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

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

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

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

then ( (\for (x,A)) \imp A in G & (\for (x,B)) \imp B in G ) by Th104;

then ((\for (x,A)) \or (\for (x,B))) \imp (A \or B) in G by Th59;

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

consider a being object such that

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

J is Subsignature of S1 by Def2;

then A4: ( the carrier of J c= the carrier of S1 & the carrier of S1 = dom (X extended_by ({}, the carrier of S1)) & dom X = the carrier of J ) by PARTFUN1:def 2, INSTALG1:10;

reconsider a = a as SortSymbol of J by A3;

reconsider b = a as SortSymbol of S1 by A4;

( x nin (vf (\for (x,A))) . b & x nin (vf (\for (x,B))) . b ) by A1, A3, Th113;

then x nin ((vf (\for (x,A))) . a) \/ ((vf (\for (x,B))) . a) by XBOOLE_0:def 3;

then x nin ((vf (\for (x,A))) (\/) (vf (\for (x,B)))) . b by PBOOLE:def 4;

then x nin (vf ((\for (x,A)) \or (\for (x,B)))) . a by A1;

hence ((\for (x,A)) \or (\for (x,B))) \imp (\for (x,(A \or B))) in G by A2, A3, Th108; :: 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)) \or (\for (x,B))) \imp (\for (x,(A \or 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)) \or (\for (x,B))) \imp (\for (x,(A \or 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)) \or (\for (x,B))) \imp (\for (x,(A \or 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)) \or (\for (x,B))) \imp (\for (x,(A \or 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)) \or (\for (x,B))) \imp (\for (x,(A \or 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)) \or (\for (x,B))) \imp (\for (x,(A \or 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)) \or (\for (x,B))) \imp (\for (x,(A \or 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)) \or (\for (x,B))) \imp (\for (x,(A \or B))) in G

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

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

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

then ( (\for (x,A)) \imp A in G & (\for (x,B)) \imp B in G ) by Th104;

then ((\for (x,A)) \or (\for (x,B))) \imp (A \or B) in G by Th59;

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

consider a being object such that

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

J is Subsignature of S1 by Def2;

then A4: ( the carrier of J c= the carrier of S1 & the carrier of S1 = dom (X extended_by ({}, the carrier of S1)) & dom X = the carrier of J ) by PARTFUN1:def 2, INSTALG1:10;

reconsider a = a as SortSymbol of J by A3;

reconsider b = a as SortSymbol of S1 by A4;

( x nin (vf (\for (x,A))) . b & x nin (vf (\for (x,B))) . b ) by A1, A3, Th113;

then x nin ((vf (\for (x,A))) . a) \/ ((vf (\for (x,B))) . a) by XBOOLE_0:def 3;

then x nin ((vf (\for (x,A))) (\/) (vf (\for (x,B)))) . b by PBOOLE:def 4;

then x nin (vf ((\for (x,A)) \or (\for (x,B)))) . a by A1;

hence ((\for (x,A)) \or (\for (x,B))) \imp (\for (x,(A \or B))) in G by A2, A3, Th108; :: thesis: verum