theorem Th61:
  f^ is total iff f"{0} = {} & f is total
proof
  thus f^ is total implies f"{0} = {} & f is total
  proof
    assume f^ is total;
    then
A1: dom (f^) = C;
    f"{0c} c= C;
    then f"{0c} c= dom f \ f"{0c} by A1,Def2;
    hence f"{0} = {} by XBOOLE_1:38;
    then C = dom f \ {} by A1,Def2;
    hence dom f = C;
  end;
  assume
A2: f"{0} = {} & f is total;
  thus dom (f^) = dom f \ f"{0c} by Def2
    .= C by A2;
end;
