let S be non empty non void ManySortedSign ; :: thesis: for U0 being MSAlgebra over S

for A being MSSubset of U0 holds (Constants U0) (\/) A c= MSSubSort A

let U0 be MSAlgebra over S; :: thesis: for A being MSSubset of U0 holds (Constants U0) (\/) A c= MSSubSort A

let A be MSSubset of U0; :: thesis: (Constants U0) (\/) A c= MSSubSort A

let i be object ; :: according to PBOOLE:def 2 :: thesis: ( not i in the carrier of S or ((Constants U0) (\/) A) . i c= (MSSubSort A) . i )

assume i in the carrier of S ; :: thesis: ((Constants U0) (\/) A) . i c= (MSSubSort A) . i

then reconsider s = i as SortSymbol of S ;

A1: for Z being set st Z in SubSort (A,s) holds

((Constants U0) (\/) A) . s c= Z

hence ((Constants U0) (\/) A) . i c= (MSSubSort A) . i by A1, SETFAM_1:5; :: thesis: verum

for A being MSSubset of U0 holds (Constants U0) (\/) A c= MSSubSort A

let U0 be MSAlgebra over S; :: thesis: for A being MSSubset of U0 holds (Constants U0) (\/) A c= MSSubSort A

let A be MSSubset of U0; :: thesis: (Constants U0) (\/) A c= MSSubSort A

let i be object ; :: according to PBOOLE:def 2 :: thesis: ( not i in the carrier of S or ((Constants U0) (\/) A) . i c= (MSSubSort A) . i )

assume i in the carrier of S ; :: thesis: ((Constants U0) (\/) A) . i c= (MSSubSort A) . i

then reconsider s = i as SortSymbol of S ;

A1: for Z being set st Z in SubSort (A,s) holds

((Constants U0) (\/) A) . s c= Z

proof

(MSSubSort A) . s = meet (SubSort (A,s))
by Def14;
let Z be set ; :: thesis: ( Z in SubSort (A,s) implies ((Constants U0) (\/) A) . s c= Z )

assume Z in SubSort (A,s) ; :: thesis: ((Constants U0) (\/) A) . s c= Z

then consider B being MSSubset of U0 such that

A2: B in SubSort A and

A3: Z = B . s by Def13;

( Constants U0 c= B & A c= B ) by A2, Th13;

then (Constants U0) (\/) A c= B by PBOOLE:16;

hence ((Constants U0) (\/) A) . s c= Z by A3; :: thesis: verum

end;assume Z in SubSort (A,s) ; :: thesis: ((Constants U0) (\/) A) . s c= Z

then consider B being MSSubset of U0 such that

A2: B in SubSort A and

A3: Z = B . s by Def13;

( Constants U0 c= B & A c= B ) by A2, Th13;

then (Constants U0) (\/) A c= B by PBOOLE:16;

hence ((Constants U0) (\/) A) . s c= Z by A3; :: thesis: verum

hence ((Constants U0) (\/) A) . i c= (MSSubSort A) . i by A1, SETFAM_1:5; :: thesis: verum