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 Th81:
  A is interval implies for x,r st x in A & x <= r & r < sup A holds r in A
proof
  assume
A1: A is interval;
  let x,r such that
A2: x in A and
A3: x <= r and
A4: r < sup A;
  per cases;
  suppose
    ex y st y in A & r < y;
    hence thesis by A1,A2,A3,Th80;
  end;
  suppose
    not ex y st y in A & r < y;
    then r is UpperBound of A by Def1;
    hence thesis by A4,Def3;
  end;
end;
