
theorem Th11:
  for A being non empty Interval, x being Real st 0 < x holds A is
  left_open_interval implies x ** A is left_open_interval
proof
  let A be non empty Interval;
  let x be Real;
  assume
A1: 0 < x;
  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:   ].c,d.] c= x ** A
      proof
        let q be object;
        assume
A9:     q in ].c,d.];
        then reconsider q as Real by A6;
        reconsider q2 = q / x as Element of REAL by XREAL_0:def 1;
        reconsider q1 = q as R_eal by XXREAL_0:def 1;
A10:    q2 in A
        proof
          reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
A12:      q3 <= b
          proof
            q1 <= d & x * q2 = q by A1,A9,XCMPLX_1:87,XXREAL_1:2;
            hence thesis by A1,A5,A6,XREAL_1:68;
          end;
          a < q3 by A4,XXREAL_0:12;
          hence thesis by A2,A12,XXREAL_1:2;
        end;
        q = x * (q / x) by A1,XCMPLX_1:87;
        hence thesis by A10,MEMBER_1:193;
      end;
      x ** A c= ].c,d.]
      proof
        let q be object;
        assume
A13:    q in x ** A;
        then reconsider q as Element of REAL;
        consider z2 being Real such that
A14:    z2 in A and
A15:    q = x * z2 by A13,INTEGRA2:39;
        reconsider q as R_eal by XXREAL_0:def 1;
A16:    -infty < q by XXREAL_0:12;
        reconsider z2 as R_eal by XXREAL_0:def 1;
        z2 <= b by A2,A14,XXREAL_1:2;
        then consider r,o being Real such that
A17:    r = z2 & o = b and
A18:    r <= o;
        x * r <= x * o by A1,A18,XREAL_1:64;
        hence thesis by A5,A7,A6,A15,A17,A16,XXREAL_1:2;
      end;
      then x ** A = ].c,d.] by A8;
      hence thesis by A6,MEASURE5:def 5;
    end;
    case
      a = -infty & b = +infty;
      hence thesis;
    end;
    case
A19:  a in REAL & b in REAL;
      then reconsider s = a as Real;
      x * s is R_eal by XXREAL_0:def 1;
      then consider d being R_eal such that
A20:  d = x * s;
      consider r being Real such that
A21:  r = b;
      x * r is R_eal by XXREAL_0:def 1;
      then consider g being R_eal such that
A22:  g = x * r;
A23:  ].d,g.] c= x ** A
      proof
        let q be object;
        assume
A24:    q in ].d,g.];
        then reconsider q as Real by A22;
        set q2 = q / x;
        reconsider q1 = q as R_eal by XXREAL_0:def 1;
A25:    q2 in A
        proof
          reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
A27:      q3 <= b
          proof
            q1 <= g by A24,XXREAL_1:2;
            then consider p,o being Real such that
A28:        p = q1 & o = g and
A29:        p <= o by A22;
            p/x <= o/x by A1,A29,XREAL_1:72;
            hence thesis by A1,A21,A22,A28,XCMPLX_1:89;
          end;
          d < q1 & x * q2 = q by A1,A24,XCMPLX_1:87,XXREAL_1:2;
          then a < q3 by A1,A20,XREAL_1:64;
          hence thesis by A2,A27,XXREAL_1:2;
        end;
        q = x * (q / x) by A1,XCMPLX_1:87;
        hence thesis by A25,MEMBER_1:193;
      end;
      x ** A c= ].d,g.]
      proof
        let q be object;
        assume
A30:    q in x ** A;
        then reconsider q as Real;
        consider z2 being Real such that
A31:    z2 in A and
A32:    q = x * z2 by A30,INTEGRA2:39;
        reconsider z2 as R_eal by XXREAL_0:def 1;
        a <= z2 by A2,A31,XXREAL_1:2;
        then consider 1o,1ra being Real such that
A33:    1o= a & 1ra = z2 and
        1o <= 1ra by A19;
        1o< 1ra by A2,A31,A33,XXREAL_1:2;
        then
A34:    x * 1o < x * 1ra by A1,XREAL_1:68;
        z2 <= b by A2,A31,XXREAL_1:2;
        then consider 2o,2r being Real such that
A35:    2o= z2 & 2r = b and
A36:    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
A37:    2o1 = x * 2o & 2r1 = x * 2r;
        2o1 <= 2r1 by A1,A36,A37,XREAL_1:64;
        hence thesis by A21,A20,A22,A32,A33,A35,A34,A37,XXREAL_1:2;
      end;
      then x ** A = ].d,g.] by A23;
      hence thesis by A22,MEASURE5:def 5;
    end;
    case
      a in REAL & b = +infty;
      hence thesis;
    end;
    case
      a = +infty & b = +infty;
      hence thesis;
    end;
  end;
  hence thesis;
end;
