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 Th34:
  A c= B implies P.A <= P.B
proof
  assume A c= B;
  then P.(B \ A) = P.B - P.A by Th33;
  then 0 <= P.B - P.A by Def8;
  then 0 + P.A <= P.B by XREAL_1:19;
  hence thesis;
end;
