let G be _Graph; :: thesis: for W being Walk of G st len W <= 3 holds
W is Path-like
let W be Walk of G; :: thesis: ( len W <= 3 implies W is Path-like )
assume A1: len W <= 3 ; :: thesis: W is Path-like
now :: thesis: W is Path-like
per cases ( len W = 1 or len W <> 1 ) ;
suppose A2: len W <> 1 ; :: thesis: W is Path-like
1 <= len W by ABIAN:12;
then 1 < len W by A2, XXREAL_0:1;
then 1 + 2 <= len W by Th1, JORDAN12:2;
then A3: len W = 3 by A1, XXREAL_0:1;
A4: now :: thesis: for m, n being odd Element of NAT st m < n & n <= len W & W . m = W . n holds
( m = 1 & n = len W )
let m, n be odd Element of NAT ; :: thesis: ( m < n & n <= len W & W . m = W . n implies ( m = 1 & n = len W ) )
assume that
A5: m < n and
A6: n <= len W and
W . m = W . n ; :: thesis: ( m = 1 & n = len W )
A7: 1 <= m by ABIAN:12;
m < (2 * 1) + 1 by A3, A5, A6, XXREAL_0:2;
then (m + 2) - 2 <= 3 - 2 by Th1;
hence m = 1 by A7, XXREAL_0:1; :: thesis: n = len W
(2 * 0) + 1 < n by A5, A7, XXREAL_0:2;
then 1 + 2 <= n by Th1;
hence n = len W by A3, A6, XXREAL_0:1; :: thesis: verum
end;
W is Trail-like by A1, Lm61;
hence W is Path-like by A4; :: thesis: verum
end;
end;
end;
hence W is Path-like ; :: thesis: verum