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 Th84:
  for W1 being Walk of G1, W2 being Walk of G2 st W1 = W2 holds W1
  .edgeSeq() = W2.edgeSeq()
proof
  let W1 be Walk of G1, W2 be Walk of G2;
  set ES1 = W1.edgeSeq(), ES2 = W2.edgeSeq();
  assume
A1: W1 = W2;
  now
    thus len ES1 = len ES1;
A2: 2 * len ES1 + 1 = len W2 by A1,Def15
      .= 2 * len ES2 + 1 by Def15;
    hence len ES2 = len ES1;
    let x be Nat;
    assume
A3: x in dom ES1;
    then
A4: x <= len ES2 by A2,FINSEQ_3:25;
A5: 1 <= x by A3,FINSEQ_3:25;
    x <= len ES1 by A3,FINSEQ_3:25;
    hence ES1.x = W2.(2*x) by A1,A5,Def15
      .= ES2.x by A5,A4,Def15;
  end;
  hence thesis by FINSEQ_2:9;
end;
