
theorem Th9:
  for A being non empty Interval, x being Real st 0 < x holds A is
  right_open_interval implies x ** A is right_open_interval
proof
  let A be non empty Interval;
  let x be Real;
  assume
A1: 0 < x;
  assume A is right_open_interval;
  then consider a being Real,b being R_eal such that
A2: A = [.a,b.[ by MEASURE5:def 4;
A3: a < b by A2,XXREAL_1:27;
  reconsider a as R_eal by XXREAL_0:def 1;
  now
    per cases by A3,Th5;
    case
      a = -infty & b = -infty;
      hence thesis;
    end;
    case
      a = -infty & b in REAL;
      hence thesis;
    end;
    case
      a = -infty & b = +infty;
      hence thesis;
    end;
    case
A4:   a in REAL & b in REAL;
      then consider r being Real such that
A5:   r = b;
      x * r is R_eal by XXREAL_0:def 1;
      then consider g being R_eal such that
A6:   g = x * r;
      consider s being Real such that
A7:   s = a;
      x * s is R_eal by XXREAL_0:def 1;
      then consider d being R_eal such that
A8:   d = x * s;
A9:  [.d,g.[ c= x ** A
      proof
        let q be object;
        assume
A10:    q in [.d,g.[;
        then reconsider q as Real by A8;
        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 < b
          proof
            q1 <= g by A10,XXREAL_1:3;
            then consider p,o being Real such that
A15:        p = q1 & o = g and
            p <= o by A6;
            p < o by A10,A15,XXREAL_1:3;
            then p/x < o/x by A1,XREAL_1:74;
            hence thesis by A1,A5,A6,A15,XCMPLX_1:89;
          end;
          a <= q3
          proof
            d <= q1 & x * q2 = q by A1,A10,XCMPLX_1:87,XXREAL_1:3;
            hence thesis by A1,A7,A8,XREAL_1:68;
          end;
          hence thesis by A2,A14,XXREAL_1:3;
        end;
        q = x * (q / x) by A1,XCMPLX_1:87;
        hence thesis by A12,MEMBER_1:193;
      end;
      x ** A c= [.d,g.[
      proof
        let q be object;
        assume
A16:    q in x ** A;
        then reconsider q as Real;
        consider z2 being Real such that
A17:    z2 in A and
A18:    q = x * z2 by A16,INTEGRA2:39;
        reconsider z2 as R_eal by XXREAL_0:def 1;
        z2 <= b by A2,A17,XXREAL_1:3;
        then consider 2o,2r being Real such that
A19:    2o= z2 & 2r = b and
        2o <= 2r by A4;
        x * 2o is R_eal & x * 2r is R_eal by XXREAL_0:def 1;
        then consider 2o1,2r1 being R_eal such that
A20:    2o1 = x * 2o & 2r1 = x * 2r;
        2o< 2r by A2,A17,A19,XXREAL_1:3;
        then
A21:    2o1 < 2r1 by A1,A20,XREAL_1:68;
        a <= z2 by A2,A17,XXREAL_1:3;
        then consider 1o,1ra being Real such that
A22:    1o= a & 1ra = z2 and
A23:    1o <= 1ra;
        x * 1o <= x * 1ra by A1,A23,XREAL_1:64;
        hence thesis by A7,A5,A8,A6,A18,A22,A19,A20,A21,XXREAL_1:3;
      end;
      then x ** A = [.d,g.[ by A9;
      hence thesis by A8,MEASURE5:def 4;
    end;
    case
A24:  a in REAL & b = +infty;
      consider s being Real such that
A25:  s = a;
      x * s is R_eal by XXREAL_0:def 1;
      then consider d being R_eal such that
A26:  d = x * s;
      consider c being R_eal such that
A27:  c = +infty;
A28:  [.d,c.[ c= x ** A
      proof
        let q be object;
        assume
A29:    q in [.d,c.[;
        then reconsider q as Real by A26;
        reconsider q2 = q / x as Element of REAL by XREAL_0:def 1;
        q is R_eal by XXREAL_0:def 1;
        then consider q1 being R_eal such that
A30:    q1 = q;
A31:    q2 in A
        proof
          reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
A33:      a <= q3
          proof
            d <= q1 & x * q2 = q by A1,A29,A30,XCMPLX_1:87,XXREAL_1:3;
            hence thesis by A1,A25,A26,A30,XREAL_1:68;
          end;
          q3 < b by A24,XXREAL_0:9;
          hence thesis by A2,A33,XXREAL_1:3;
        end;
        q = x * (q / x) by A1,XCMPLX_1:87;
        hence thesis by A31,MEMBER_1:193;
      end;
      x ** A c= [.d,c.[
      proof
        let q be object;
        assume
A34:    q in x ** A;
        then reconsider q as Element of REAL;
        consider z2 being Real such that
A35:    z2 in A and
A36:    q = x * z2 by A34,INTEGRA2:39;
        reconsider q as R_eal by XXREAL_0:def 1;
A37:    q < +infty by XXREAL_0:9;
        reconsider z2 as R_eal by XXREAL_0:def 1;
        a <= z2 by A2,A35,XXREAL_1:3;
        then consider o,r being Real such that
A38:    o = a & r = z2 and
A39:    o <= r;
        x * o <= x * r by A1,A39,XREAL_1:64;
        hence thesis by A25,A27,A26,A36,A38,A37,XXREAL_1:3;
      end;
      then x ** A = [.d,c.[ by A28;
      hence thesis by A26,MEASURE5:def 4;
    end;
    case
      a = +infty & b = +infty;
      hence thesis;
    end;
  end;
  hence thesis;
end;
