
theorem Th47:
  for G1 being _Graph, G2 being Subgraph of G1
  for W1 being Walk of G1, W2 being Walk of G2
  st W1 = W2 & W1 is minlength holds W2 is minlength
proof
  let G1 be _Graph, G2 be Subgraph of G1, W1 be Walk of G1, W2 be Walk of G2;
  assume A1: W1 = W2 & W1 is minlength;
  now
    given W9 being Walk of G2 such that
      A2: W9 is_Walk_from W2.first(),W2.last() & len W9 < len W2;
    reconsider W8 = W9 as Walk of G1 by GLIB_001:167;
    W8 is_Walk_from W2.first(),W2.last() by A2, GLIB_001:19;
    then W8 is_Walk_from W1.first(),W2.last() by A1;
    then W8 is_Walk_from W1.first(),W1.last() by A1;
    hence contradiction by A1, A2, CHORD:def 2;
  end;
  hence thesis by CHORD:def 2;
end;
