
theorem
  for A being non empty Interval, x being Real st 0 < x for B
  being non empty Interval st B = x ** A holds A = [.inf A,sup A.[ implies (B =
[.inf B,sup B.[ & for s,t being Real st s = inf A & t = sup A holds inf B = x *
  s & sup B = x * t)
proof
  let A be non empty Interval;
  let x be Real;
  assume
A1: 0 < x;
  let B be non empty Interval;
  assume
A2: B = x ** A;
A3: inf A <= sup A by XXREAL_2:40;
  assume
A4: A = [.inf A,sup A.[;
  then inf A <> sup A;
  then inf A < sup A by A3,XXREAL_0:1;
  then inf A in A by A4,XXREAL_1:3;
  then reconsider a = inf A as Real;
A5: for s,t being Real st s = inf A & t = sup A holds inf B = x * s & sup B
  = x * t & B is right_open_interval
  proof
    let s,t be Real;
    assume that
A6: s = inf A and
A7: t = sup A;
    inf B = x * s & sup B = x * t & B is right_open_interval
    proof
      s <= t by A6,A7,XXREAL_2:40;
      then
A8:   x * s <= x * t by A1,XREAL_1:64;
      x * s is R_eal by XXREAL_0:def 1;
      then consider d being R_eal such that
A9:   d = x * s;
      x * t is R_eal by XXREAL_0:def 1;
      then consider g being R_eal such that
A10:  g = x * t;
A11:  [.d,g.[ c= x ** A
      proof
        let q be object;
        assume
A12:    q in [.d,g.[;
        then reconsider q as Real by A9;
        reconsider q2 = q / x as Element of REAL by XREAL_0:def 1;
        reconsider q1 = q as R_eal by XXREAL_0:def 1;
A13:    q1 < g by A12,XXREAL_1:3;
A14:    q2 in A
        proof
          reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
          inf A <= q3 & q3 < sup A & q3 in REAL
          proof
A15:        q3 < sup A
            proof
              consider p,o being Real such that
A16:          p = q1 & o = g and
              p <= o by A10,A13;
              q1 < g by A12,XXREAL_1:3;
              then p/x < o/x by A1,A16,XREAL_1:74;
              hence thesis by A1,A7,A10,A16,XCMPLX_1:89;
            end;
            d <= q1 & x * q2 = q by A1,A12,XCMPLX_1:87,XXREAL_1:3;
            hence thesis by A1,A6,A9,A15,XREAL_1:68;
          end;
          hence thesis by A4,XXREAL_1:3;
        end;
        q = x * (q / x) by A1,XCMPLX_1:87;
        hence thesis by A14,MEMBER_1:193;
      end;
      x ** A c= [.d,g.[
      proof
        let q be object;
        assume
A17:    q in x ** A;
        then reconsider q as Real;
        consider z2 being Real such that
A18:    z2 in A and
A19:    q = x * z2 by A17,INTEGRA2:39;
        reconsider z2 as R_eal by XXREAL_0:def 1;
        z2 <= sup A by A4,A18,XXREAL_1:3;
        then consider 2o,2r being Real such that
A20:    2o= z2 & 2r = sup A and
        2o <= 2r by A7;
        x * 2o is R_eal & x * 2r is R_eal by XXREAL_0:def 1;
        then consider 2o1,2r1 being R_eal such that
A21:    2o1 = x * 2o & 2r1 = x * 2r;
        2o< 2r by A4,A18,A20,XXREAL_1:3;
        then
A22:    2o1 < 2r1 by A1,A21,XREAL_1:68;
        inf A <= z2 by A4,A18,XXREAL_1:3;
        then consider 1o,1ra being Real such that
A23:    1o= inf A & 1ra = z2 and
A24:    1o <= 1ra by A6;
        x * 1o <= x * 1ra by A1,A24,XREAL_1:64;
        hence thesis by A6,A7,A9,A10,A19,A23,A20,A21,A22,XXREAL_1:3;
      end;
      then x ** A = [.d,g.[ by A11;
      hence thesis by A2,A9,A10,A8,MEASURE5:def 4,MEASURE6:11,15;
    end;
    hence thesis;
  end;
  A = [.a,sup A.[ by A4;
  then A is right_open_interval by MEASURE5:def 4;
  then x ** A is right_open_interval by A1,Th9;
  hence thesis by A2,A5,MEASURE6:18;
end;
