let G be _Graph; :: thesis: for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
(len (W .remove m,n)) + n = (len W) + m
let W be Walk of G; :: thesis: for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
(len (W .remove m,n)) + n = (len W) + m
let m, n be odd Element of NAT ; :: thesis: ( m <= n & n <= len W & W . m = W . n implies (len (W .remove m,n)) + n = (len W) + m )
set W1 = W .cut 1,m;
set W2 = W .cut n,(len W);
assume that
A1: m <= n and
A2: n <= len W and
A3: W . m = W . n ; :: thesis: (len (W .remove m,n)) + n = (len W) + m
A4: W .remove m,n = (W .cut 1,m) .append (W .cut n,(len W)) by A1, A2, A3, Def12;
A5: (len (W .cut n,(len W))) + n = (len W) + 1 by A2, Lm15;
A6: W . n = (W .cut n,(len W)) .first() by A2, Lm16;
A7: 1 <= m by HEYTING3:1;
A8: m <= len W by A1, A2, XXREAL_0:2;
then A9: (len (W .cut 1,m)) + 1 = m + 1 by A7, Lm15, JORDAN12:3;
(W .cut 1,m) .last() = W . n by A3, A7, A8, Lm16, JORDAN12:3;
then (len ((W .cut 1,m) .append (W .cut n,(len W)))) + 1 = m + (((len W) + 1) + (- n)) by A6, A9, A5, Lm9
.= (((len W) + m) + (- n)) + 1 ;
hence (len (W .remove m,n)) + n = (len W) + m by A4; :: thesis: verum