let S be non empty non void all-with_const_op ManySortedSign ; :: thesis: for U0 being non-empty MSAlgebra of S
for U1, U2 being non-empty MSSubAlgebra of U0 holds the Sorts of U1 /\ the Sorts of U2 is non-empty
let U0 be non-empty MSAlgebra of S; :: thesis: for U1, U2 being non-empty MSSubAlgebra of U0 holds the Sorts of U1 /\ the Sorts of U2 is non-empty
let U1, U2 be non-empty MSSubAlgebra of U0; :: thesis: the Sorts of U1 /\ the Sorts of U2 is non-empty
Constants U0 is V8() MSSubset of U2 by Th11;
then A1: Constants U0 c= the Sorts of U2 by PBOOLE:def 18;
Constants U0 is V8() MSSubset of U1 by Th11;
then Constants U0 c= the Sorts of U1 by PBOOLE:def 18;
then A2: (Constants U0) /\ (Constants U0) c= the Sorts of U1 /\ the Sorts of U2 by A1, PBOOLE:21;

now

let i be set ; :: thesis: ( i in the carrier of S implies not ( the Sorts of U1 /\ the Sorts of U2) . i is empty )
assume i in the carrier of S ; :: thesis: not ( the Sorts of U1 /\ the Sorts of U2) . i is empty
then reconsider s = i as SortSymbol of S ;
( the Sorts of U1 /\ the Sorts of U2) . s = ( the Sorts of U1 . s) /\ ( the Sorts of U2 . s) by PBOOLE:def 5;
then A3: (Constants U0) . s c= ( the Sorts of U1 . s) /\ ( the Sorts of U2 . s) by A2, PBOOLE:def 2;
ex a being set st a in (Constants U0) . s by XBOOLE_0:def 1;
hence not ( the Sorts of U1 /\ the Sorts of U2) . i is empty by A3, PBOOLE:def 5; :: thesis: verum

end;

hence the Sorts of U1 /\ the Sorts of U2 is non-empty by PBOOLE:def 13; :: thesis: verum