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 G being QC-theory of L
for A being Formula of L
for x being Element of Union X st L is subst-correct holds
A \imp (\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 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 G being QC-theory of L
for A being Formula of L
for x being Element of Union X st L is subst-correct holds
A \imp (\ex (x,A)) in G
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 G being QC-theory of L
for A being Formula of L
for x being Element of Union X st L is subst-correct holds
A \imp (\ex (x,A)) in G
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 G being QC-theory of L
for A being Formula of L
for x being Element of Union X st L is subst-correct holds
A \imp (\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 being Element of Union X st L is subst-correct holds
A \imp (\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 being Element of Union X st L is subst-correct holds
A \imp (\ex (x,A)) in G
let G be QC-theory of L; :: thesis: for A being Formula of L
for x being Element of Union X st L is subst-correct holds
A \imp (\ex (x,A)) in G
let A be Formula of L; :: thesis: for x being Element of Union X st L is subst-correct holds
A \imp (\ex (x,A)) in G
let x be Element of Union X; :: thesis: ( L is subst-correct implies A \imp (\ex (x,A)) in G )
set Y = X extended_by ({}, the carrier of S1);
assume A1: L is subst-correct ; :: thesis: A \imp (\ex (x,A)) in G
consider a being object such that
A2: ( a in dom X & x in X . a ) by CARD_5:2;
reconsider a = a as SortSymbol of J by A2;
A3: ( x in X . a & a is SortSymbol of S1 ) by A2, Th8;
then A4: ( x in (X extended_by ({}, the carrier of S1)) . a & dom (X extended_by ({}, the carrier of S1)) = the carrier of S1 ) by Th2, PARTFUN1:def 2;
then reconsider x0 = x as Element of Union (X extended_by ({}, the carrier of S1)) by A3, CARD_5:2;
(A / (x0,x0)) \imp (\ex (x,A)) in G by A1, A2, Th110;
hence A \imp (\ex (x,A)) in G by A1, A3, A4; :: thesis: verum