reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem
  G1 == G2 & x is VertexSeq of G1 implies x is VertexSeq of G2
proof
  assume that
A1: G1 == G2 and
A2: x is VertexSeq of G1;
  reconsider x2 = x as FinSequence of the_Vertices_of G2 by A1,A2,
GLIB_000:def 34;
  now
    let n be Element of NAT;
    assume that
A3: 1 <= n and
A4: n < len x2;
    consider e being set such that
A5: e Joins x2.n, x2.(n+1), G1 by A2,A3,A4,Def1;
    e Joins x2.n, x2.(n+1), G2 by A1,A5,GLIB_000:88;
    hence ex e being set st e Joins x2.n, x2.(n+1), G2;
  end;
  hence thesis by Def1;
end;
