
theorem Th32:
  for G being _Graph, W being Walk of G st
    W is open & W is Path-like holds W is vertex-distinct
proof
  let G be _Graph, W be Walk of G;
  assume that
A1: W is open and
A2: W is Path-like;
  now
    let m,n be odd Element of NAT such that
A3: m <= len W and
A4: n <= len W and
A5: W.m = W.n;
    assume
A6: m <> n;
    per cases by A6,XXREAL_0:1;
    suppose
A7:   m < n;
      then
A8:   W.n = W.last() by A2,A4,A5;
      W.m = W.first() by A2,A4,A5,A7;
      hence contradiction by A1,A5,A8;
    end;
    suppose
A9:   m > n;
      then
A10:  W.n = W.first() by A2,A3,A5;
      W.m = W.last() by A2,A3,A5,A9;
      hence contradiction by A1,A5,A10;
    end;
  end;
  hence thesis;
end;
