the disjoint sets class proof from lemma we havec(mnrc(mt nt rc(mb nb smt ( )nt ( and by theorem (mt nt rmt nt log thusc(mnrmt nt log (mb nb smt ( )nt ( rearranging and combining terms yields (mnrc(mb nb mt ( log )nt ( so choose log clearlythis choice implies that ( log and thus we obtain (mnrc(mb nb log mt ( rearranging as in theorem we obtain (mnr (mb nb log mb ( this timelet (mnrc(mnr mthen (mnrd(mb nb log ( which implies (mnrn logr this yields (mnr logr an om (mnbound not surprisinglywe can now use theorem to improve theorem theorem (mnr log* proof following the steps in the proof of theorem we have (mnrc(mt nt rc(mb nb smt ( )nt ( and by theorem (mt nt mt nt logr thusc(mnr mt nt logr (mb nb smt ( )nt ( rearranging and combining terms yields (mnrc(mb nb mt ( logr )nt ( so choose logr to obtain (mnrc(mb nb logr mt (