
theorem Th31:
  for f being Function
  for a being set, y being object holds [a,y] in Trace f
  iff a in dom f & y in f.a & for b being set st b in dom f & b c= a & y in f.b
  holds a = b
proof
  let f be Function, a9 be set, y9 be object;
  now
    given a, y being set such that
A1: [a9,y9] = [a,y] and
A2: a in dom f & y in f.a & for b being set st b in dom f & b c= a & y
    in f. b holds a = b;
    a9 = a & y9 = y by A1,XTUPLE_0:1;
    hence a9 in dom f & y9 in f.a9 & for b being set st b in dom f & b c= a9 &
    y9 in f.b holds a9 = b by A2;
  end;
  hence thesis by Def17;
end;
