let n be non empty Nat; :: thesis: for S being non empty non void n PC-correct PCLangSignature

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds (A \or B) \imp (B \or A) in F

let S be non empty non void n PC-correct PCLangSignature ; :: thesis: for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds (A \or B) \imp (B \or A) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B being Formula of L holds (A \or B) \imp (B \or A) in F

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds (A \or B) \imp (B \or A) in F

let A, B be Formula of L; :: thesis: (A \or B) \imp (B \or A) in F

A1: (A \imp (B \or A)) \imp ((B \imp (B \or A)) \imp ((A \or B) \imp (B \or A))) in F by Def38;

A2: ( A \imp (B \or A) in F & B \imp (B \or A) in F ) by Def38;

(B \imp (B \or A)) \imp ((A \or B) \imp (B \or A)) in F by A1, A2, Def38;

hence (A \or B) \imp (B \or A) in F by A2, Def38; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds (A \or B) \imp (B \or A) in F

let S be non empty non void n PC-correct PCLangSignature ; :: thesis: for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds (A \or B) \imp (B \or A) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B being Formula of L holds (A \or B) \imp (B \or A) in F

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds (A \or B) \imp (B \or A) in F

let A, B be Formula of L; :: thesis: (A \or B) \imp (B \or A) in F

A1: (A \imp (B \or A)) \imp ((B \imp (B \or A)) \imp ((A \or B) \imp (B \or A))) in F by Def38;

A2: ( A \imp (B \or A) in F & B \imp (B \or A) in F ) by Def38;

(B \imp (B \or A)) \imp ((A \or B) \imp (B \or A)) in F by A1, A2, Def38;

hence (A \or B) \imp (B \or A) in F by A2, Def38; :: thesis: verum