
theorem Th13:
  for G being _finite natural-weighted WGraph, EL being
FF:ELabeling of G, W being Walk of G st W is non trivial & W is_augmenting_wrt
  EL holds 0 < W.tolerance(EL)
proof
  let G be _finite natural-weighted WGraph, EL being FF:ELabeling of G, W be
  Walk of G such that
A1: W is non trivial and
A2: W is_augmenting_wrt EL;
  set T = W.tolerance(EL);
  T in rng (W.flowSeq(EL)) by A1,A2,Def16;
  then consider n being Nat such that
A3: n in dom (W.flowSeq(EL)) and
A4: T = (W.flowSeq(EL)).n by FINSEQ_2:10;
  reconsider n as Element of NAT by ORDINAL1:def 12;
  dom (W.flowSeq(EL)) = dom W.edgeSeq() by A2,Def15;
  then
A5: 2*n in dom W by A3,GLIB_001:78;
  then 1 <= 2*n by FINSEQ_3:25;
  then reconsider 2n1 = 2*n-1 as odd Element of NAT by INT_1:5;
  2*n <= len W by A5,FINSEQ_3:25;
  then
A6: 2*n-1 < len W - 0 by XREAL_1:15;
  set v1 = W.(2n1), e = W.(2*n), v2 = W.(2*n+1);
A7: 2*n-1 + 2 = 2*n+1;
A8: 2*n-1 + 1 = 2*n;
  now
    per cases;
    suppose
A9:   e DJoins v1,v2,G;
      then
A10:  T = (the_Weight_of G).e - EL.e by A2,A3,A4,Def15;
      EL.e < (the_Weight_of G).e by A2,A6,A8,A7,A9;
      then EL.e - EL.e < T by A10,XREAL_1:14;
      hence thesis;
    end;
    suppose
A11:  not e DJoins v1,v2,G;
      then T = EL.e by A2,A3,A4,Def15;
      hence thesis by A2,A6,A8,A7,A11;
    end;
  end;
  hence thesis;
end;
