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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

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

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

assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: ( not A \imp B in G or (\ex (x,A)) \imp (\ex (x,B)) in G )

assume A2: A \imp B in G ; :: thesis: (\ex (x,A)) \imp (\ex (x,B)) in G

(A \imp B) \imp ((\not B) \imp (\not A)) in G by Th57;

then (\not B) \imp (\not A) in G by A2, Def38;

then A3: (\for (x,(\not B))) \imp (\for (x,(\not A))) in G by A1, Th115;

((\for (x,(\not B))) \imp (\for (x,(\not A)))) \imp ((\not (\for (x,(\not A)))) \imp (\not (\for (x,(\not B))))) in G by Th57;

then A4: (\not (\for (x,(\not A)))) \imp (\not (\for (x,(\not B)))) in G by A3, Def38;

(\ex (x,A)) \iff (\not (\for (x,(\not A)))) in G by Th105;

then A5: (\ex (x,A)) \imp (\not (\for (x,(\not B)))) in G by A4, Th92;

(\ex (x,B)) \iff (\not (\for (x,(\not B)))) in G by Th105;

then (\not (\for (x,(\not B)))) \iff (\ex (x,B)) in G by Th90;

hence (\ex (x,A)) \imp (\ex (x,B)) in G by A5, Th93; :: 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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

(\ex (x,A)) \imp (\ex (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 & A \imp B in G holds

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

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

assume A1: ( L is subst-correct & L is vf-qc-correct ) ; :: thesis: ( not A \imp B in G or (\ex (x,A)) \imp (\ex (x,B)) in G )

assume A2: A \imp B in G ; :: thesis: (\ex (x,A)) \imp (\ex (x,B)) in G

(A \imp B) \imp ((\not B) \imp (\not A)) in G by Th57;

then (\not B) \imp (\not A) in G by A2, Def38;

then A3: (\for (x,(\not B))) \imp (\for (x,(\not A))) in G by A1, Th115;

((\for (x,(\not B))) \imp (\for (x,(\not A)))) \imp ((\not (\for (x,(\not A)))) \imp (\not (\for (x,(\not B))))) in G by Th57;

then A4: (\not (\for (x,(\not A)))) \imp (\not (\for (x,(\not B)))) in G by A3, Def38;

(\ex (x,A)) \iff (\not (\for (x,(\not A)))) in G by Th105;

then A5: (\ex (x,A)) \imp (\not (\for (x,(\not B)))) in G by A4, Th92;

(\ex (x,B)) \iff (\not (\for (x,(\not B)))) in G by Th105;

then (\not (\for (x,(\not B)))) \iff (\ex (x,B)) in G by Th90;

hence (\ex (x,A)) \imp (\ex (x,B)) in G by A5, Th93; :: thesis: verum