reconsider B = {} as Subset of Omega by XBOOLE_1:2;
take B ; :: thesis: ex A being set st
( A in Sigma & B c= A & P . A = 0 )
take A = {} ; :: thesis: ( A in Sigma & B c= A & P . A = 0 )
thus A in Sigma by PROB_1:4; :: thesis: ( B c= A & P . A = 0 )
thus B c= A ; :: thesis: P . A = 0
thus P . A = 0 by VALUED_0:def 19; :: thesis: verum