reserve X for set;
reserve UN for Universe;

theorem Th91:
  for S being Sequence st dom S = NAT holds S.NAT = {} & last S = {}
  proof
    let S be Sequence;
    assume
A1: dom S = NAT;
    not NAT in NAT;
    hence S.NAT = {} by A1,FUNCT_1:def 2;
    hence thesis by A1,Th90;
  end;
