let G be _Graph; :: thesis: for W being Walk of G
for e, v being set st W is Path-like & e Joins W .last() ,v,G & not e in W .edges() & ( W is trivial or not W is closed ) & ( for n being odd Element of NAT st 1 < n & n <= len W holds
W . n <> v ) holds
W .addEdge e is Path-like
let W be Walk of G; :: thesis: for e, v being set st W is Path-like & e Joins W .last() ,v,G & not e in W .edges() & ( W is trivial or not W is closed ) & ( for n being odd Element of NAT st 1 < n & n <= len W holds
W . n <> v ) holds
W .addEdge e is Path-like
let e, v be set ; :: thesis: ( W is Path-like & e Joins W .last() ,v,G & not e in W .edges() & ( W is trivial or not W is closed ) & ( for n being odd Element of NAT st 1 < n & n <= len W holds
W . n <> v ) implies W .addEdge e is Path-like )
assume that
A1: W is Path-like and
A2: e Joins W .last() ,v,G and
A3: not e in W .edges() and
A4: ( W is trivial or not W is closed ) and
A5: for n being odd Element of NAT st 1 < n & n <= len W holds
W . n <> v ; :: thesis: W .addEdge e is Path-like
reconsider lenW = len W as odd Element of NAT ;
set W2 = W .addEdge e;
A6: e in (W .last()) .edgesInOut() by A2, GLIB_000:65;

now

W is Trail-like by A1, Def28;
hence W .addEdge e is Trail-like by A3, A6, Lm60; :: thesis: for m, n being odd Element of NAT st m < n & n <= len (W .addEdge e) & (W .addEdge e) . m = (W .addEdge e) . n holds
( m = 1 & n = len (W .addEdge e) )
let m, n be odd Element of NAT ; :: thesis: ( m < n & n <= len (W .addEdge e) & (W .addEdge e) . m = (W .addEdge e) . n implies ( m = 1 & n = len (W .addEdge e) ) )
assume that
A7: m < n and
A8: n <= len (W .addEdge e) and
A9: (W .addEdge e) . m = (W .addEdge e) . n ; :: thesis: ( m = 1 & n = len (W .addEdge e) )

now

per cases ( not W is closed or W is trivial ) by A4;

suppose A10: not W is closed ; :: thesis: ( m = 1 & n = len (W .addEdge e) )

now

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

suppose A11: n <= len W ; :: thesis: ( m = 1 & n = len (W .addEdge e) )

A12: 1 <= m by ABIAN:12;
m <= len W by A7, A11, XXREAL_0:2;
then m in dom W by A12, FINSEQ_3:27;
then A13: (W .addEdge e) . m = W . m by A2, Lm38;
1 <= n by ABIAN:12;
then n in dom W by A11, FINSEQ_3:27;
then A14: W . m = W . n by A2, A9, A13, Lm38;
then m = 1 by A1, A7, A11, Def28;
then W .first() = W .last() by A1, A7, A11, A14, Def28;
hence ( m = 1 & n = len (W .addEdge e) ) by A10, Def24; :: thesis: verum

end;

suppose n > len W ; :: thesis: ( m = 1 & n = len (W .addEdge e) )

then lenW + 1 <= n by NAT_1:13;
then lenW + 1 < n by XXREAL_0:1;
then (lenW + 1) + 1 <= n by NAT_1:13;
then (len W) + (1 + 1) <= n ;
then A15: len (W .addEdge e) <= n by A2, Lm37;
then n = len (W .addEdge e) by A8, XXREAL_0:1;
then (W .addEdge e) . n = (W .addEdge e) . ((len W) + 2) by A2, Lm37;
then A16: (W .addEdge e) . n = v by A2, Lm38;
m < len (W .addEdge e) by A7, A8, A15, XXREAL_0:1;
then m < (len W) + (1 + 1) by A2, Lm37;
then m < ((len W) + 1) + 1 ;
then m <= lenW + 1 by NAT_1:13;
then m < lenW + 1 by XXREAL_0:1;
then A17: m <= len W by NAT_1:13;
1 <= m by ABIAN:12;
then m in dom W by A17, FINSEQ_3:27;
then A18: W . m = v by A2, A9, A16, Lm38;

now

A19: 1 <= m by ABIAN:12;
assume m <> 1 ; :: thesis: contradiction
then 1 < m by A19, XXREAL_0:1;
hence contradiction by A5, A17, A18; :: thesis: verum

end;

hence m = 1 ; :: thesis: n = len (W .addEdge e)
thus n = len (W .addEdge e) by A8, A15, XXREAL_0:1; :: thesis: verum

end;

end;

end;

hence ( m = 1 & n = len (W .addEdge e) ) ; :: thesis: verum

end;

suppose W is trivial ; :: thesis: ( m = 1 & n = len (W .addEdge e) )

then ex v being Vertex of G st W = G .walkOf v by Lm56;
then len W = 1 by FINSEQ_1:56;
then A20: len (W .addEdge e) = 1 + 2 by A2, Lm37;
A21: m + 1 <= n by A7, NAT_1:13;
A22: 1 <= m by ABIAN:12;
then 1 + 1 <= m + 1 by XREAL_1:9;
then 2 * 1 <= n by A21, XXREAL_0:2;
then 2 * 1 < n by XXREAL_0:1;
then A23: len (W .addEdge e) <= n by A20, NAT_1:13;
then n = len (W .addEdge e) by A8, XXREAL_0:1;
then (m + 1) - 1 <= 3 - 1 by A7, A20, NAT_1:13;
then m < 2 * 1 by XXREAL_0:1;
then m + 1 <= 2 by NAT_1:13;
then (m + 1) - 1 <= 2 - 1 by XREAL_1:15;
hence ( m = 1 & n = len (W .addEdge e) ) by A8, A22, A23, XXREAL_0:1; :: thesis: verum

end;

end;

end;

hence ( m = 1 & n = len (W .addEdge e) ) ; :: thesis: verum

end;

hence W .addEdge e is Path-like by Def28; :: thesis: verum