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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) in G

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

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) in G

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) in G

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

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

then A \imp (\ex (x,A)) in G by Th112;

then ( \for (y,(A \imp (\ex (x,A)))) in G & (\for (y,(A \imp (\ex (x,A))))) \imp ((\for (y,A)) \imp (\for (y,(\ex (x,A))))) in G ) by A1, Def39, Th109;

then (\for (y,A)) \imp (\for (y,(\ex (x,A)))) in G by Def38;

then A2: \for (x,((\for (y,A)) \imp (\for (y,(\ex (x,A)))))) in G by Def39;

consider a being object such that

A3: ( a in dom X & y in X . a ) by CARD_5:2;

consider b being object such that

A4: ( b in dom X & x in X . b ) 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 ) by PARTFUN1:def 2, INSTALG1:10;

then reconsider a = a, b = b as Element of S1 by A3, A4;

reconsider c = b as Element of J by A4;

vf (\for (y,(\ex (x,A)))) = (vf (\ex (x,A))) (\) (a -singleton y) by A1, A3;

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

then x nin (vf (\for (y,(\ex (x,A))))) . c by A1, A4, Th114;

then (\for (x,((\for (y,A)) \imp (\for (y,(\ex (x,A))))))) \imp ((\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A))))) in G by A1, A4, Th120;

hence (\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) in G by A2, 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) 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 being Formula of L

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) in G

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

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) in G

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

(\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) in G

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

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

then A \imp (\ex (x,A)) in G by Th112;

then ( \for (y,(A \imp (\ex (x,A)))) in G & (\for (y,(A \imp (\ex (x,A))))) \imp ((\for (y,A)) \imp (\for (y,(\ex (x,A))))) in G ) by A1, Def39, Th109;

then (\for (y,A)) \imp (\for (y,(\ex (x,A)))) in G by Def38;

then A2: \for (x,((\for (y,A)) \imp (\for (y,(\ex (x,A)))))) in G by Def39;

consider a being object such that

A3: ( a in dom X & y in X . a ) by CARD_5:2;

consider b being object such that

A4: ( b in dom X & x in X . b ) 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 ) by PARTFUN1:def 2, INSTALG1:10;

then reconsider a = a, b = b as Element of S1 by A3, A4;

reconsider c = b as Element of J by A4;

vf (\for (y,(\ex (x,A)))) = (vf (\ex (x,A))) (\) (a -singleton y) by A1, A3;

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

then x nin (vf (\for (y,(\ex (x,A))))) . c by A1, A4, Th114;

then (\for (x,((\for (y,A)) \imp (\for (y,(\ex (x,A))))))) \imp ((\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A))))) in G by A1, A4, Th120;

hence (\ex (x,(\for (y,A)))) \imp (\for (y,(\ex (x,A)))) in G by A2, Def38; :: thesis: verum