
theorem Th35:
  for f being non-empty Function, X being set, i being object st i in dom f
  holds f +* (i,X) is non-empty iff X is non empty
proof
  let f be non-empty Function, X be set, i be object;
  assume A1: i in dom f;
  hereby
    assume A2: f +* (i,X) is non-empty;
    i in dom(f +* (i,X)) by A1, FUNCT_7:30;
    then (f +* (i,X)).i <> {} by A2, FUNCT_1:def 9;
    hence X is non empty by A1, FUNCT_7:31;
  end;
  assume A3: X is non empty;
  for x being object st x in dom(f +* (i,X)) holds (f +* (i,X)).x is non empty
  proof
    let x be object;
    assume A4: x in dom(f +* (i,X));
    A5: x in dom f by A4, FUNCT_7:30;
    x <> i implies (f +* (i,X)).x = f.x by FUNCT_7:32;
    hence thesis by A5, FUNCT_1:def 9, A3, FUNCT_7:31;
  end;
  hence thesis by FUNCT_1:def 9;
end;
