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 Th74:
  for W1 being Walk of G1, W2 being Walk of G2 st W1 = W2 holds W1
  .vertexSeq() = W2.vertexSeq()
proof
  let W1 be Walk of G1, W2 be Walk of G2;
  set VS1 = W1.vertexSeq(), VS2 = W2.vertexSeq();
  assume
A1: W1 = W2;
  now
    thus len VS1 = len VS1;
A2: 2 * len VS1 = len W2 + 1 by A1,Def14
      .= 2 * len VS2 by Def14;
    hence len VS2 = len VS1;
    let x be Nat;
    assume
A3: x in dom VS1;
    then
A4: x <= len VS2 by A2,FINSEQ_3:25;
A5: 1 <= x by A3,FINSEQ_3:25;
    x <= len VS1 by A3,FINSEQ_3:25;
    hence VS1.x = W2.(2*x - 1) by A1,A5,Def14
      .= VS2.x by A5,A4,Def14;
  end;
  hence thesis by FINSEQ_2:9;
end;
