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
(W .cut (1,m)) .last() = (W .cut (n,(len W))) .first()
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
(W .cut (1,m)) .last() = (W .cut (n,(len W))) .first()
let m, n be odd Element of NAT ; :: thesis: ( m <= n & n <= len W & W . m = W . n implies (W .cut (1,m)) .last() = (W .cut (n,(len W))) .first() )
assume that
A1: m <= n and
A2: n <= len W and
A3: W . m = W . n ; :: thesis: (W .cut (1,m)) .last() = (W .cut (n,(len W))) .first()
A4: 1 <= m by ABIAN:12;
m <= len W by A1, A2, XXREAL_0:2;
hence (W .cut (1,m)) .last() = W . n by A3, A4, Lm16, JORDAN12:2
.= (W .cut (n,(len W))) .first() by A2, Lm16 ;
:: thesis: verum