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 empty-yielding GeneratorSet of T
for S1 being non empty non void b₁ -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 (\ex (x,A))) . a
let J be non empty non void Signature; :: thesis: for T being non-empty MSAlgebra over J
for X being empty-yielding 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 (\ex (x,A))) . a
let T be non-empty MSAlgebra over J; :: thesis: for X being empty-yielding 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 (\ex (x,A))) . a
let X be empty-yielding 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 (\ex (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 (\ex (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 (\ex (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 (\ex (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 (\ex (x,A))) . a )
assume A1: L is vf-qc-correct ; :: thesis: for a being SortSymbol of S1 st x in X . a holds
x nin (vf (\ex (x,A))) . a
let a be SortSymbol of S1; :: thesis: ( x in X . a implies x nin (vf (\ex (x,A))) . a )
assume x in X . a ; :: thesis: x nin (vf (\ex (x,A))) . a
then vf (\ex (x,A)) = (vf A) (\) (a -singleton x) by A1;
then A2: (vf (\ex (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 (\ex (x,A))) . a by A2, XBOOLE_0:def 5; :: thesis: verum