set W2 = W .cut (m,n);

now :: thesis: W .cut (m,n) is vertex-distinct

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) is vertex-distinct

then reconsider m9 = m as odd Element of NAT ;

now :: thesis: for a, b being odd Element of NAT st a <= len (W .cut (m,n)) & b <= len (W .cut (m,n)) & (W .cut (m,n)) . a = (W .cut (m,n)) . b holds
a = b

let a, b be odd Element of NAT ; :: thesis: ( a <= len (W .cut (m,n)) & b <= len (W .cut (m,n)) & (W .cut (m,n)) . a = (W .cut (m,n)) . b implies a = b )
assume that
A2: a <= len (W .cut (m,n)) and
A3: b <= len (W .cut (m,n)) and
A4: (W .cut (m,n)) . a = (W .cut (m,n)) . b ; :: thesis: a = b
reconsider aaa1 = a - 1, baa1 = b - 1 as even Element of NAT by ABIAN:12, INT_1:5;
A5: baa1 < (len (W .cut (m,n))) - 0 by A3, XREAL_1:15;
then A6: (W .cut (m,n)) . (baa1 + 1) = W . (m + baa1) by A1, Lm15;
A7: aaa1 < (len (W .cut (m,n))) - 0 by A2, XREAL_1:15;
then m + aaa1 in dom W by A1, Lm15;
then A8: m9 + aaa1 <= len W by FINSEQ_3:25;
m + baa1 in dom W by A1, A5, Lm15;
then A9: m9 + baa1 <= len W by FINSEQ_3:25;
(W .cut (m,n)) . (aaa1 + 1) = W . (m + aaa1) by A1, A7, Lm15;
then aaa1 + m9 = baa1 + m9 by A4, A6, A8, A9, Def29;
hence a = b ; :: thesis: verum

end;

hence W .cut (m,n) is vertex-distinct ; :: 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 vertex-distinct

hence W .cut (m,n) is vertex-distinct by Def11; :: thesis: verum

end;

end;

end;

hence W .cut (m,n) is vertex-distinct ; :: thesis: verum