
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 & B is open_interval
    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 & B is open_interval
      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;
          set q2 = q / x;
          q is R_eal by XXREAL_0:def 1;
          then consider q1 being R_eal such that
A12:      q1 = q;
A13:      q1 < g by A11,A12,MEASURE5:def 1;
A14:      q2 in A
          proof
            reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
A16:        q3 < sup A
            proof
              consider p,o being Real such that
A17:          p = q1 & o = g and
              p <= o by A9,A12,A13;
              p/x < o/x by A1,A13,A17,XREAL_1:74;
              hence thesis by A1,A6,A9,A12,A17,XCMPLX_1:89;
            end;
            d < q1 & x * q2 = q by A1,A11,A12,MEASURE5:def 1,XCMPLX_1:87;
            then inf A < q3 by A1,A5,A8,A12,XREAL_1:64;
            hence thesis by A3,A16,MEASURE5:def 1;
          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
A18:      q in x ** A;
          then reconsider q as Real;
          consider z2 being Real such that
A19:      z2 in A and
A20:      q = x * z2 by A18,INTEGRA2:39;
          reconsider z2 as R_eal by XXREAL_0:def 1;
          inf A <= z2 by A3,A19,MEASURE5:def 1;
          then consider 1o,1ra being Real such that
A21:      1o= inf A & 1ra = z2 and
          1o <= 1ra by A5;
          1o< 1ra by A3,A19,A21,MEASURE5:def 1;
          then
A22:      x * 1o < x * 1ra by A1,XREAL_1:68;
          z2 <= sup A by A3,A19,MEASURE5:def 1;
          then consider 2o,2r being Real such that
A23:      2o= z2 & 2r = sup A and
          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
A24:      2o1 = x * 2o & 2r1 = x * 2r;
          2o< 2r by A3,A19,A23,MEASURE5:def 1;
          then 2o1 < 2r1 by A1,A24,XREAL_1:68;
          hence thesis by A5,A6,A8,A9,A20,A21,A23,A22,A24,MEASURE5:def 1;
        end;
        then x ** A = ].d,g.[ by A10;
        hence thesis by A2,A8,A9,A7,MEASURE5:def 2,MEASURE6:8,12;
      end;
      hence thesis;
    end;
    A is open_interval by A3,MEASURE5:def 2;
    then x ** A is open_interval by A1,Th7;
    hence thesis by A2,A4,MEASURE6:16;
  end;
  hence thesis;
end;
