reserve X,x,y,z for set;
reserve n,m,k,k9,d9 for Nat;
reserve d for non zero Nat;
reserve i,i0,i1 for Element of Seg d;
reserve l,r,l9,r9,l99,r99,x,x9,l1,r1,l2,r2 for Element of REAL d;
reserve Gi for non trivial finite Subset of REAL;
reserve li,ri,li9,ri9,xi,xi9 for Real;
reserve G for Grating of d;

theorem Th18:
  [li,ri] is Gap of Gi & [li9,ri] is Gap of Gi implies li = li9
proof
A1: li <= li9 & li9 <= li implies li = li9 by XXREAL_0:1;
  assume that
A2: [li,ri] is Gap of Gi and
A3: [li9,ri] is Gap of Gi;
A4: li in Gi by A2,Th13;
A5: li9 in Gi by A3,Th13;
  per cases by A2,Th13;
  suppose
A6: li < ri & for xi st xi in Gi holds not (li < xi & xi < ri);
    li9 <= li or li < li9 & li9 < ri or ri <= li9;
    hence thesis by A1,A3,A4,A5,A6,Th13;
  end;
  suppose
A7: ri < li & for xi st xi in Gi holds not (li < xi or xi < ri);
    li9 < ri or ri <= li9 & li9 <= li or li < li9;
    hence thesis by A1,A3,A4,A5,A7,Th13;
  end;
end;
