now :: thesis: W .remove (m,n) is directed

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

suppose ( m is odd & n is odd & m <= n & n <= len W & W . m = W . n ) ; :: thesis: W .remove (m,n) is directed

then W .remove (m,n) = (W .cut (1,m)) .append (W .cut (n,(len W))) by Def12;
hence W .remove (m,n) is directed ; :: thesis: verum

end;

suppose ( not m is odd or not n is odd or not m <= n or not n <= len W or not W . m = W . n ) ; :: thesis: W .remove (m,n) is directed

hence W .remove (m,n) is directed by Def12; :: thesis: verum

end;

end;

end;

hence W .remove (m,n) is directed ; :: thesis: verum