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 Th81:
 for e,x,y being object holds
  e Joins x,y,G iff G.walkOf(x,e,y).edgeSeq() = <*e*>
proof let e,x,y be object;
  set W = G.walkOf(x,e,y);
  hereby
    assume
A1: e Joins x,y, G;
    then len W = 3 by Th13;
    then
A2: 2+1 = 2*len W.edgeSeq()+1 by Def15;
A3: W = <*x,e,y*> by A1,Def5;
A4: now
      let k be Nat;
      assume that
A5:   1 <= k and
A6:   k <= len W.edgeSeq();
A7:   k = 1 by A2,A5,A6,XXREAL_0:1;
      then W.edgeSeq().k = W.(2*1) by A6,Def15
        .= e by A3;
      hence W.edgeSeq().k = <*e*>.k by A7;
    end;
    len W.edgeSeq() = len <*e*> by A2,FINSEQ_1:39;
    hence W.edgeSeq() = <*e*> by A4,FINSEQ_1:14;
  end;
  assume W.edgeSeq() = <*e*>;
  then len W.edgeSeq() = 1 by FINSEQ_1:39;
  then
A8: len W = 2*1+1 by Def15;
  now
    assume not e Joins x,y,G;
    then W = G.walkOf(the Element of the_Vertices_of G) by Def5;
    hence contradiction by A8,Th12;
  end;
  hence thesis;
end;
