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 \or B) \and (A \or C)) \imp (A \or (B \and C)) 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 \or B) \and (A \or C)) \imp (A \or (B \and C)) 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 \or B) \and (A \or C)) \imp (A \or (B \and C)) in F

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

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

( B \imp (C \imp (B \and C)) in F & A \imp A in F ) by Def38, Th34;

then ( (A \or B) \imp (A \or (C \imp (B \and C))) in F & (A \or (C \imp (B \and C))) \imp ((A \or C) \imp (A \or (B \and C))) in F ) by Th59, Th79;

then A1: (A \or B) \imp ((A \or C) \imp (A \or (B \and C))) in F by Th45;

((A \or B) \imp ((A \or C) \imp (A \or (B \and C)))) \imp (((A \or B) \and (A \or C)) \imp (A \or (B \and C))) in F by Th48;

hence ((A \or B) \and (A \or C)) \imp (A \or (B \and C)) in F by A1, Def38; :: 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 \or B) \and (A \or C)) \imp (A \or (B \and C)) 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 \or B) \and (A \or C)) \imp (A \or (B \and C)) 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 \or B) \and (A \or C)) \imp (A \or (B \and C)) in F

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

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

( B \imp (C \imp (B \and C)) in F & A \imp A in F ) by Def38, Th34;

then ( (A \or B) \imp (A \or (C \imp (B \and C))) in F & (A \or (C \imp (B \and C))) \imp ((A \or C) \imp (A \or (B \and C))) in F ) by Th59, Th79;

then A1: (A \or B) \imp ((A \or C) \imp (A \or (B \and C))) in F by Th45;

((A \or B) \imp ((A \or C) \imp (A \or (B \and C)))) \imp (((A \or B) \and (A \or C)) \imp (A \or (B \and C))) in F by Th48;

hence ((A \or B) \and (A \or C)) \imp (A \or (B \and C)) in F by A1, Def38; :: thesis: verum