
theorem Th12:
  for A being non empty Interval, x being Real st x < 0 holds A is
  left_open_interval implies x ** A is right_open_interval
proof
  let A be non empty Interval;
  let x be Real;
  assume
A1: x < 0;
  assume A is left_open_interval;
  then consider a being R_eal,b being Real such that
A2: A = ].a,b.] by MEASURE5:def 5;
A3: a < b by A2,XXREAL_1:26;
  reconsider b as R_eal by XXREAL_0:def 1;
  now
    per cases by A3,Th5;
    case
      a = -infty & b = -infty;
      hence thesis;
    end;
    case
A4:   a = -infty & b in REAL;
      consider s being Real such that
A5:   s = b;
      x * s is R_eal by XXREAL_0:def 1;
      then consider d being R_eal such that
A6:   d = x * s;
      consider c being R_eal such that
A7:   c = +infty;
A8:   [.d,c.[ c= x ** A
      proof
        let q be object;
        assume
A9:     q in [.d,c.[;
        then reconsider q as Element of REAL by A6,XREAL_0:def 1;
        consider q2 being Real such that
A10:    q2 = q / x;
        reconsider q2 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
A11:    q1 = q;
A12:    q2 in A
        proof
          q2 is R_eal by XXREAL_0:def 1;
          then consider q3 being R_eal such that
A13:      q3 = q2;
A14:      q3 <= b
          proof
            d <= q1 & x * q2 = q by A1,A9,A11,A10,XCMPLX_1:87,XXREAL_1:3;
            hence thesis by A1,A5,A6,A11,A13,XREAL_1:69;
          end;
          a < q3 by A4,A13,XXREAL_0:12;
          hence thesis by A2,A13,A14,XXREAL_1:2;
        end;
        q = x * q2 by A1,A10,XCMPLX_1:87;
        hence thesis by A12,MEMBER_1:193;
      end;
      x ** A c= [.d,c.[
      proof
        let q be object;
        assume
A15:    q in x ** A;
        then reconsider q as Element of REAL;
        consider z2 being Real such that
A16:    z2 in A and
A17:    q = x * z2 by A15,INTEGRA2:39;
        reconsider q as R_eal by XXREAL_0:def 1;
A18:    q < +infty by XXREAL_0:9;
        reconsider z2 as R_eal by XXREAL_0:def 1;
        z2 <= b by A2,A16,XXREAL_1:2;
        then consider r,o being Real such that
A19:    r = z2 & o = b and
A20:    r <= o;
        x * o <= x * r by A1,A20,XREAL_1:65;
        hence thesis by A5,A7,A6,A17,A19,A18,XXREAL_1:3;
      end;
      then x ** A = [.d,c.[ by A8;
      hence thesis by A6,MEASURE5:def 4;
    end;
    case
      a = -infty & b = +infty;
      hence thesis;
    end;
    case
A21:  a in REAL & b in REAL;
      then reconsider s = a, r = b as Real;
      x * s is R_eal by XXREAL_0:def 1;
      then consider d being R_eal such that
A22:  d = x * s;
      x * r is R_eal by XXREAL_0:def 1;
      then consider g being R_eal such that
A23:  g = x * r;
A24:  [.g,d.[ c= x ** A
      proof
        let q be object;
        assume
A25:    q in [.g,d.[;
        then reconsider q as Real by A23;
        consider q2 being Real such that
A26:    q2 = q / x;
        q is R_eal by XXREAL_0:def 1;
        then consider q1 being R_eal such that
A27:    q1 = q;
A28:    q1 < d by A25,A27,XXREAL_1:3;
A29:    q2 in A
        proof
          q2 is R_eal by XXREAL_0:def 1;
          then consider q3 being R_eal such that
A30:      q3 = q2;
A31:      q3 <= b
          proof
            g <= q1 by A25,A27,XXREAL_1:3;
            then consider p,o being Real such that
A32:        p = g & o = q1 and
A33:        p <= o by A23,A27;
            o/x <= p/x by A1,A33,XREAL_1:73;
            hence thesis by A1,A23,A27,A26,A30,A32,XCMPLX_1:89;
          end;
          x * q2 = q by A1,A26,XCMPLX_1:87;
          then a < q3 by A1,A22,A27,A28,A30,XREAL_1:65;
          hence thesis by A2,A30,A31,XXREAL_1:2;
        end;
        q = x * (q / x) by A1,XCMPLX_1:87;
        hence thesis by A26,A29,MEMBER_1:193;
      end;
      x ** A c= [.g,d.[
      proof
        let q be object;
        assume
A34:    q in x ** A;
        then reconsider q as Real;
        consider z2 being Real such that
A35:    z2 in A and
A36:    q = x * z2 by A34,INTEGRA2:39;
        reconsider z2 as R_eal by XXREAL_0:def 1;
        a <= z2 by A2,A35,XXREAL_1:2;
        then consider 1o,1ra being Real such that
A37:    1o= a & 1ra = z2 and
        1o <= 1ra by A21;
        1o< 1ra by A2,A35,A37,XXREAL_1:2;
        then
A38:    x * 1ra < x * 1o by A1,XREAL_1:69;
        z2 <= b by A2,A35,XXREAL_1:2;
        then consider 2o,2r being Real such that
A39:    2o= z2 & 2r = b and
A40:    2o <= 2r;
        x * 2o is R_eal & x * 2r is R_eal by XXREAL_0:def 1;
        then consider 2o1,2r1 being R_eal such that
A41:    2o1 = x * 2o & 2r1 = x * 2r;
        2r1 <= 2o1 by A1,A40,A41,XREAL_1:65;
        hence thesis by A22,A23,A36,A37,A39,A38,A41,XXREAL_1:3;
      end;
      then x ** A = [.g,d.[ by A24;
      hence thesis by A23,MEASURE5:def 4;
    end;
    case
      a in REAL & b = +infty;
      hence thesis;
    end;
    case
      a = +infty & b = +infty;
      hence thesis;
    end;
  end;
  hence thesis;
end;
