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 Th18:
  1 <= k & k+1 <= len f & f is_sequence_on G & [i,j] in Indices G
  & [i+1,j] in Indices G & f/.k = G*(i+1,j) & f/.(k+1) = G*(i,j) implies
  left_cell(f,k,G) = cell(G,i,j-'1)
proof
A1: i < i+1 & i+1 <= i+1+1 by XREAL_1:29;
  assume 1 <= k & k+1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i
  +1,j] in Indices G & f/.k = G*(i+1,j) & f/.(k+1) = G*(i,j);
  hence thesis by A1,Def2;
end;
