
theorem Th27:
  for A being Interval, x being Real holds
    A is closed_interval iff x ++ A is closed_interval
proof
  let A be Interval;
  let x be Real;
A1: for B being Interval, y being Real st B is closed_interval
    holds y ++ B is closed_interval
  proof
    let B be Interval;
    let y be Real;
    reconsider y as Real;
    reconsider z = y as R_eal by XXREAL_0:def 1;
    assume B is closed_interval;
    then consider a1,b1 being Real such that
A2: B = [.a1,b1.] by MEASURE5:def 3;
    reconsider a=a1,b=b1 as R_eal by XXREAL_0:def 1;
    reconsider s = z + a, t = z + b as R_eal;
 y ++ B = [.s,t.]
    proof
      thus y ++ B c= [.s,t.]
      proof
        let c be object;
        assume
A3:     c in y ++ B;
        then reconsider c as Real;
        consider d being Real such that
A4:     d in B and
A5:     c = y + d by A3,Lm1;
        reconsider d1 = d as R_eal by XXREAL_0:def 1;
        a <= d1 by A2,A4,XXREAL_1:1; then
A6:     s <= z + d1 by XXREAL_3:36;
        d1 <= b by A2,A4,XXREAL_1:1;
        then
A7:     z + d1 <= t by XXREAL_3:36;
        z + d1 = c by A5,SUPINF_2:1;
        hence thesis by A6,A7,XXREAL_1:1;
      end;
      reconsider a,b as R_eal;
      let c be object;
      assume
A8:  c in [.s,t.];
      then reconsider c as Real;
      reconsider c1 = c as R_eal by XXREAL_0:def 1;
A9:  c = y + (c - y);
      c1 <= z + b by A8,XXREAL_1:1;
      then c1 - z <= (b + z) - z by XXREAL_3:36;
      then
A10:  c1 - z <= b by XXREAL_3:22;
      z + a <= c1 by A8,XXREAL_1:1;
      then (a + z) - z <= c1 - z by XXREAL_3:36;
      then
A11:  a <= c1 - z by XXREAL_3:22;
      c1 - z = c - y by SUPINF_2:3;
      then c - y in B by A2,A11,A10;
      hence thesis by A9,Lm1;
    end;
    hence thesis by MEASURE5:def 3;
  end;
  x ++ A is closed_interval implies A is closed_interval
  proof
    reconsider y = -x as Real;
    assume
A12: x ++ A is closed_interval;
    then reconsider B = x ++ A as Interval;
    y ++ B = A by Th23;
    hence thesis by A1,A12;
  end;
  hence thesis by A1;
end;
