
theorem Th33:
  for f being U-continuous Function st dom f is subset-closed for
  a, y being set st [a,y] in Trace f holds a is finite
proof
  let f be U-continuous Function such that
A1: dom f is subset-closed;
  let a, y be set;
  assume
A2: [a,y] in Trace f;
  then
A3: a in dom f by Th31;
  y in f.a by A2,Th31;
  then consider b be set such that
A4: b is finite and
A5: b c= a and
A6: y in f.b by A1,A3,Th21;
  b in dom f by A1,A3,A5;
  hence thesis by A2,A4,A5,A6,Th31;
end;
