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 EdgeSeq of G1 implies x is EdgeSeq of G2
proof
  assume that
A1: G1 == G2 and
A2: x is EdgeSeq of G1;
  reconsider es = x as EdgeSeq of G1 by A2;
  reconsider es2 = es as FinSequence of the_Edges_of G2 by A1,GLIB_000:def 34;
  consider vs being FinSequence of the_Vertices_of G1 such that
A3: len vs = len es + 1 and
A4: for n being Element of NAT st 1 <= n & n <= len es holds es.n Joins
  vs.n,vs.(n+1),G1 by Def2;
  now
    reconsider vs as FinSequence of the_Vertices_of G2 by A1,GLIB_000:def 34;
    take vs;
    thus len vs = len es + 1 by A3;
    let n be Element of NAT;
    assume that
A5: 1 <= n and
A6: n <= len es2;
    es2.n Joins vs.n,vs.(n+1),G1 by A4,A5,A6;
    hence es2.n Joins vs.n,vs.(n+1),G2 by A1,GLIB_000:88;
  end;
  hence thesis by Def2;
end;
