reserve Omega for set;
reserve X, Y, Z, p,x,y,z for set;
reserve D, E for Subset of Omega;
reserve f for Function;
reserve m,n for Nat;
reserve r,r1 for Real;
reserve seq for Real_Sequence;
reserve F for Field_Subset of X;
reserve ASeq,BSeq for SetSequence of Omega;
reserve A1 for SetSequence of X;
reserve Sigma for SigmaField of Omega;
reserve Si for SigmaField of X;
reserve A, B for Event of Sigma,
  ASeq for SetSequence of Sigma;
reserve P for Function of Sigma,REAL;
reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve A, B for Event of Sigma,
  ASeq for SetSequence of Sigma;
reserve P for Function of Sigma,REAL;
reserve D, E for Subset of Omega;
reserve BSeq for SetSequence of Omega;
reserve P for Probability of Sigma;

theorem Th31:
  P.(([#] Sigma) \ A) + P.A = 1
proof
A1: (([#] Sigma) \ A) \/ A = A` \/ A .= [#] Omega by SUBSET_1:10
    .= Omega;
  (([#] Sigma) \ A) misses A by XBOOLE_1:79;
  then P.(([#] Sigma) \ A) + P.A = P.Omega by A1,Def8
    .= 1 by Def8;
  hence thesis;
end;
