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 Th14:
 for e,x,y being object holds
  e Joins x,y,G implies G.walkOf(x,e,y).first() = x & G.walkOf(x,e
  ,y).last() = y & G.walkOf(x,e,y) is_Walk_from x,y
proof let e,x,y be object;
  set W = G.walkOf(x,e,y);
  assume e Joins x,y,G;
  then
A1: W = <*x,e,y*> by Def5;
  hence
A2: W.first() = x;
  len W = 3 by A1,FINSEQ_1:45;
  hence W.last() = y by A1;
  hence thesis by A2;
end;
