reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;
reserve A,B for ext-real-membered set;

theorem
  A is interval implies for r st inf A < r & r < sup A holds r in A
proof
  assume
A1: A is interval;
  let r such that
A2: inf A < r and
A3: r < sup A;
  per cases;
  suppose
    ex y st y in A & r > y;
    hence thesis by A1,A3,Th81;
  end;
  suppose
    not ex y st y in A & r > y;
    then r is LowerBound of A by Def2;
    hence thesis by A2,Def4;
  end;
end;
