
theorem Th7:
  for A being non empty Interval, x being Real st x<>0 holds A is
  open_interval implies x ** A is open_interval
proof
  let A be non empty Interval;
  let x be Real;
  assume
A1: x <> 0;
  assume
A2: A is open_interval;
  then consider a,b being R_eal such that
A3: A = ].a,b.[ by MEASURE5:def 2;
A4: a < b by A3,XXREAL_1:28;
  now
    per cases;
    case
A5:   x < 0;
      now
        per cases by A4,Th5;
        case
          a = -infty & b = -infty;
          then ].a,b.[ = {};
          then x ** A = {} by A3;
          hence thesis;
        end;
        case
A6:       a = -infty & b in REAL;
          then reconsider s = b as Real;
          set c = +infty;
          x * s is R_eal by XXREAL_0:def 1;
          then consider d being R_eal such that
A7:       d = x * s;
A8:       ].d,c.[ c= x ** A
          proof
            let q be object;
            assume
A9:        q in ].d,c.[;
            then reconsider q as Real;
            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:        d < q1 by A9,MEASURE5:def 1;
A11:        q2 in A
            proof
              reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
              x * q2 = q by A1,XCMPLX_1:87;
              then
A12:          q3 < b by A5,A7,A10,XREAL_1:65;
              a < q3 by A6,XXREAL_0:12;
              hence thesis by A3,A12,MEASURE5:def 1;
            end;
            q = x * (q / x) by A1,XCMPLX_1:87;
            hence thesis by A11,MEMBER_1:193;
          end;
          x ** A c= ].d,c.[
          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:        q < +infty by XXREAL_0:9;
            reconsider z2 as R_eal by XXREAL_0:def 1;
            z2 < b by A3,A14,MEASURE5:def 1;
            then consider r,o being Real such that
A17:        r = z2 & o = b and
            r <= o by A6;
            reconsider o1 = x * o, r1 = x * r as R_eal by XXREAL_0:def 1;
            r < o by A3,A14,A17,MEASURE5:def 1;
            then o1 < r1 by A5,XREAL_1:69;
            hence thesis by A7,A15,A17,A16,MEASURE5:def 1;
          end;
          then x ** A = ].d,c.[ by A8;
          hence thesis by MEASURE5:def 2;
        end;
        case
          a = -infty & b = +infty;
          hence thesis by A2,A3,A5,Th2,XXREAL_1:224;
        end;
        case
A18:      a in REAL & b in REAL;
          then reconsider s = a, r = b as Real;
          reconsider d = x * s, g = x * r as R_eal by XXREAL_0:def 1;
A19:      ].g,d.[ c= x ** A
          proof
            let q be object;
            assume
A20:        q in ].g,d.[;
            then reconsider q as Real;
            set q2 = q / x;
            reconsider q1 = q as R_eal by XXREAL_0:def 1;
A21:        q1 < d by A20,MEASURE5:def 1;
A22:        g < q1 by A20,MEASURE5:def 1;
A23:        q2 in A
            proof
              reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
              x * q2 = q by A1,XCMPLX_1:87;
              then
A24:          a < q3 by A5,A21,XREAL_1:65;
              q/x < (x * r)/x by A5,A22,XREAL_1:75;
              then q3 < b by A5,XCMPLX_1:89;
              hence thesis by A3,A24,MEASURE5:def 1;
            end;
            q = x * (q / x) by A1,XCMPLX_1:87;
            hence thesis by A23,MEMBER_1:193;
          end;
          x ** A c= ].g,d.[
          proof
            let q be object;
            assume
A25:        q in x ** A;
            then reconsider q as Real;
            consider z2 being Real such that
A26:        z2 in A and
A27:        q = x * z2 by A25,INTEGRA2:39;
            reconsider z2 as R_eal by XXREAL_0:def 1;
            a < z2 by A3,A26,MEASURE5:def 1;
            then consider 1o,1ra being Real such that
A28:        1o= a & 1ra = z2 and
            1o <= 1ra by A18;
            z2 < b by A3,A26,MEASURE5:def 1;
            then consider 2o,2r being Real such that
A29:        2o= z2 & 2r = b and
            2o <= 2r by A18;
            reconsider 1o1 = x * 1o, 1r1 = x * 1ra, 2o1 = x * 2o, 2r1 = x * 2r
            as R_eal by XXREAL_0:def 1;
            2o< 2r by A3,A26,A29,MEASURE5:def 1;
            then
A30:        2r1 < 2o1 by A5,XREAL_1:69;
            1o < 1ra by A3,A26,A28,MEASURE5:def 1;
            then 1r1 < 1o1 by A5,XREAL_1:69;
            hence thesis by A27,A28,A29,A30,MEASURE5:def 1;
          end;
          then x ** A = ].g,d.[ by A19;
          hence thesis by MEASURE5:def 2;
        end;
        case
A31:      a in REAL & b = +infty;
          then reconsider s = a as Real;
          set c = -infty;
          reconsider d = x * s as R_eal by XXREAL_0:def 1;
A32:      ].c,d.[ c= x ** A
          proof
            let q be object;
            assume
A33:        q in ].c,d.[;
            then reconsider q as Real;
            reconsider q2 = q / x as Element of REAL by XREAL_0:def 1;
            reconsider q1 = q as R_eal by XXREAL_0:def 1;
A34:        q = x * (q / x) by A1,XCMPLX_1:87;
            q2 in A
            proof
              reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
              q1 <= d by A33,MEASURE5:def 1;
              then x * q2 < x * s by A33,A34,MEASURE5:def 1;
              then
A35:          a < q3 by A5,XREAL_1:65;
              q3 < b by A31,XXREAL_0:9;
              hence thesis by A3,A35,MEASURE5:def 1;
            end;
            hence thesis by A34,MEMBER_1:193;
          end;
          x ** A c= ].c,d.[
          proof
            let q be object;
            assume
A36:        q in x ** A;
            then reconsider q as Element of REAL;
            consider z2 being Real such that
A37:        z2 in A and
A38:        q = x * z2 by A36,INTEGRA2:39;
            reconsider z2,q as R_eal by XXREAL_0:def 1;
            a < z2 by A3,A37,MEASURE5:def 1;
            then consider o,r being Real such that
A39:        o = a & r = z2 and
            o <= r by A31;
            reconsider o1 = x * o, r1 = x * r as R_eal by XXREAL_0:def 1;
A40:        -infty < q by XXREAL_0:12;
            o < r by A3,A37,A39,MEASURE5:def 1;
            then r1 < o1 by A5,XREAL_1:69;
            hence thesis by A38,A39,A40,MEASURE5:def 1;
          end;
          then x ** A = ].c,d.[ by A32;
          hence thesis by MEASURE5:def 2;
        end;
        case
          a = +infty & b = +infty;
          then ].a,b.[ = {};
          then x ** A = {} by A3;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
    case
      x = 0;
      hence thesis by A1;
    end;
    case
A41:  0 < x;
      now
        per cases by A4,Th5;
        case
          a = -infty & b = -infty;
          then ].a,b.[ = {};
          then x ** A = {} by A3;
          hence thesis;
        end;
        case
A42:      a = -infty & b in REAL;
          then reconsider s = b as Real;
          set c = -infty;
          reconsider d = x * s as R_eal by XXREAL_0:def 1;
A43:      ].c,d.[ c= x ** A
          proof
            let q be object;
            assume
A44:        q in ].c,d.[;
            then reconsider q as Real;
            reconsider q2 = q / x as Element of REAL by XREAL_0:def 1;
            reconsider q1 = q as R_eal by XXREAL_0:def 1;
A45:        q1 < d by A44,MEASURE5:def 1;
A46:        q2 in A
            proof
              reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
              x * q2 = q by A1,XCMPLX_1:87;
              then
A47:          q3 < b by A41,A45,XREAL_1:64;
              a < q3 by A42,XXREAL_0:12;
              hence thesis by A3,A47,MEASURE5:def 1;
            end;
            q = x * (q / x) by A1,XCMPLX_1:87;
            hence thesis by A46,MEMBER_1:193;
          end;
          x ** A c= ].c,d.[
          proof
            let q be object;
            assume
A48:        q in x ** A;
            then reconsider q as Element of REAL;
            consider z2 being Real such that
A49:        z2 in A and
A50:        q = x * z2 by A48,INTEGRA2:39;
            reconsider z2,q as R_eal by XXREAL_0:def 1;
            z2 < b by A3,A49,MEASURE5:def 1;
            then consider r,o being Real such that
A51:        r = z2 & o = b and
            r <= o by A42;
            reconsider o1 = x * o, r1 = x * r as R_eal by XXREAL_0:def 1;
A52:        -infty < q by XXREAL_0:12;
            r < o by A3,A49,A51,MEASURE5:def 1;
            then r1 < o1 by A41,XREAL_1:68;
            hence thesis by A50,A51,A52,MEASURE5:def 1;
          end;
          then x ** A = ].c,d.[ by A43;
          hence thesis by MEASURE5:def 2;
        end;
        case
          a = -infty & b = +infty;
          hence thesis by A2,A3,A41,Th2,XXREAL_1:224;
        end;
        case
A53:      a in REAL & b in REAL;
          then reconsider s = a, r = b as Real;
          reconsider d = x * s as R_eal by XXREAL_0:def 1;
          reconsider g = x * r as R_eal by XXREAL_0:def 1;
A54:      ].d,g.[ c= x ** A
          proof
            let q be object;
            assume
A55:        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
A56:        q1 = q;
A57:        q1 < g by A55,A56,MEASURE5:def 1;
A58:        d < q1 by A55,A56,MEASURE5:def 1;
A59:        q2 in A
            proof
              reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
              x * q2 = q by A1,XCMPLX_1:87;
              then
A60:          a < q3 by A41,A56,A58,XREAL_1:64;
              q/x < (x * r)/x by A41,A56,A57,XREAL_1:74;
              then q3 < b by A41,XCMPLX_1:89;
              hence thesis by A3,A60,MEASURE5:def 1;
            end;
            q = x * (q / x) by A1,XCMPLX_1:87;
            hence thesis by A59,MEMBER_1:193;
          end;
          x ** A c= ].d,g.[
          proof
            let q be object;
            assume
A61:        q in x ** A;
            then reconsider q as Real;
            consider z2 being Real such that
A62:        z2 in A and
A63:        q = x * z2 by A61,INTEGRA2:39;
            reconsider z2 as R_eal by XXREAL_0:def 1;
            z2 < b by A3,A62,MEASURE5:def 1;
            then consider 2o,2r being Real such that
A64:        2o= z2 & 2r = b and
            2o <= 2r by A53;
            reconsider 2o1 = x * 2o, 2r1 = x * 2r as R_eal by XXREAL_0:def 1;
            2o< 2r by A3,A62,A64,MEASURE5:def 1;
            then
A65:        2o1 < 2r1 by A41,XREAL_1:68;
            a < z2 by A3,A62,MEASURE5:def 1;
            then consider 1o,1ra being Real such that
A66:        1o= a & 1ra = z2 and
            1o <= 1ra by A53;
            reconsider 1o1 = x * 1o, 1r1 = x * 1ra as R_eal by XXREAL_0:def 1;
            1o< 1ra by A3,A62,A66,MEASURE5:def 1;
            then 1o1 < 1r1 by A41,XREAL_1:68;
            hence thesis by A63,A66,A64,A65,MEASURE5:def 1;
          end;
          then x ** A = ].d,g.[ by A54;
          hence thesis by MEASURE5:def 2;
        end;
        case
A67:      a in REAL & b = +infty;
          then reconsider s = a as Element of REAL;
          set c = +infty;
          reconsider d = x * s as R_eal by XXREAL_0:def 1;
A68:      x ** A c= ].d,c.[
          proof
            let q be object;
            assume
A69:        q in x ** A;
            then reconsider q as Element of REAL;
            consider z2 being Real such that
A70:        z2 in A and
A71:        q = x * z2 by A69,INTEGRA2:39;
            reconsider q as R_eal by XXREAL_0:def 1;
A72:        q < +infty by XXREAL_0:9;
            reconsider z2 as R_eal by XXREAL_0:def 1;
            a < z2 by A3,A70,MEASURE5:def 1;
            then consider o,r being Real such that
A73:        o = a & r = z2 and
            o <= r by A67;
            reconsider o1 = x * o, r1 = x * r as R_eal by XXREAL_0:def 1;
            o < r by A3,A70,A73,MEASURE5:def 1;
            then o1 < r1 by A41,XREAL_1:68;
            hence thesis by A71,A73,A72,MEASURE5:def 1;
          end;
          ].d,c.[ c= x ** A
          proof
            let q be object;
            assume
A74:        q in ].d,c.[;
            then reconsider q as Real;
            reconsider q2 = q / x as Element of REAL by XREAL_0:def 1;
            reconsider q1 = q as R_eal by XXREAL_0:def 1;
A75:        q = x * (q / x) by A1,XCMPLX_1:87;
A76:        d < q1 by A74,MEASURE5:def 1;
            q2 in A
            proof
              reconsider q3 = q2 as R_eal by XXREAL_0:def 1;
              a < q3 & q3 < b by A41,A67,A76,A75,XREAL_1:64,XXREAL_0:9;
              hence thesis by A3,MEASURE5:def 1;
            end;
            hence thesis by A75,MEMBER_1:193;
          end;
          then x ** A = ].d,c.[ by A68;
          hence thesis by MEASURE5:def 2;
        end;
        case
          a = +infty & b = +infty;
          then ].a,b.[ = {};
          then x ** A = {} by A3;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
