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 Th58:
  for C1,C2 being Chain of (k + 1),G holds del(C1 + C2) = del C1 + del C2
proof
  let C1,C2 be Chain of (k + 1),G;
  now
    let A be Cell of k,G;
    A1: star
 A /\ (C1 \+\ C2) = (star A /\ C1) \+\ (star A /\ C2) by XBOOLE_1:112;
A2: A in del(C1 + C2) iff k + 1 <= d & card(star A /\ (C1 \+\ C2)) is odd
    by Th48;
A3: A in del(C1) iff k + 1 <= d & card(star A /\ C1) is odd by Th48;
    A in del(C2) iff k + 1 <= d & card(star A /\ C2) is odd by Th48;
    hence A in del(C1 + C2) iff A in del C1 + del C2 by A1,A2,A3,Th7,XBOOLE_0:1
;
  end;
  hence thesis by SUBSET_1:3;
end;
