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 Th106:
 for e,x,y being object holds
  e Joins x,y,G iff G.walkOf(x,e,y).edges() = {e}
proof let e,x,y be object;
  set W = G.walkOf(x,e,y);
  hereby
    assume e Joins x,y,G;
    then W.edgeSeq() = <*e*> by Th81;
    hence W.edges() = {e} by FINSEQ_1:39;
  end;
  assume W.edges() = {e};
  then e in W.edges() by TARSKI:def 1;
  then consider n being even Element of NAT such that
A1: 1 <= n and
A2: n <= len W and
  W.n = e by Lm46;
A3: 2*0+1 < n by A1,XXREAL_0:1;
  now
    assume not e Joins x,y,G;
    then W = G.walkOf(the Element of the_Vertices_of G) by Def5;
    hence contradiction by A2,A3,Th12;
  end;
  hence thesis;
end;
