
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 sup A in A by A4,XXREAL_1:2;
  then reconsider b = sup 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
  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
    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 A10;
        set q2 = q / x;
        reconsider q1 = q as R_eal by XXREAL_0:def 1;
A13:    q2 in A
        proof
          reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
A14:      q3 <= sup A
          proof
            q1 <= g by A12,XXREAL_1:2;
            then consider p,o being Real such that
A15:        p = q1 & o = g and
A16:        p <= o by A10;
            p/x <= o/x by A1,A16,XREAL_1:72;
            hence thesis by A1,A7,A10,A15,XCMPLX_1:89;
          end;
          d < q1 & x * q2 = q by A1,A12,XCMPLX_1:87,XXREAL_1:2;
          then inf A < q3 by A1,A6,A9,XREAL_1:64;
          hence thesis by A4,A14,XXREAL_1:2;
        end;
        q = x * q2 by A1,XCMPLX_1:87;
        hence thesis by A13,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;
        reconsider q as R_eal by XXREAL_0:def 1;
        inf A <= z2 by A4,A18,XXREAL_1:2;
        then consider 1o,1ra being Real such that
A20:    1o= inf A & 1ra = z2 and
        1o <= 1ra by A6;
        1o< 1ra by A4,A18,A20,XXREAL_1:2;
        then
A21:    d < q by A1,A6,A9,A19,A20,XREAL_1:68;
        z2 <= sup A by A4,A18,XXREAL_1:2;
        then consider 2o,2r being Real such that
A22:    2o= z2 & 2r = sup A and
A23:    2o <= 2r by A7;
        x * 2o <= x * 2r by A1,A23,XREAL_1:64;
        hence thesis by A7,A10,A19,A22,A21,XXREAL_1:2;
      end;
      then x ** A = ].d,g.] by A11;
      hence thesis by A2,A9,A10,A8,MEASURE6:9,13;
    end;
    hence thesis;
  end;
  A = ].inf A,b.] by A4;
  then A is left_open_interval by MEASURE5:def 5;
  then B is left_open_interval by A1,A2,Th11;
  hence thesis by A5,MEASURE6:19;
end;
