reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;
reserve i for Nat;
reserve m for Nat,
        D for non empty set;
reserve l for Nat;

theorem
  not m in dom p implies not m+1 in dom p
proof
  assume not m in dom p; then
A1: m >= card p by Lm1;
  m+1 >= m by NAT_1:11;
  hence thesis by Lm1,A1,XXREAL_0:2;
end;
