reserve i,j,k,n for Nat,
  X,Y,a,b,c,x for set,
  r,s for Real;
reserve f,g for FinSequence of TOP-REAL 2;
reserve f for non constant standard special_circular_sequence,
  p,p1,p2,q for Point of TOP-REAL 2;
reserve G for Go-board;

theorem Th22:
  for f,k st 1<=k & k+1<=len f holds left_cell(f,k,GoB f) is
  convex & right_cell(f,k,GoB f) is convex
proof
  let f,k such that
A1: 1<=k and
A2: k+1<=len f;
  left_cell(f,k) = left_cell(f,k,GoB f) by A1,A2,JORDAN1H:21;
  hence left_cell(f,k,GoB f) is convex by A1,A2,Th21;
  k <= len f by A2,NAT_1:13;
  then
A3: len f-'k+k = len f by XREAL_1:235;
  then
A4: len f-'k >= 1 by A2,XREAL_1:6;
  then
A5: right_cell(f,k) = left_cell(Rev f,len f-'k) by A1,A3,GOBOARD9:10;
  len f = len Rev f & len f-'k+1 <= len f by A1,A3,FINSEQ_5:def 3,XREAL_1:6;
  then left_cell(Rev f,len f-'k) is convex by A4,Th21;
  hence thesis by A1,A2,A5,JORDAN1H:23;
end;
