let G be _Graph; :: thesis: for W being Walk of G
for m, n being Element of NAT st W is Path-like holds
W .cut (m,n) is Path-like
let W be Walk of G; :: thesis: for m, n being Element of NAT st W is Path-like holds
W .cut (m,n) is Path-like
let m, n be Element of NAT ; :: thesis: ( W is Path-like implies W .cut (m,n) is Path-like )
assume A1: W is Path-like ; :: thesis: W .cut (m,n) is Path-like

now :: thesis: W .cut (m,n) is Path-like

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 A2: ( m is odd & n is odd & m <= n & n <= len W ) ; :: thesis: W .cut (m,n) is Path-like

then reconsider m9 = m as odd Element of NAT ;

now :: thesis: ( W .cut (m,n) is Trail-like & ( for x, y being odd Element of NAT st x < y & y <= len (W .cut (m,n)) & (W .cut (m,n)) . x = (W .cut (m,n)) . y holds
( x = 1 & y = len (W .cut (m,n)) ) ) )

W is Trail-like by A1;
hence W .cut (m,n) is Trail-like by Lm59; :: thesis: for x, y being odd Element of NAT st x < y & y <= len (W .cut (m,n)) & (W .cut (m,n)) . x = (W .cut (m,n)) . y holds
( x = 1 & y = len (W .cut (m,n)) )
let x, y be odd Element of NAT ; :: thesis: ( x < y & y <= len (W .cut (m,n)) & (W .cut (m,n)) . x = (W .cut (m,n)) . y implies ( x = 1 & y = len (W .cut (m,n)) ) )
assume that
A3: x < y and
A4: y <= len (W .cut (m,n)) and
A5: (W .cut (m,n)) . x = (W .cut (m,n)) . y ; :: thesis: ( x = 1 & y = len (W .cut (m,n)) )
reconsider xaa1 = x - 1 as even Element of NAT by ABIAN:12, INT_1:5;
reconsider yaa1 = y - 1 as even Element of NAT by ABIAN:12, INT_1:5;
x - 1 < y - 1 by A3, XREAL_1:14;
then A6: xaa1 + m < yaa1 + m by XREAL_1:8;
x <= len (W .cut (m,n)) by A3, A4, XXREAL_0:2;
then x - 1 < (len (W .cut (m,n))) - 0 by XREAL_1:15;
then A7: (W .cut (m,n)) . (xaa1 + 1) = W . (m + xaa1) by A2, Lm15;
A8: y - 1 < (len (W .cut (m,n))) - 0 by A4, XREAL_1:15;
then A9: (W .cut (m,n)) . (yaa1 + 1) = W . (m + yaa1) by A2, Lm15;
m + yaa1 in dom W by A2, A8, Lm15;
then A10: m9 + yaa1 <= len W by FINSEQ_3:25;
then A11: m9 + yaa1 = len W by A1, A5, A7, A9, A6;

A12: now :: thesis: not xaa1 <> 0

assume A13: xaa1 <> 0 ; :: thesis: contradiction
m >= 1 by A2, ABIAN:12;
then 1 + 0 < m + xaa1 by A13, XREAL_1:8;
hence contradiction by A1, A5, A7, A9, A6, A10; :: thesis: verum

end;

then (m + 1) - 1 = 1 by A1, A5, A7, A9, A6, A10;
then A14: (len (W .cut (m,n))) + 1 = n + 1 by A2, Lm15;
thus x = 1 by A12; :: thesis: y = len (W .cut (m,n))
m9 + xaa1 = 1 by A1, A5, A7, A9, A6, A10;
hence y = len (W .cut (m,n)) by A2, A4, A11, A12, A14, XXREAL_0:1; :: thesis: verum

end;

hence W .cut (m,n) is Path-like ; :: 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) is Path-like

hence W .cut (m,n) is Path-like by A1, Def11; :: thesis: verum

end;

end;

end;

hence W .cut (m,n) is Path-like ; :: thesis: verum