp A in G by Def38;

then A2: (\for (x,(A \and B))) \imp (\for (x,A)) in G by A1, Th115;

(A \and B) \imp B in G by Def38;

then A3: (\for (x,(A \and B))) \imp (\for (x,B)) in G by A1, Th115;

((\for (x,(A \and B))) \imp (\for (x,A))) \imp (((\for (x,(A \and B))) \imp (\for (x,B))) \imp ((\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B))))) in G by Th49;

then ((\for (x,(A \and B))) \imp (\for (x,B))) \imp ((\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B)))) in G by A2, Def38;

hence (\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B))) in G by A3, Def38; :: 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 \and B))) \imp ((\for (x,A)) \and (\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 \and B))) \imp ((\for (x,A)) \and (\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 \and B))) \imp ((\for (x,A)) \and (\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 \and B))) \imp ((\for (x,A)) \and (\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 \and B))) \imp ((\for (x,A)) \and (\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 \and B))) \imp ((\for (x,A)) \and (\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 \and B))) \imp ((\for (x,A)) \and (\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 \and B))) \imp ((\for (x,A)) \and (\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 \and B))) \imp ((\for (x,A)) \and (\for (x,B))) in G )

assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: (\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B))) in G

(A \and B) \imp A in G by Def38;

then A2: (\for (x,(A \and B))) \imp (\for (x,A)) in G by A1, Th115;

(A \and B) \imp B in G by Def38;

then A3: (\for (x,(A \and B))) \imp (\for (x,B)) in G by A1, Th115;

((\for (x,(A \and B))) \imp (\for (x,A))) \imp (((\for (x,(A \and B))) \imp (\for (x,B))) \imp ((\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B))))) in G by Th49;

then ((\for (x,(A \and B))) \imp (\for (x,B))) \imp ((\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B)))) in G by A2, Def38;

hence (\for (x,(A \and B))) \imp ((\for (x,A)) \and (\for (x,B))) in G by A3, Def38; :: thesis: verum