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 x, y being Element of Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

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 x, y being Element of Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

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 x, y being Element of Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

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 x, y being Element of Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

let S1 be non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X; :: thesis: for x, y being Element of Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

let x, y be Element of Union X; :: thesis: for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis: for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

let G1 be QC-theory_with_equality of L; :: thesis: for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

let s be SortSymbol of S1; :: thesis: ( L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y implies \for (x,(\ex (y,(x '=' (y,L))))) in G1 )

assume that

A0: ( L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct ) and

A1: ( x in X . s & y in X . s & x <> y ) ; :: thesis: \for (x,(\ex (y,(x '=' (y,L))))) in G1

A2: ( s in dom X & dom X = the carrier of J ) by A1, FUNCT_1:def 2, PARTFUN1:def 2;

then ( X . s c= the Sorts of T . s & the Sorts of T . s = the Sorts of L . s ) by Th16, PBOOLE:def 2, PBOOLE:def 18;

then reconsider t1 = x, t2 = y as Element of L,s by A1;

reconsider j = s as SortSymbol of J by A2;

reconsider q1 = t1, q2 = t2 as Element of T,j by Th16;

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

reconsider y0 = y, x0 = x as Element of Union (X extended_by ({}, the carrier of S1)) by Th24;

dom (X extended_by ({}, the carrier of S1)) = the carrier of S1 by PARTFUN1:def 2;

then A4: ( X . s = (X extended_by ({}, the carrier of S1)) . s & the Sorts of L . the formula-sort of S1 <> {} & X extended_by ({}, the carrier of S1) is ManySortedSubset of the Sorts of L ) by A2, Th1, Th23;

vf t1 = s -singleton x0 by A0, A1;

then (vf t1) . s = {x0} by AOFA_A00:6;

then B2: y0 nin (vf t1) . s by A1, TARSKI:def 1;

A3: ((t1 '=' (t2,L)) / (y0,x0)) \imp (\ex (y,(t1 '=' (t2,L)))) in G1 by A0, A1, A2, Th110;

A5: (t1 '=' (t2,L)) / (y0,x0) = (t1 '=' (t2,L)) / (y0,t1) by A1, A4, Th14

.= (t1 / (y0,t1)) '=' ((t2 / (y0,t1)),L) by A0, A1

.= t1 '=' ((t2 / (y0,t1)),L) by B2, A0, A1, A4

.= t1 '=' (t1,L) by A0, A1, A4 ;

q1 '=' (q1,L) in G1 by Def42;

then \ex (y,(x '=' (y,L))) in G1 by A3, A5, Def38;

hence \for (x,(\ex (y,(x '=' (y,L))))) in G1 by Def39; :: 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 x, y being Element of Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

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 x, y being Element of Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

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 x, y being Element of Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

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 x, y being Element of Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

let S1 be non empty non void J -extension n PC-correct QC-correct QCLangSignature over Union X; :: thesis: for x, y being Element of Union X

for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

let x, y be Element of Union X; :: thesis: for L being non-empty Language of X extended_by ({}, the carrier of S1),S1

for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

let L be non-empty Language of X extended_by ({}, the carrier of S1),S1; :: thesis: for G1 being QC-theory_with_equality of L

for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

let G1 be QC-theory_with_equality of L; :: thesis: for s being SortSymbol of S1 st L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y holds

\for (x,(\ex (y,(x '=' (y,L))))) in G1

let s be SortSymbol of S1; :: thesis: ( L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct & x in X . s & y in X . s & x <> y implies \for (x,(\ex (y,(x '=' (y,L))))) in G1 )

assume that

A0: ( L is subst-correct & L is subst-eq-correct & L is subst-correct3 & L is vf-eq-correct ) and

A1: ( x in X . s & y in X . s & x <> y ) ; :: thesis: \for (x,(\ex (y,(x '=' (y,L))))) in G1

A2: ( s in dom X & dom X = the carrier of J ) by A1, FUNCT_1:def 2, PARTFUN1:def 2;

then ( X . s c= the Sorts of T . s & the Sorts of T . s = the Sorts of L . s ) by Th16, PBOOLE:def 2, PBOOLE:def 18;

then reconsider t1 = x, t2 = y as Element of L,s by A1;

reconsider j = s as SortSymbol of J by A2;

reconsider q1 = t1, q2 = t2 as Element of T,j by Th16;

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

reconsider y0 = y, x0 = x as Element of Union (X extended_by ({}, the carrier of S1)) by Th24;

dom (X extended_by ({}, the carrier of S1)) = the carrier of S1 by PARTFUN1:def 2;

then A4: ( X . s = (X extended_by ({}, the carrier of S1)) . s & the Sorts of L . the formula-sort of S1 <> {} & X extended_by ({}, the carrier of S1) is ManySortedSubset of the Sorts of L ) by A2, Th1, Th23;

vf t1 = s -singleton x0 by A0, A1;

then (vf t1) . s = {x0} by AOFA_A00:6;

then B2: y0 nin (vf t1) . s by A1, TARSKI:def 1;

A3: ((t1 '=' (t2,L)) / (y0,x0)) \imp (\ex (y,(t1 '=' (t2,L)))) in G1 by A0, A1, A2, Th110;

A5: (t1 '=' (t2,L)) / (y0,x0) = (t1 '=' (t2,L)) / (y0,t1) by A1, A4, Th14

.= (t1 / (y0,t1)) '=' ((t2 / (y0,t1)),L) by A0, A1

.= t1 '=' ((t2 / (y0,t1)),L) by B2, A0, A1, A4

.= t1 '=' (t1,L) by A0, A1, A4 ;

q1 '=' (q1,L) in G1 by Def42;

then \ex (y,(x '=' (y,L))) in G1 by A3, A5, Def38;

hence \for (x,(\ex (y,(x '=' (y,L))))) in G1 by Def39; :: thesis: verum