theorem
  c in dom f & c in X implies f/.c in rng (f|X)
proof
  assume that
A1: c in dom f and
A2: c in X;
  (f qua Function).c in rng (f|X) by A1,A2,FUNCT_1:50;
  hence thesis by A1,PARTFUN1:def 6;
end;
