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

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 :: thesis: W is Trail-like end;

hence
W is Trail-like
; :: thesis: verumper cases
( len W = 1 or len W <> 1 )
;

end;

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

then
W is trivial
by Lm55;

then ex v being Vertex of G st W = G .walkOf v by Lm56;

hence W is Trail-like by Lm4; :: thesis: verum

end;then ex v being Vertex of G st W = G .walkOf v by Lm56;

hence W is Trail-like by Lm4; :: thesis: verum

suppose A2:
len W <> 1
; :: thesis: W is Trail-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;

end;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;

now :: thesis: for m, n being even Element of NAT st 1 <= m & m < n & n <= len W holds

W . m <> W . n

hence
W is Trail-like
by Lm57; :: thesis: verumW . m <> W . n

let m, n be even Element of NAT ; :: thesis: ( 1 <= m & m < n & n <= len W implies W . m <> W . n )

assume that

A4: 1 <= m and

A5: m < n and

A6: n <= len W ; :: thesis: W . m <> W . n

(2 * 0) + 1 < m by A4, XXREAL_0:1;

then A7: 1 + 1 <= m by NAT_1:13;

n < 2 + 1 by A3, A6, XXREAL_0:1;

then n <= 2 by NAT_1:13;

hence W . m <> W . n by A5, A7, XXREAL_0:2; :: thesis: verum

end;assume that

A4: 1 <= m and

A5: m < n and

A6: n <= len W ; :: thesis: W . m <> W . n

(2 * 0) + 1 < m by A4, XXREAL_0:1;

then A7: 1 + 1 <= m by NAT_1:13;

n < 2 + 1 by A3, A6, XXREAL_0:1;

then n <= 2 by NAT_1:13;

hence W . m <> W . n by A5, A7, XXREAL_0:2; :: thesis: verum