reserve FT for non empty RelStr,
  A,B,C for Subset of FT;

theorem
  for A being Subset of FT,x1,x2 being Element of FT st ex f being
  FinSequence of FT st f is continuous & rng f c=A & f.1=x1 & f.len f=x2 ex g
  being FinSequence of FT st g is_minimum_path_in A,x1,x2
proof
  let A be Subset of FT,x1,x2 be Element of FT;
  given f being FinSequence of FT such that
A1: f is continuous & rng f c=A & f.1=x1 & f.(len f)=x2;
  consider g2 being FinSequence of FT such that
A2: g2 is continuous & rng g2 c=A & g2.1=x1 & g2.(len g2)=x2 & for h
being FinSequence of FT st h is continuous & rng h c=A & h.1= x1 & h.(len h)=x2
  holds len g2 <= len h by A1,Lm4;
  g2 is_minimum_path_in A,x1,x2 by A2;
  hence thesis;
end;
