reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem Th67:
  p - A = p iff A misses rng p
proof
  thus p - A = p implies A misses rng p
  proof
    assume that
A1: p - A = p and
A2: not A misses rng p;
    len(p - A) <> len p by A2,Th59;
    hence contradiction by A1;
  end;
  assume A misses rng p;
  then p " A = {} by RELAT_1:138;
  then Sgm(Seg(len p) \ p " A) = idseq(len p) by Th46;
  then p * Sgm(Seg(len p) \ p " A) = p by FINSEQ_2:54;
  hence thesis by FINSEQ_1:def 3;
end;
