
theorem Th36:
  for f being non-empty Function, X being set, i being object st i in dom f
  holds product(f +* (i,X)) = {} iff X is empty
proof
  let f be non-empty Function, X be set, i be object;
  assume A1: i in dom f;
  then A2: i in dom(f +* (i,X)) by FUNCT_7:30;
  hereby
    assume product(f +* (i,X)) = {};
    then f +* (i,X) is non non-empty;
    hence X is empty by A1, Th35;
  end;
  assume X is empty;
  then (f +* (i,X)).i = {} by A1, FUNCT_7:31;
  then {} in rng (f +* (i,X)) by A2, FUNCT_1:def 3;
  hence thesis by CARD_3:26;
end;
