reserve Omega, I for non empty set;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve D, E, F for Subset-Family of Omega;
reserve  B, sB for non empty Subset of Sigma;
reserve b for Element of B;
reserve a for Element of Sigma;
reserve p, q, u, v for Event of Sigma;
reserve n, m for Element of NAT;
reserve S, S9, X, x, y, z, i, j for set;

theorem Th1:
  for f being Function, X being set st X c= dom f holds X <> {}
  implies rng (f|X) <> {}
proof
  let f be Function, X be set;
  assume
A1: X c= dom f;
  set x = the Element of X;
  assume X <> {};
  then x in X;
  hence thesis by A1,FUNCT_1:50;
end;
