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 Th130:
 for e,x being object holds
  e Joins W.last(),x,G implies W.addEdge(e) is non trivial
proof let e,x be object;
  assume e Joins W.last(), x, G;
  then
A1: len W.addEdge(e) = len W + 2 by Lm37;
  1 + 0 < len W + 2 by XREAL_1:8;
  hence thesis by A1,Lm55;
end;
