
theorem Th51:
  for f being Function st f.{} = {} for x,y being object st {x} in
  dom f & y in f.{x} holds [x,y] in LinTrace f
proof
  let f be Function;
  assume
A1: f.{} = {};
  let x,y be object;
  set a = {x};
  [x,y] in LinTrace f iff [{x},y] in Trace f by Th50;
  then
  [x,y] in LinTrace f iff {x} in dom f & y in f.{x} & for b being set st b
  in dom f & b c= a & y in f.b holds a = b by Th31;
  hence thesis by A1,ZFMISC_1:33;
end;
