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

now

per cases ( ( not m is even & not n is even & m <= n & n <= len W ) or m is even or n is even or not m <= n or not n <= len W ) ;

suppose A1: ( not m is even & not n is even & m <= n & n <= len W ) ; :: thesis: (W .cut (m,n)) .edges() c= W .edges()

then reconsider m9 = m as odd Element of NAT ;

now

let e be set ; :: thesis: ( e in (W .cut (m,n)) .edges() implies e in W .edges() )
assume e in (W .cut (m,n)) .edges() ; :: thesis: e in W .edges()
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 A2, INT_1:5;
A5: xaa1 < (len (W .cut (m,n))) - 0 by A3, XREAL_1:15;
then A6: m + xaa1 in dom W by A1, Lm15;
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 A6, FINSEQ_3:25;
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 ( m is even or n is even 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