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
  for E being finite non empty set, A being Event of E holds prob(A) <= 1
proof
  let E be finite non empty set, A be Event of E;
  0 < card E by Lm1;
  then card A * (card E)" <= card E * (card E)" by NAT_1:43,XREAL_1:64;
  then card A / card E <= card E * (card E)" by XCMPLX_0:def 9;
  then prob([#] E) = card E / card E & prob(A) <= card E / card E by
XCMPLX_0:def 9;
  hence thesis by XCMPLX_1:60;
end;
