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 Th66:
  p - A = {} iff rng p c= A
proof
  thus p - A = {} implies rng p c= A
  proof
    assume that
A1: p - A = {} and
A2: not rng p c= A;
    rng p \ A <> {} by A2,XBOOLE_1:37;
    then rng(p - A) <> {} by Th63;
    hence thesis by A1;
  end;
  assume
A3: rng p c= A;
  rng(p - A) = rng p \ A by Th63;
  hence thesis by A3,XBOOLE_1:37;
end;
