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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

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

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

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

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

( (A \iff B) \imp (A \imp B) in G & (A \iff B) \imp (B \imp A) in G ) by Def38;

then ( A \imp B in G & B \imp A in G ) by A2, Def38;

then ( (\ex (x,A)) \imp (\ex (x,B)) in G & (\ex (x,B)) \imp (\ex (x,A)) in G ) by A1, Th128;

then A3: ((\ex (x,A)) \imp (\ex (x,B))) \and ((\ex (x,B)) \imp (\ex (x,A))) in G by Th35;

(((\ex (x,A)) \imp (\ex (x,B))) \and ((\ex (x,B)) \imp (\ex (x,A)))) \imp ((\ex (x,A)) \iff (\ex (x,B))) in G by Def38;

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

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

(\ex (x,A)) \iff (\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 \iff B in G holds

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

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

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

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

( (A \iff B) \imp (A \imp B) in G & (A \iff B) \imp (B \imp A) in G ) by Def38;

then ( A \imp B in G & B \imp A in G ) by A2, Def38;

then ( (\ex (x,A)) \imp (\ex (x,B)) in G & (\ex (x,B)) \imp (\ex (x,A)) in G ) by A1, Th128;

then A3: ((\ex (x,A)) \imp (\ex (x,B))) \and ((\ex (x,B)) \imp (\ex (x,A))) in G by Th35;

(((\ex (x,A)) \imp (\ex (x,B))) \and ((\ex (x,B)) \imp (\ex (x,A)))) \imp ((\ex (x,A)) \iff (\ex (x,B))) in G by Def38;

hence (\ex (x,A)) \iff (\ex (x,B)) in G by A3, Def38; :: thesis: verum