reserve i,j,k,n for Nat,
  D for non empty set,
  f, g for FinSequence of D;
reserve G for Go-board,
  f, g for FinSequence of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  r, s for Real,
  x for set;

theorem Th7:
  n <= len f & x in L~(f/^n)
  implies ex i being Nat st n+1 <= i & i+1 <= len f & x in LSeg(f,i)
proof
  assume that
A1: n <= len f and
A2: x in L~(f/^n);
  consider j being Nat such that
A3: 1 <= j and
A4: j+1 <= len(f/^n) and
A5: x in LSeg(f/^n,j) by A2,SPPOL_2:13;
  j+1 <= len f - n by A1,A4,RFINSEQ:def 1;
  then
A6: n+(j+1) <= len f by XREAL_1:19;
  take n+j;
  j+1 <= len f - n by A1,A4,RFINSEQ:def 1;
  hence thesis by A3,A5,A6,SPPOL_2:5,XREAL_1:6;
end;
