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 Th50:
  k + 1 <= d implies for A being Cell of k,G, B being Cell of (k + 1),G holds
  A in del {B} iff A c= B
proof
  assume
A1: k + 1 <= d;
  let A be Cell of k,G, B be Cell of (k + 1),G;
  set X = star A /\ {B};
  card X is odd iff B in star A
  proof
    per cases;
    suppose
A2:   B in star A;
      now
        let B9 be object;
        B9 in {B} iff B9 = B by TARSKI:def 1;
        hence B9 in X iff B9 = B by A2,XBOOLE_0:def 4;
      end;
      then X = {B} by TARSKI:def 1;
      then card X = 2* 0 + 1 by CARD_1:30;
      hence thesis by A2;
    end;
    suppose
A3:   not B in star A;
      now
        let B9 be object;
        B9 = B or not B9 in {B} by TARSKI:def 1;
        hence not B9 in X by A3,XBOOLE_0:def 4;
      end;
      then card X = 2* 0 by CARD_1:27,XBOOLE_0:def 1;
      hence thesis by A3;
    end;
  end;
  hence thesis by A1,Th47,Th48;
end;
