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 Th71:
  n in dom W.vertexSeq() iff 2*n-1 in dom W
proof
  hereby
    assume
A1: n in dom W.vertexSeq();
    then
A2: 1 <= n by FINSEQ_3:25;
    then 1 <= n+n by NAT_1:12;
    then
A3: 2*n-1 is Element of NAT by INT_1:5;
    n <= len W.vertexSeq() by A1,FINSEQ_3:25;
    then 2*n <= 2*len W.vertexSeq() by XREAL_1:64;
    then 2*n <= len W + 1 by Def14;
    then
A4: 2*n-1 <= len W + 1 - 1 by XREAL_1:13;
    2*1 <= 2*n by A2,XREAL_1:64;
    then 2-1 <= 2*n-1 by XREAL_1:13;
    hence 2*n-1 in dom W by A4,A3,FINSEQ_3:25;
  end;
  assume
A5: 2*n-1 in dom W;
  then reconsider 2naa1=2*n-1 as Element of NAT;
  1 <= 2naa1 by A5,FINSEQ_3:25;
  then 1+1 <= 2*n-1+1 by XREAL_1:7;
  then 2*1 <= 2*n;
  then
A6: 1 <= n by XREAL_1:68;
  2naa1 <= len W by A5,FINSEQ_3:25;
  then 2*n-1+1 <= len W+1 by XREAL_1:7;
  then 2*n <= 2 * len W.vertexSeq() by Def14;
  then n <= len W.vertexSeq() by XREAL_1:68;
  hence thesis by A6,FINSEQ_3:25;
end;
