
theorem Th2:
  for G being real-weighted WGraph, EL be FF:ELabeling of G, W
  being Walk of G, m,n be Nat st W is_augmenting_wrt EL holds W.cut(m,n)
  is_augmenting_wrt EL
proof
  let G be real-weighted WGraph, EL be FF:ELabeling of G, W being Walk of G, m
  ,n be Nat;
  set W2 = W.cut(m,n);
  assume
A1: W is_augmenting_wrt EL;
  now
    per cases;
    suppose
A2:   m is odd & n is odd & m <= n & n <= len W;
      then reconsider m9=m, n9 = n as odd Element of NAT by ORDINAL1:def 12;
      now
        let x be odd Nat;
        reconsider x9 = x as Element of NAT by ORDINAL1:def 12;
        set v1b = W2.x, eb = W2.(x+1), v2b = W2.(x+2);
        assume
A3:     x < len W2;
        then
A4:     x9 in dom W2 by GLIB_001:12;
A5:     m9 <= n9 by A2;
A6:     x9+2 in dom W2 by A3,GLIB_001:12;
        then
A7:     W2.(x9+2) = W.(m9+(x9+2)-1) by A2,A5,GLIB_001:47;
        x9+1 in dom W2 by A3,GLIB_001:12;
        then
A8:     W2.(x9+1) = W.(m9+(x9+1)-1) by A2,A5,GLIB_001:47;
        (m9+x9-1) in dom W by A2,A4,A5,GLIB_001:47;
        then reconsider
        a = m9+x-1,a2=m+(x+2)-1 as Element of NAT by A8;
        reconsider a as odd Element of NAT;
        set v1a = W.a, ea = W.(a+1), v2a = W.(a+2);
        (m9+(x9+2)-1) in dom W by A2,A6,A5,GLIB_001:47;
        then a2 <= len W by FINSEQ_3:25;
        then
A9:     m+(x+2)-1-2 < len W - 0 by XREAL_1:15;
        hereby
          assume eb DJoins v1b,v2b,G;
          then ea DJoins v1a,v2a,G by A2,A4,A5,A8,A7,GLIB_001:47;
          hence EL.eb < (the_Weight_of G).eb by A1,A8,A9;
        end;
        assume not eb DJoins v1b,v2b,G;
        then not ea DJoins v1a,v2a,G by A2,A4,A5,A8,A7,GLIB_001:47;
        hence 0 < EL.eb by A1,A8,A9;
      end;
      hence thesis;
    end;
    suppose
      not (m is odd & n is odd & m <= n & n <= len W);
      hence thesis by A1,GLIB_001:def 11;
    end;
  end;
  hence thesis;
end;
