reserve i,j,k,n,m for Nat;
reserve p,q for Point of TOP-REAL 2;
reserve G for Go-board;
reserve C for Subset of TOP-REAL 2;

theorem Th27:
  for f being non constant standard special_circular_sequence
  holds i_w_n f < i_e_n f
proof
  let f be non constant standard special_circular_sequence;
  set G = GoB f;
A1: i_w_n f <= len G by JORDAN5D:45;
A2: G*(i_w_n f,width G) = N-min L~f by JORDAN5D:def 7;
  assume
A3: i_w_n f >= i_e_n f;
A4: 1 <= i_e_n f by JORDAN5D:45;
A5: (N-min L~f)`1 < (N-max L~f)`1 by SPRECT_2:51;
A6: G*(i_e_n f,width G) = N-max L~f by JORDAN5D:def 8;
  then i_w_n f <> i_e_n f by A5,JORDAN5D:def 7;
  then
A7: i_w_n f > i_e_n f by A3,XXREAL_0:1;
  width G > 1 by GOBOARD7:33;
  hence contradiction by A1,A2,A4,A6,A5,A7,GOBOARD5:3;
end;
