
theorem Th10:
  for A being non empty Interval, x being Real st x < 0 holds A is
  right_open_interval implies x ** A is left_open_interval
proof
  let A be non empty Interval;
  let x be Real;
  assume
A1: x < 0;
  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:  ].g,d.] c= x ** A
      proof
        let q be object;
        assume
A10:    q in ].g,d.];
        then reconsider q as Real by A8;
        set q2 = q / x;
        q is R_eal by XXREAL_0:def 1;
        then consider q1 being R_eal such that
A11:    q1 = q;
A12:    g < q1 by A10,A11,XXREAL_1:2;
A13:    q2 in A
        proof
          reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
A15:      q3 < b
          proof
            consider p,o being Real such that
A16:        p = g & o = q1 and
            p <= o by A6,A11,A12;
            g < q1 by A10,A11,XXREAL_1:2;
            then o/x < p/x by A1,A16,XREAL_1:75;
            hence thesis by A1,A5,A6,A11,A16,XCMPLX_1:89;
          end;
          a <= q3
          proof
            q1 <= d & x * q2 = q by A1,A10,A11,XCMPLX_1:87,XXREAL_1:2;
            hence thesis by A1,A7,A8,A11,XREAL_1:69;
          end;
          hence thesis by A2,A15,XXREAL_1:3;
        end;
        q = x * (q / x) by A1,XCMPLX_1:87;
        hence thesis by A13,MEMBER_1:193;
      end;
      x ** A c= ].g,d.]
      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;
        z2 <= b by A2,A18,XXREAL_1:3;
        then consider 2o,2r being Real such that
A20:    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
A21:    2o1 = x * 2o & 2r1 = x * 2r;
        2o< 2r by A2,A18,A20,XXREAL_1:3;
        then
A22:    2r1 < 2o1 by A1,A21,XREAL_1:69;
        a <= z2 by A2,A18,XXREAL_1:3;
        then consider 1o,1ra being Real such that
A23:    1o= a & 1ra = z2 and
A24:    1o <= 1ra;
        x * 1ra <= x * 1o by A1,A24,XREAL_1:65;
        hence thesis by A7,A5,A8,A6,A19,A23,A20,A21,A22,XXREAL_1:2;
      end;
      then x ** A = ].g,d.] by A9;
      hence thesis by A8,MEASURE5:def 5;
    end;
    case
A25:  a in REAL & b = +infty;
      consider s being Real such that
A26:  s = a;
      x * s is R_eal by XXREAL_0:def 1;
      then consider d being R_eal such that
A27:  d = x * s;
      consider c being R_eal such that
A28:  c = -infty;
A29:  ].c,d.] c= x ** A
      proof
        let q be object;
        assume
A30:    q in ].c,d.];
        then reconsider q as Real by A27;
        reconsider q2 = q / x as Element of REAL by XREAL_0:def 1;
        reconsider q1 = q as R_eal by XXREAL_0:def 1;
A31:    q2 in A
        proof
          reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
A32:      a <= q3
          proof
            q1 <= d & x * q2 = q by A1,A30,XCMPLX_1:87,XXREAL_1:2;
            hence thesis by A1,A26,A27,XREAL_1:69;
          end;
          q3 < b by A25,XXREAL_0:9;
          hence thesis by A2,A32,XXREAL_1:3;
        end;
        q = x * q2 by A1,XCMPLX_1:87;
        hence thesis by A31,MEMBER_1:193;
      end;
      x ** A c= ].c,d.]
      proof
        let q be object;
        assume
A33:    q in x ** A;
        then reconsider q as Element of REAL;
        consider z2 being Real such that
A34:    z2 in A and
A35:    q = x * z2 by A33,INTEGRA2:39;
        reconsider q as R_eal by XXREAL_0:def 1;
A36:    -infty < q by XXREAL_0:12;
        reconsider z2 as R_eal by XXREAL_0:def 1;
        a <= z2 by A2,A34,XXREAL_1:3;
        then consider o,r being Real such that
A37:    o = a & r = z2 and
A38:    o <= r;
        x * r <= x * o by A1,A38,XREAL_1:65;
        hence thesis by A26,A28,A27,A35,A37,A36,XXREAL_1:2;
      end;
      then x ** A = ].c,d.] by A29;
      hence thesis by A27,MEASURE5:def 5;
    end;
    case
      a = +infty & b = +infty;
      hence thesis;
    end;
  end;
  hence thesis;
end;
