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 Walk of G1 implies x is Walk of G2
proof
  assume that
A1: G1 == G2 and
A2: x is Walk of G1;
A3: the_Vertices_of G1 = the_Vertices_of G2 by A1,GLIB_000:def 34;
  then reconsider
  W = x as FinSequence of the_Vertices_of G2 \/ the_Edges_of G2 by A1,A2,
GLIB_000:def 34;
A4: now
    let n be odd Element of NAT;
    assume n < len W;
    then W.(n+1) Joins W.n, W.(n+2), G1 by A2,Def3;
    hence W.(n+1) Joins W.n, W.(n+2), G2 by A1,GLIB_000:88;
  end;
  W.1 in the_Vertices_of G2 by A2,A3,Def3;
  hence thesis by A2,A4,Def3;
end;
