let G be _Graph; :: thesis: for W being Walk of G
for n, m being Element of NAT holds (W .cut (m,n)) .edges() c= W .edges()

let W be Walk of G; :: thesis: for n, m being Element of NAT holds (W .cut (m,n)) .edges() c= W .edges()
let n, m be Element of NAT ; :: thesis: (W .cut (m,n)) .edges() c= W .edges()
now :: thesis: (W .cut (m,n)) .edges() c= W .edges()
per cases ( ( m is odd & n is odd & m <= n & n <= len W ) or not m is odd or not n is odd or not m <= n or not n <= len W ) ;
suppose A1: ( m is odd & n is odd & m <= n & n <= len W ) ; :: thesis: (W .cut (m,n)) .edges() c= W .edges()
then reconsider m9 = m as odd Element of NAT ;
now :: thesis: for e being object st e in (W .cut (m,n)) .edges() holds
e in W .edges()
let e be object ; :: thesis: ( e in (W .cut (m,n)) .edges() implies e in W .edges() )
assume e in (W .cut (m,n)) .edges() ; :: thesis:
then consider x being even Element of NAT such that
A2: 1 <= x and
A3: x <= len (W .cut (m,n)) and
A4: (W .cut (m,n)) . x = e by Lm46;
reconsider xaa1 = x - 1 as odd Element of NAT by ;
A5: xaa1 < (len (W .cut (m,n))) - 0 by ;
then A6: m + xaa1 in dom W by ;
then A7: m9 + xaa1 <= len W by FINSEQ_3:25;
xaa1 + 1 = x ;
then A8: e = W . (m + xaa1) by A1, A4, A5, Lm15;
1 <= m9 + xaa1 by ;
hence e in W .edges() by A8, A7, Lm46; :: thesis: verum
end;
hence (W .cut (m,n)) .edges() c= W .edges() by TARSKI:def 3; :: thesis: verum
end;
suppose ( not m is odd or not n is odd or not m <= n or not n <= len W ) ; :: thesis: (W .cut (m,n)) .edges() c= W .edges()
hence (W .cut (m,n)) .edges() c= W .edges() by Def11; :: thesis: verum
end;
end;
end;
hence (W .cut (m,n)) .edges() c= W .edges() ; :: thesis: verum