let G be _Graph; :: thesis: for C being Path of G st C is Cycle-like & C .length() > 3 holds
for x being Vertex of G st x in C .vertices() holds
ex m, n being odd Nat st
( m + 2 < n & n <= len C & ( not m = 1 or not n = len C ) & ( not m = 1 or not n = (len C) - 2 ) & ( not m = 3 or not n = len C ) & C . m <> C . n & C . m in G .AdjacentSet {x} & C . n in G .AdjacentSet {x} )

let C be Path of G; :: thesis: ( C is Cycle-like & C .length() > 3 implies for x being Vertex of G st x in C .vertices() holds
ex m, n being odd Nat st
( m + 2 < n & n <= len C & ( not m = 1 or not n = len C ) & ( not m = 1 or not n = (len C) - 2 ) & ( not m = 3 or not n = len C ) & C . m <> C . n & C . m in G .AdjacentSet {x} & C . n in G .AdjacentSet {x} ) )

assume that
A1: C is Cycle-like and
A2: C .length() > 3 ; :: thesis: for x being Vertex of G st x in C .vertices() holds
ex m, n being odd Nat st
( m + 2 < n & n <= len C & ( not m = 1 or not n = len C ) & ( not m = 1 or not n = (len C) - 2 ) & ( not m = 3 or not n = len C ) & C . m <> C . n & C . m in G .AdjacentSet {x} & C . n in G .AdjacentSet {x} )

C .length() >= 3 + 1 by ;
then 2 * () >= 2 * 4 by XREAL_1:64;
then (2 * ()) + 1 >= 8 + 1 by XREAL_1:7;
then A3: len C >= 9 by GLIB_001:112;
let x be Vertex of G; :: thesis: ( x in C .vertices() implies ex m, n being odd Nat st
( m + 2 < n & n <= len C & ( not m = 1 or not n = len C ) & ( not m = 1 or not n = (len C) - 2 ) & ( not m = 3 or not n = len C ) & C . m <> C . n & C . m in G .AdjacentSet {x} & C . n in G .AdjacentSet {x} ) )

assume x in C .vertices() ; :: thesis: ex m, n being odd Nat st
( m + 2 < n & n <= len C & ( not m = 1 or not n = len C ) & ( not m = 1 or not n = (len C) - 2 ) & ( not m = 3 or not n = len C ) & C . m <> C . n & C . m in G .AdjacentSet {x} & C . n in G .AdjacentSet {x} )

then consider n being odd Element of NAT such that
A4: n <= len C and
A5: C . n = x by GLIB_001:87;
A6: 0 + 1 <= n by ABIAN:12;
per cases ( n = 1 or n = len C or ( not n = 1 & not n = len C ) ) ;
suppose A7: ( n = 1 or n = len C ) ; :: thesis: ex m, n being odd Nat st
( m + 2 < n & n <= len C & ( not m = 1 or not n = len C ) & ( not m = 1 or not n = (len C) - 2 ) & ( not m = 3 or not n = len C ) & C . m <> C . n & C . m in G .AdjacentSet {x} & C . n in G .AdjacentSet {x} )

reconsider i = (2 * 1) + 1 as odd Nat ;
reconsider k = (2 * 0) + 1 as odd Nat ;
A8: (len C) + 0 > 9 + (- 6) by ;
then reconsider Ci = C . i as Vertex of G by GLIB_001:7;
A9: now :: thesis: x = C . k
per cases ( n = 1 or n = len C ) by A7;
suppose n = 1 ; :: thesis: x = C . k
hence x = C . k by A5; :: thesis: verum
end;
end;
end;
then A10: x <> Ci by ;
(len C) + (- 2) >= 9 + (- 2) by ;
then (len C) - (2 * 1) >= 0 by XXREAL_0:2;
then (len C) - 2 is odd Element of NAT by INT_1:3;
then reconsider j = (len C) - 2 as odd Nat ;
take i ; :: thesis: ex n being odd Nat st
( i + 2 < n & n <= len C & ( not i = 1 or not n = len C ) & ( not i = 1 or not n = (len C) - 2 ) & ( not i = 3 or not n = len C ) & C . i <> C . n & C . i in G .AdjacentSet {x} & C . n in G .AdjacentSet {x} )

take j ; :: thesis: ( i + 2 < j & j <= len C & ( not i = 1 or not j = len C ) & ( not i = 1 or not j = (len C) - 2 ) & ( not i = 3 or not j = len C ) & C . i <> C . j & C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
A11: (len C) + (- 2) >= 9 + (- 2) by ;
hence i + 2 < j by XXREAL_0:2; :: thesis: ( j <= len C & ( not i = 1 or not j = len C ) & ( not i = 1 or not j = (len C) - 2 ) & ( not i = 3 or not j = len C ) & C . i <> C . j & C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
A12: (len C) + 0 > (len C) + (- 2) by XREAL_1:8;
hence j <= len C ; :: thesis: ( ( not i = 1 or not j = len C ) & ( not i = 1 or not j = (len C) - 2 ) & ( not i = 3 or not j = len C ) & C . i <> C . j & C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
thus ( ( not i = 1 or not j = len C ) & ( not i = 1 or not j = (len C) - 2 ) & ( not i = 3 or not j = len C ) ) ; :: thesis: ( C . i <> C . j & C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
hereby :: thesis: ( C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
assume A13: C . i = C . j ; :: thesis: contradiction
(i + 2) + (- 2) < j + 0 by ;
hence contradiction by A12, A13, GLIB_001:def 28; :: thesis: verum
end;
(len C) + 0 > 9 + (- 8) by ;
then C . (k + 1) Joins C . k,C . i,G by GLIB_001:def 3;
then x,Ci are_adjacent by A9;
hence C . i in G .AdjacentSet {x} by ; :: thesis:
A14: j in NAT by ORDINAL1:def 12;
then reconsider Cj = C . j as Vertex of G by ;
A15: now :: thesis: x = C . (j + 2)
per cases ( n = 1 or n = len C ) by A7;
suppose n = 1 ; :: thesis: x = C . (j + 2)
then x = C .first() by A5;
then x = C .last() by ;
hence x = C . (j + 2) ; :: thesis: verum
end;
suppose n = len C ; :: thesis: x = C . (j + 2)
hence x = C . (j + 2) by A5; :: thesis: verum
end;
end;
end;
A16: now :: thesis: not x = Cj
assume x = Cj ; :: thesis: contradiction
then A17: j = 1 by ;
j + 2 = len C ;
hence contradiction by A3, A17; :: thesis: verum
end;
C . (j + 1) Joins Cj,x,G by ;
then x,Cj are_adjacent by Def3;
hence C . j in G .AdjacentSet {x} by ; :: thesis: verum
end;
suppose A18: ( not n = 1 & not n = len C ) ; :: thesis: ex m, n being odd Nat st
( m + 2 < n & n <= len C & ( not m = 1 or not n = len C ) & ( not m = 1 or not n = (len C) - 2 ) & ( not m = 3 or not n = len C ) & C . m <> C . n & C . m in G .AdjacentSet {x} & C . n in G .AdjacentSet {x} )

then (2 * 0) + 1 < n by ;
then 1 + 2 <= n by Th4;
then 3 + (- 2) <= n + (- 2) by XREAL_1:7;
then 0 <= n - (2 * 1) ;
then n - 2 is odd Element of NAT by INT_1:3;
then reconsider i = n - 2 as odd Nat ;
A19: i + 0 < i + 2 by XREAL_1:8;
then reconsider Ci = C . i as Vertex of G by ;
reconsider j = n + 2 as odd Nat ;
A20: n < len C by ;
then A21: n + 2 <= ((len C) - 2) + 2 by Th4;
then reconsider Cj = C . j as Vertex of G by GLIB_001:7;
A22: now :: thesis: not Cj = xend;
take i ; :: thesis: ex n being odd Nat st
( i + 2 < n & n <= len C & ( not i = 1 or not n = len C ) & ( not i = 1 or not n = (len C) - 2 ) & ( not i = 3 or not n = len C ) & C . i <> C . n & C . i in G .AdjacentSet {x} & C . n in G .AdjacentSet {x} )

take j ; :: thesis: ( i + 2 < j & j <= len C & ( not i = 1 or not j = len C ) & ( not i = 1 or not j = (len C) - 2 ) & ( not i = 3 or not j = len C ) & C . i <> C . j & C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
n + 0 < n + 2 by XREAL_1:8;
hence ( i + 2 < j & j <= len C ) by ; :: thesis: ( ( not i = 1 or not j = len C ) & ( not i = 1 or not j = (len C) - 2 ) & ( not i = 3 or not j = len C ) & C . i <> C . j & C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
A24: now :: thesis: ( i = 1 implies not j = len C )
assume that
A25: i = 1 and
A26: j = len C ; :: thesis: contradiction
j = i + 4 ;
hence contradiction by A3, A25, A26; :: thesis: verum
end;
hence ( not i = 1 or not j = len C ) ; :: thesis: ( ( not i = 1 or not j = (len C) - 2 ) & ( not i = 3 or not j = len C ) & C . i <> C . j & C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
hereby :: thesis: ( ( not i = 3 or not j = len C ) & C . i <> C . j & C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
assume that
A27: i = 1 and
A28: j = (len C) - 2 ; :: thesis: contradiction
(len C) + (- 2) >= 9 + (- 3) by ;
hence contradiction by A27, A28; :: thesis: verum
end;
hereby :: thesis: ( C . i <> C . j & C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
assume that
A29: i = 3 and
A30: j = len C ; :: thesis: contradiction
j = i + 4 ;
hence contradiction by A3, A29, A30; :: thesis: verum
end;
A31: i in NAT by ORDINAL1:def 12;
A32: now :: thesis: not Ci = xend;
A34: n + 2 <= ((len C) - 2) + 2 by ;
hereby :: thesis: ( C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )
A35: (i + 2) + (- 2) < j + 0 by XREAL_1:8;
assume C . i = C . j ; :: thesis: contradiction
hence contradiction by A31, A34, A24, A35, GLIB_001:def 28; :: thesis: verum
end;
i < len C by ;
then C . (i + 1) Joins C . i,C . (i + 2),G by ;
then x,Ci are_adjacent by ;
hence C . i in G .AdjacentSet {x} by ; :: thesis:
C . (n + 1) Joins C . n,C . j,G by ;
then x,Cj are_adjacent by A5;
hence C . j in G .AdjacentSet {x} by ; :: thesis: verum
end;
end;