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
  for A being left_end right_end interval ext-real-membered set holds
  A = [.min A, max A.]
proof
  let A be left_end right_end interval ext-real-membered set;
  let x;
A1: max A in A by Def6;
  thus x in A implies x in [.min A, max A.]
  proof
    assume
A2: x in A;
    then
A3: x <= max A by Th4;
    min A <= x by A2,Th3;
    hence thesis by A3,XXREAL_1:1;
  end;
  min A in A by Def5;
  then [.min A, max A.] c= A by A1,Def12;
  hence thesis;
end;
