reserve i,i1,i2,i9,i19,j,j1,j2,j9,j19,k,k1,k2,l,m,n for Nat;
reserve r,s,r9,s9 for Real;
reserve D for non empty set, f for FinSequence of D;
reserve f for FinSequence of TOP-REAL 2, G for Go-board;

theorem
  1 <= k & k+1 <= len f & f is_sequence_on G implies right_cell(f,k,G)
  is closed
proof
  assume
A1: 1 <= k & k+1 <= len f & f is_sequence_on G;
  then consider i1,j1,i2,j2 being Nat such that
A2: [i1,j1] in Indices G & f/.k = G*(i1,j1) & [i2,j2] in Indices G & f/.
( k+1 ) = G*(i2,j2) &( i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1
  & j1 = j2 or i1 = i2 & j1 = j2+1) by JORDAN8:3;
  i1 = i2 & j1+1 = j2 & right_cell(f,k,G) = cell(G,i1,j1) or i1+1 = i2 &
  j1 = j2 & right_cell(f,k,G) = cell(G,i1,j1-'1) or i1 = i2+1 & j1 = j2 &
right_cell(f,k,G) = cell(G,i2,j2) or i1 = i2 & j1 = j2+1 & right_cell(f,k,G) =
  cell(G,i1-'1,j2) by A1,A2,Def1;
  hence thesis by GOBRD11:33;
end;
