
theorem Th61:
  for G2 being _Graph, v,e,w being object, G1 being addAdjVertex of G2,v,e,w
  for W1 being Walk of G1, W2 being Walk of G2
  st W1 = W2 & W2 is minlength holds W1 is minlength
proof
  let G2 be _Graph, v,e,w be object, G1 be addAdjVertex of G2,v,e,w;
  let W1 be Walk of G1, W2 be Walk of G2;
  assume A1: W1 = W2 & W2 is minlength;
  then W1.first() = W2.first() & W1.last() = W2.last();
  then A2: W1.first() in the_Vertices_of G2 & W1.last() in the_Vertices_of G2;
  per cases;
  suppose A3: not e in the_Edges_of G2 & not v in the_Vertices_of G2 &
      w in the_Vertices_of G2;
    now
      let W3 be Walk of G1;
      assume A4: W3 is_Walk_from W1.first(),W1.last();
      set T = the Trail of W3;
      A5: T is_Walk_from W1.first(),W1.last() by A4, GLIB_001:160;
      then A6: T.first()=W1.first() & T.last()=W1.last() by GLIB_001:def 23;
      then A7: not e in T.edges() by A2, A3, GLIB_006:147;
      T is Walk of G2
      proof
        per cases;
        suppose T is trivial;
          then consider v0 being Vertex of G1 such that
            A8: T = G1.walkOf(v0) by GLIB_001:128;
          A9: v0 in the_Vertices_of G2 by A2, A6, A8;
          T.vertices() = {v0} by A8, GLIB_001:90;
          then not v in T.vertices() by A3, A9, TARSKI:def 1;
          hence thesis by A3, GLIB_006:146;
        end;
        suppose T is non trivial;
          hence thesis by A3, A7, GLIB_006:146;
        end;
      end;
      then reconsider W4 = T as Walk of G2;
      W4 is_Walk_from W1.first(),W1.last() by A5, GLIB_001:19;
      then W4 is_Walk_from W2.first(),W1.last() by A1;
      then W4 is_Walk_from W2.first(),W2.last() by A1;
      then A10: len T >= len W1 by A1, CHORD:def 2;
      len W3 >= len T by GLIB_001:162;
      hence len W3 >= len W1 by A10, XXREAL_0:2;
    end;
    hence thesis by CHORD:def 2;
  end;
  suppose A11: not e in the_Edges_of G2 & not w in the_Vertices_of G2 &
      v in the_Vertices_of G2;
    now
      let W3 be Walk of G1;
      assume A12: W3 is_Walk_from W1.first(),W1.last();
      set T = the Trail of W3;
      A13: T is_Walk_from W1.first(),W1.last() by A12, GLIB_001:160;
      then A14: T.first()=W1.first() & T.last()=W1.last() by GLIB_001:def 23;
      then A15: not e in T.edges() by A2, A11, GLIB_006:147;
      T is Walk of G2
      proof
        per cases;
        suppose T is trivial;
          then consider v0 being Vertex of G1 such that
            A16: T = G1.walkOf(v0) by GLIB_001:128;
          A17: v0 in the_Vertices_of G2 by A2, A14, A16;
          T.vertices() = {v0} by A16, GLIB_001:90;
          then not w in T.vertices() by A11, A17, TARSKI:def 1;
          hence thesis by A11, GLIB_006:145;
        end;
        suppose T is non trivial;
          hence thesis by A11, A15, GLIB_006:145;
        end;
      end;
      then reconsider W4 = T as Walk of G2;
      W4 is_Walk_from W1.first(),W1.last() by A13, GLIB_001:19;
      then W4 is_Walk_from W2.first(),W1.last() by A1;
      then W4 is_Walk_from W2.first(),W2.last() by A1;
      then A18: len T >= len W1 by A1, CHORD:def 2;
      len W3 >= len T by GLIB_001:162;
      hence len W3 >= len W1 by A18, XXREAL_0:2;
    end;
    hence thesis by CHORD:def 2;
  end;
  suppose not((not e in the_Edges_of G2 & not v in the_Vertices_of G2 &
      w in the_Vertices_of G2) or (not e in the_Edges_of G2 &
      not w in the_Vertices_of G2 & v in the_Vertices_of G2));
    then G1 == G2 by GLIB_006:def 12;
    hence thesis by A1, Th49;
  end;
end;
