reserve E for non empty set;
reserve a for Element of E;
reserve A, B for Subset of E;
reserve Y for set;
reserve p for FinSequence;
reserve e, e1, e2 for Singleton of E;

theorem Th19:
  for E being finite non empty set, A,B being Event of E st A c= B
  holds prob(A) <= prob(B)
proof
  let E be finite non empty set, A,B be Event of E;
  assume
A1: A c= B;
  0 < card E by Lm1;
  then card A * (card E)" <= card B * (card E)" by A1,NAT_1:43,XREAL_1:64;
  then card A / card E <= card B * (card E)" by XCMPLX_0:def 9;
  hence thesis by XCMPLX_0:def 9;
end;
