theorem Th31:
  for c being with_endpoints Curve of T
  holds the_first_point_of c in rng c & the_last_point_of c in rng c
  proof
    let c be with_endpoints Curve of T;
A1: inf dom c <= sup dom c by XXREAL_2:40;
    dom c = [.inf dom c,sup dom c.] by Th27;
    then inf dom c in dom c & sup dom c in dom c by A1,XXREAL_1:1;
    hence thesis by FUNCT_1:3;
  end;
