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 Th102:
  e in W.edges() iff ex n being Element of NAT st n in dom W
  .edgeSeq() & W.edgeSeq().n = e
proof
  hereby
    assume e in W.edges();
    then consider n being object such that
A1: n in dom W.edgeSeq() and
A2: W.edgeSeq().n = e by FUNCT_1:def 3;
    reconsider n as Element of NAT by A1;
    take n;
    thus n in dom W.edgeSeq() & W.edgeSeq().n = e by A1,A2;
  end;
  given n being Element of NAT such that
A3: n in dom W.edgeSeq() and
A4: W.edgeSeq().n = e;
  thus thesis by A3,A4,FUNCT_1:def 3;
end;
