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 Th7:
  for n being even Element of NAT st n in dom W holds W.n in the_Edges_of G
proof
  let n be even Element of NAT;
  assume
A1: n in dom W;
  then 1 <= n by FINSEQ_3:25;
  then reconsider naa1 = n-1 as odd Element of NAT by INT_1:5;
  n <= len W by A1,FINSEQ_3:25;
  then naa1 < len W - 0 by XREAL_1:15;
  then W.(naa1+1) Joins W.naa1, W.(naa1+2), G by Def3;
  hence thesis by GLIB_000:def 13;
end;
