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

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

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

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

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

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

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

( (A \imp B) \imp (A \imp B) in F & ((A \imp B) \imp (A \imp B)) \imp (((A \imp B) \and A) \imp B) in F ) by Th34, Th48;

then ( ((A \imp B) \and A) \imp B in F & B \imp (C \or B) in F ) by Def38;

then ( ((A \imp B) \and A) \imp (C \or B) in F & (((A \imp B) \and A) \imp (C \or B)) \imp ((A \imp B) \imp (A \imp (C \or B))) in F ) by Th45, Th47;

then (A \imp B) \imp (A \imp (C \or B)) in F by Def38;

hence (A \imp B) \imp ((C \or A) \imp (C \or B)) in F by A1, Th45; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

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

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

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

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

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

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

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

( (A \imp B) \imp (A \imp B) in F & ((A \imp B) \imp (A \imp B)) \imp (((A \imp B) \and A) \imp B) in F ) by Th34, Th48;

then ( ((A \imp B) \and A) \imp B in F & B \imp (C \or B) in F ) by Def38;

then ( ((A \imp B) \and A) \imp (C \or B) in F & (((A \imp B) \and A) \imp (C \or B)) \imp ((A \imp B) \imp (A \imp (C \or B))) in F ) by Th45, Th47;

then (A \imp B) \imp (A \imp (C \or B)) in F by Def38;

hence (A \imp B) \imp ((C \or A) \imp (C \or B)) in F by A1, Th45; :: thesis: verum