reserve i,j,k,n for Nat;

theorem Th17:
  for f being clockwise_oriented non constant standard
special_circular_sequence for G being Go-board st f is_sequence_on G for i,j,k
being Nat st 1 <= k & k+1 <= len f & [i,j] in Indices G & [i+1,j] in
Indices G & f/.k = G*(i+1,j) & f/.(k+1) = G*(i,j) holds (f/.k)`2 <> N-bound L~f
proof
  let f be clockwise_oriented non constant standard special_circular_sequence;
  let G be Go-board;
  assume
A1: f is_sequence_on G;
  let i,j,k be Nat;
  assume that
A2: 1 <= k & k+1 <= len f and
A3: [i,j] in Indices G and
A4: [i+1,j] in Indices G and
A5: f/.k = G*(i+1,j) and
A6: f/.(k+1) = G*(i,j) and
A7: (f/.k)`2 = N-bound L~f;
A8: right_cell(f,k,G) = cell(G,i,j) by A1,A2,A3,A4,A5,A6,GOBRD13:26;
A9: j <= width G by A4,MATRIX_0:32;
A10: 1 <= i+1 & 1 <= j by A4,MATRIX_0:32;
  set p = 1/2*(G*(i,j)+G*(i+1,j+1));
A11: 0+1 <= i & 1 <= j by A3,MATRIX_0:32;
A12: j <= width G by A3,MATRIX_0:32;
A13: i+1 <= len G by A4,MATRIX_0:32;
  per cases by A12,XXREAL_0:1;
  suppose
    j = width G;
    hence contradiction by A1,A2,A3,A4,A5,A6,Th13;
  end;
  suppose
A14: j < width G;
    i < len G by A13,NAT_1:13;
    then
A15: Int cell(G,i,j) = {|[r,s]| where r,s is Real:
    G*(i,1)`1 < r & r < G*
    (i+1,1)`1 & G*(1,j)`2 < s & s < G*(1,j+1)`2 } by A11,A14,GOBOARD6:26;
    j+1 <= width G by A14,NAT_1:13;
    then
A16: p in Int right_cell(f,k,G) by A11,A13,A8,GOBOARD6:31;
    then consider r,s be Real such that
A17: p = |[r,s]| and
    G*(i,1)`1 < r and
    r < G*(i+1,1)`1 and
A18: G*(1,j)`2 < s and
    s < G*(1,j+1)`2 by A8,A15;
    p`2 = s by A17,EUCLID:52;
    then p`2 > N-bound L~f by A5,A7,A13,A10,A9,A18,GOBOARD5:1;
    then
A19: p in LeftComp f by Th12;
    Int right_cell(f,k,G) c= RightComp f by A1,A2,JORDAN1H:25;
    then p in LeftComp f /\ RightComp f by A16,A19,XBOOLE_0:def 4;
    then LeftComp f meets RightComp f by XBOOLE_0:def 7;
    hence contradiction by GOBRD14:14;
  end;
end;
