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 A2, NAT_1:13;

then 2 * (C .length()) >= 2 * 4 by XREAL_1:64;

then (2 * (C .length())) + 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;

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 A2, NAT_1:13;

then 2 * (C .length()) >= 2 * 4 by XREAL_1:64;

then (2 * (C .length())) + 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 ) )
;

end;

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} )

( 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 A3, XREAL_1:8;

then reconsider Ci = C . i as Vertex of G by GLIB_001:7;

then A10: x <> Ci by A8, GLIB_001:def 28;

(len C) + (- 2) >= 9 + (- 2) by A3, XREAL_1:7;

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 A3, XREAL_1:7;

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} )

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 A10, Th51; :: thesis: C . j in G .AdjacentSet {x}

A14: j in NAT by ORDINAL1:def 12;

then reconsider Cj = C . j as Vertex of G by A12, GLIB_001:7;

then x,Cj are_adjacent by Def3;

hence C . j in G .AdjacentSet {x} by A16, Th51; :: thesis: verum

end;reconsider k = (2 * 0) + 1 as odd Nat ;

A8: (len C) + 0 > 9 + (- 6) by A3, XREAL_1:8;

then reconsider Ci = C . i as Vertex of G by GLIB_001:7;

then A10: x <> Ci by A8, GLIB_001:def 28;

(len C) + (- 2) >= 9 + (- 2) by A3, XREAL_1:7;

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 A3, XREAL_1:7;

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} )

(len C) + 0 > 9 + (- 8)
by A3, XREAL_1:8;assume A13:
C . i = C . j
; :: thesis: contradiction

(i + 2) + (- 2) < j + 0 by A11, XXREAL_0:2;

hence contradiction by A12, A13, GLIB_001:def 28; :: thesis: verum

end;(i + 2) + (- 2) < j + 0 by A11, XXREAL_0:2;

hence contradiction by A12, A13, GLIB_001:def 28; :: thesis: verum

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 A10, Th51; :: thesis: C . j in G .AdjacentSet {x}

A14: j in NAT by ORDINAL1:def 12;

then reconsider Cj = C . j as Vertex of G by A12, GLIB_001:7;

A16: now :: thesis: not x = Cj

C . (j + 1) Joins Cj,x,G
by A14, A12, A15, GLIB_001:def 3;assume
x = Cj
; :: thesis: contradiction

then A17: j = 1 by A14, A12, A15, GLIB_001:def 28;

j + 2 = len C ;

hence contradiction by A3, A17; :: thesis: verum

end;then A17: j = 1 by A14, A12, A15, GLIB_001:def 28;

j + 2 = len C ;

hence contradiction by A3, A17; :: thesis: verum

then x,Cj are_adjacent by Def3;

hence C . j in G .AdjacentSet {x} by A16, Th51; :: thesis: verum

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} )

( 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 A6, XXREAL_0:1;

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 A4, GLIB_001:7, XXREAL_0:2;

reconsider j = n + 2 as odd Nat ;

A20: n < len C by A4, A18, XXREAL_0:1;

then A21: n + 2 <= ((len C) - 2) + 2 by Th4;

then reconsider Cj = C . j as Vertex of G by GLIB_001:7;

( 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 A20, Th4; :: 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} )

then C . (i + 1) Joins C . i,C . (i + 2),G by A31, GLIB_001:def 3;

then x,Ci are_adjacent by A5, Def3;

hence C . i in G .AdjacentSet {x} by A32, Th51; :: thesis: C . j in G .AdjacentSet {x}

C . (n + 1) Joins C . n,C . j,G by A20, GLIB_001:def 3;

then x,Cj are_adjacent by A5;

hence C . j in G .AdjacentSet {x} by A22, Th51; :: thesis: verum

end;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 A4, GLIB_001:7, XXREAL_0:2;

reconsider j = n + 2 as odd Nat ;

A20: n < len C by A4, A18, XXREAL_0:1;

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 = x

take
i
; :: thesis: ex n being odd Nat st A23:
n + 2 > n + 0
by XREAL_1:8;

assume Cj = x ; :: thesis: contradiction

hence contradiction by A5, A18, A21, A23, GLIB_001:def 28; :: thesis: verum

end;assume Cj = x ; :: thesis: contradiction

hence contradiction by A5, A18, A21, A23, GLIB_001:def 28; :: thesis: verum

( 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 A20, Th4; :: 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 )

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} )assume that

A25: i = 1 and

A26: j = len C ; :: thesis: contradiction

j = i + 4 ;

hence contradiction by A3, A25, A26; :: thesis: verum

end;A25: i = 1 and

A26: j = len C ; :: thesis: contradiction

j = i + 4 ;

hence contradiction by A3, A25, A26; :: thesis: verum

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 A3, XREAL_1:7;

hence contradiction by A27, A28; :: thesis: verum

end;A27: i = 1 and

A28: j = (len C) - 2 ; :: thesis: contradiction

(len C) + (- 2) >= 9 + (- 3) by A3, XREAL_1:7;

hence contradiction by A27, A28; :: thesis: verum

hereby :: thesis: ( C . i <> C . j & C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )

A31:
i in NAT
by ORDINAL1:def 12;assume that

A29: i = 3 and

A30: j = len C ; :: thesis: contradiction

j = i + 4 ;

hence contradiction by A3, A29, A30; :: thesis: verum

end;A29: i = 3 and

A30: j = len C ; :: thesis: contradiction

j = i + 4 ;

hence contradiction by A3, A29, A30; :: thesis: verum

A32: now :: thesis: not Ci = x

A34:
n + 2 <= ((len C) - 2) + 2
by A20, Th4;A33:
n + 0 > n + (- 2)
by XREAL_1:8;

assume Ci = x ; :: thesis: contradiction

hence contradiction by A4, A5, A18, A31, A33, GLIB_001:def 28; :: thesis: verum

end;assume Ci = x ; :: thesis: contradiction

hence contradiction by A4, A5, A18, A31, A33, GLIB_001:def 28; :: thesis: verum

hereby :: thesis: ( C . i in G .AdjacentSet {x} & C . j in G .AdjacentSet {x} )

i < len C
by A4, A19, XXREAL_0:2;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;assume C . i = C . j ; :: thesis: contradiction

hence contradiction by A31, A34, A24, A35, GLIB_001:def 28; :: thesis: verum

then C . (i + 1) Joins C . i,C . (i + 2),G by A31, GLIB_001:def 3;

then x,Ci are_adjacent by A5, Def3;

hence C . i in G .AdjacentSet {x} by A32, Th51; :: thesis: C . j in G .AdjacentSet {x}

C . (n + 1) Joins C . n,C . j,G by A20, GLIB_001:def 3;

then x,Cj are_adjacent by A5;

hence C . j in G .AdjacentSet {x} by A22, Th51; :: thesis: verum