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;
reserve D for set,
  f,f1,f2 for FinSequence of D,
  G for Matrix of D;

theorem
  1 <= k & k+2 <= n & f|n turns_left k,G implies f turns_left k,G
proof
  assume that
A1: 1 <= k & k+2 <= n and
A2: f|n turns_left k,G;
  per cases;
  suppose
    len f <= n;
    hence thesis by A2,FINSEQ_1:58;
  end;
  suppose
A3: n < len f;
    let i19,j19,i29,j29 be Nat;
A4: len(f|n) = n by A3,FINSEQ_1:59;
    then k+1 in dom(f|n) by A1,SEQ_4:135;
    then
A5: (f|n)/.(k+1) = f/.(k+1) by FINSEQ_4:70;
    k+2 in dom(f|n) by A1,A4,SEQ_4:135;
    then
A6: (f|n)/.(k+2) = f/.(k+2 ) by FINSEQ_4:70;
    k in dom(f|n) by A1,A4,SEQ_4:135;
    then (f|n)/.k = f/.k by FINSEQ_4:70;
    hence thesis by A2,A5,A6;
  end;
end;
