
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;
  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
    assume
A3: A = [.inf A,sup A.];
A4: 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
A5:   s = inf A and
A6:   t = sup A;
      inf B = x * s & sup B = x * t
      proof
        s <= t by A5,A6,XXREAL_2:40;
        then
A7:     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
A8:     d = x * s;
        x * t is R_eal by XXREAL_0:def 1;
        then consider g being R_eal such that
A9:     g = x * t;
A10:    [.d,g.] c= x ** A
        proof
          let q be object;
          assume
A11:      q in [.d,g.];
          then reconsider q as Real by A8,A9;
          set q2 = q / x;
          reconsider q1 = q as R_eal by XXREAL_0:def 1;
A12:      q2 in A
          proof
            reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
A14:        q3 <= sup A
            proof
              q1 <= g by A11,XXREAL_1:1;
              then consider p,o being Real such that
A15:          p = q1 & o = g and
A16:          p <= o by A9;
              p/x <= o/x by A1,A16,XREAL_1:72;
              hence thesis by A1,A6,A9,A15,XCMPLX_1:89;
            end;
            inf A <= q3
            proof
              d <= q1 & x * q2 = q by A1,A11,XCMPLX_1:87,XXREAL_1:1;
              hence thesis by A1,A5,A8,XREAL_1:68;
            end;
            hence thesis by A3,A14,XXREAL_1:1;
          end;
          q = x * q2 by A1,XCMPLX_1:87;
          hence thesis by A12,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;
          inf A <= z2 by A3,A18,XXREAL_1:1;
          then consider 1o,1ra being Real such that
A20:      1o= inf A & 1ra = z2 and
A21:      1o <= 1ra by A5;
A22:      x * 1o <= x * 1ra by A1,A21,XREAL_1:64;
          z2 <= sup A by A3,A18,XXREAL_1:1;
          then consider 2o,2r being Real such that
A23:      2o= z2 & 2r = sup A and
A24:      2o <= 2r by A6;
          x * 2o is R_eal & x * 2r is R_eal by XXREAL_0:def 1;
          then consider 2o1,2r1 being R_eal such that
A25:      2o1 = x * 2o & 2r1 = x * 2r;
          2o1 <= 2r1 by A1,A24,A25,XREAL_1:64;
          hence thesis by A5,A6,A8,A9,A19,A20,A23,A22,A25,XXREAL_1:1;
        end;
        then x ** A = [.d,g.] by A10;
        hence thesis by A2,A8,A9,A7,MEASURE6:10,14;
      end;
      hence thesis;
    end;
    inf A <= sup A by XXREAL_2:40;
    then inf A in A & sup A in A by A3,XXREAL_1:1;
    then A is closed_interval by A3,MEASURE5:def 3;
    then x ** A is closed_interval by A1,Th8;
    hence thesis by A2,A4,MEASURE6:17;
  end;
  hence thesis;
end;
