theorem Th40:
  for C being thin of P holds (COM P).C = 0
proof
  let C be thin of P;
  consider A being set such that
A1: A in Sigma and
A2: C c= A and
A3: P.A = 0 by Def4;
  reconsider A as Event of Sigma by A1;
A4: (COM P).A = 0 by A3,Th39;
  Sigma c= COM(Sigma,P) by Th28;
  then reconsider A as Event of COM(Sigma,P);
  (COM P).C <= (COM P).A by A2,PROB_1:34;
  hence thesis by A4,PROB_1:def 8;
end;
