
theorem Th14:
  for G being WGraph, x,y,e being set st e Joins x,y,G holds G
  .walkOf(x,e,y).weightSeq() = <* (the_Weight_of G).e *>
proof
  let G be WGraph, x,y,e be set;
  set W = G.walkOf(x,e,y);
  assume e Joins x,y,G;
  then
A1: W.edgeSeq() = <*e*> by GLIB_001:83;
  then len W.edgeSeq() = 1 by FINSEQ_1:39;
  then
A2: len W.weightSeq() = 1 by Def18;
A3: now
    let n be Nat;
    assume that
A4: 1 <= n and
A5: n <= len W.weightSeq();
A6: n = 1 by A2,A4,A5,XXREAL_0:1;
    hence W.weightSeq().n = (the_Weight_of G).(<*e*>.1) by A1,A5,Def18
      .= (the_Weight_of G).e
      .= <* (the_Weight_of G).e *>.n by A6;
  end;
  len W.weightSeq() = len <* (the_Weight_of G).e *> by A2,FINSEQ_1:39;
  hence thesis by A3,FINSEQ_1:14;
end;
