
theorem
  for f being U-continuous Function st dom f is subset-closed holds
  Trace f c= graph f
proof
  let f be U-continuous Function such that
A1: dom f is subset-closed;
  let x,z be object;
  assume [x,z] in Trace f;
  then consider a, y being set such that
A2: [x,z] = [a,y] and
A3: a in dom f and
A4: y in f.a and
A5: for b being set st b in dom f & b c= a & y in f.b holds a = b by Def17;
  consider b being set such that
A6: b is finite and
A7: b c= a and
A8: y in f.b by A1,A3,A4,Th21;
  b in dom f by A1,A3,A7;
  then a = b by A5,A7,A8;
  hence thesis by A2,A3,A4,A6,Th24;
end;
