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 holds
( W .cut (m,n) is V5() iff m = n )
let W be Walk of G; :: thesis: for m, n being odd Element of NAT st m <= n & n <= len W holds
( W .cut (m,n) is V5() iff m = n )
let m, n be odd Element of NAT ; :: thesis: ( m <= n & n <= len W implies ( W .cut (m,n) is V5() iff m = n ) )
assume that
A1: m <= n and
A2: n <= len W ; :: thesis: ( W .cut (m,n) is V5() iff m = n )
A3: (len (W .cut (m,n))) + m = n + 1 by A1, A2, Lm15;
hereby :: thesis: ( m = n implies W .cut (m,n) is V5() )
assume W .cut (m,n) is V5() ; :: thesis: m = n
then 1 = (n - m) + 1 by A3, Lm55;
hence m = n ; :: thesis: verum
end;
assume m = n ; :: thesis: W .cut (m,n) is V5()
hence W .cut (m,n) is V5() by A3, Lm55; :: thesis: verum