let S be non empty non void ManySortedSign ; :: thesis: for U0 being non-empty MSAlgebra of S
for U1, U2 being strict MSSubAlgebra of U0 holds (U1 /\ U2) "\/" U2 = U2
let U0 be non-empty MSAlgebra of S; :: thesis: for U1, U2 being strict MSSubAlgebra of U0 holds (U1 /\ U2) "\/" U2 = U2
let U1, U2 be strict MSSubAlgebra of U0; :: thesis: (U1 /\ U2) "\/" U2 = U2
reconsider u12 = the Sorts of (U1 /\ U2), u2 = the Sorts of U2 as MSSubset of U0 by Def10;
( u12 c= the Sorts of U0 & u2 c= the Sorts of U0 ) by PBOOLE:def 18;
then u12 \/ u2 c= the Sorts of U0 by PBOOLE:16;
then reconsider A = u12 \/ u2 as MSSubset of U0 by PBOOLE:def 18;
u12 = the Sorts of U1 /\ the Sorts of U2 by Def17;
then u12 c= u2 by PBOOLE:15;
then A1: A = u2 by PBOOLE:22;
(U1 /\ U2) "\/" U2 = GenMSAlg A by Def19;
hence (U1 /\ U2) "\/" U2 = U2 by A1, Th23; :: thesis: verum