let G1, G2 be _Graph; :: thesis: for W1 being Walk of G1

for W2 being Walk of G2

for m, n being Element of NAT st W1 = W2 holds

W1 .cut (m,n) = W2 .cut (m,n)

let W1 be Walk of G1; :: thesis: for W2 being Walk of G2

for m, n being Element of NAT st W1 = W2 holds

W1 .cut (m,n) = W2 .cut (m,n)

let W2 be Walk of G2; :: thesis: for m, n being Element of NAT st W1 = W2 holds

W1 .cut (m,n) = W2 .cut (m,n)

let m, n be Element of NAT ; :: thesis: ( W1 = W2 implies W1 .cut (m,n) = W2 .cut (m,n) )

assume A1: W1 = W2 ; :: thesis: W1 .cut (m,n) = W2 .cut (m,n)

for W2 being Walk of G2

for m, n being Element of NAT st W1 = W2 holds

W1 .cut (m,n) = W2 .cut (m,n)

let W1 be Walk of G1; :: thesis: for W2 being Walk of G2

for m, n being Element of NAT st W1 = W2 holds

W1 .cut (m,n) = W2 .cut (m,n)

let W2 be Walk of G2; :: thesis: for m, n being Element of NAT st W1 = W2 holds

W1 .cut (m,n) = W2 .cut (m,n)

let m, n be Element of NAT ; :: thesis: ( W1 = W2 implies W1 .cut (m,n) = W2 .cut (m,n) )

assume A1: W1 = W2 ; :: thesis: W1 .cut (m,n) = W2 .cut (m,n)

now :: thesis: W1 .cut (m,n) = W2 .cut (m,n)end;

hence
W1 .cut (m,n) = W2 .cut (m,n)
; :: thesis: verumper cases
( ( m is odd & n is odd & m <= n & n <= len W1 ) or not m is odd or not n is odd or not m <= n or not n <= len W1 )
;

end;