let G be _Graph; :: thesis: for W being Walk of G st len W <= 3 holds
W is Trail-like
let W be Walk of G; :: thesis: ( len W <= 3 implies W is Trail-like )
assume A1: len W <= 3 ; :: thesis: W is Trail-like

now

per cases ( len W = 1 or len W <> 1 ) ;

suppose len W = 1 ; :: thesis: W is Trail-like

then W is trivial by Lm55;
then consider v being Vertex of G such that
A2: W = G .walkOf v by Lm56;
thus W is Trail-like by A2, Lm4; :: thesis: verum

end;

suppose A3: len W <> 1 ; :: thesis: W is Trail-like

1 <= len W by HEYTING3:1;
then 1 < len W by A3, XXREAL_0:1;
then 1 + 2 <= len W by Th1, JORDAN12:3;
then A4: len W = 3 by A1, XXREAL_0:1;

now

let m, n be even Element of NAT ; :: thesis: ( 1 <= m & m < n & n <= len W implies W . m <> W . n )
assume A5: ( 1 <= m & m < n & n <= len W ) ; :: thesis: W . m <> W . n
(2 * 0 ) + 1 < m by A5, XXREAL_0:1;
then A6: 1 + 1 <= m by NAT_1:13;
n < 2 + 1 by A4, A5, XXREAL_0:1;
then n <= 2 by NAT_1:13;
hence W . m <> W . n by A5, A6, XXREAL_0:2; :: thesis: verum

end;

hence W is Trail-like by Lm57; :: thesis: verum

end;

end;

end;

hence W is Trail-like ; :: thesis: verum