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 A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

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 A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

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 A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

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 A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

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 A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis: for A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

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

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

let x be Element of Union X; :: thesis: ( L is vf-qc-correct implies for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a )

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

assume A1: L is vf-qc-correct ; :: thesis: for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

let a be SortSymbol of S1; :: thesis: ( x in X . a implies x nin (vf (\for (x,A))) . a )

assume x in X . a ; :: thesis: x nin (vf (\for (x,A))) . a

then vf (\for (x,A)) = (vf A) (\) (a -singleton x) by A1;

then A2: (vf (\for (x,A))) . a = ((vf A) . a) \ ((a -singleton x) . a) by PBOOLE:def 6

.= ((vf A) . a) \ {x} by AOFA_A00:6 ;

x in {x} by TARSKI:def 1;

hence x nin (vf (\for (x,A))) . a by A2, XBOOLE_0:def 5; :: 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 A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

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 A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

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 A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

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 A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

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 A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis: for A being Formula of L

for x being Element of Union X st L is vf-qc-correct holds

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

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

for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

let x be Element of Union X; :: thesis: ( L is vf-qc-correct implies for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a )

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

assume A1: L is vf-qc-correct ; :: thesis: for a being SortSymbol of S1 st x in X . a holds

x nin (vf (\for (x,A))) . a

let a be SortSymbol of S1; :: thesis: ( x in X . a implies x nin (vf (\for (x,A))) . a )

assume x in X . a ; :: thesis: x nin (vf (\for (x,A))) . a

then vf (\for (x,A)) = (vf A) (\) (a -singleton x) by A1;

then A2: (vf (\for (x,A))) . a = ((vf A) . a) \ ((a -singleton x) . a) by PBOOLE:def 6

.= ((vf A) . a) \ {x} by AOFA_A00:6 ;

x in {x} by TARSKI:def 1;

hence x nin (vf (\for (x,A))) . a by A2, XBOOLE_0:def 5; :: thesis: verum