let A be non empty Interval; :: thesis: for x being Real st x < 0 & A is right_open_interval holds
x ** A is left_open_interval
let x be Real; :: thesis: ( x < 0 & A is right_open_interval implies x ** A is left_open_interval )
assume A1: x < 0 ; :: thesis: ( not A is right_open_interval or x ** A is left_open_interval )
assume A is right_open_interval ; :: thesis: x ** A is left_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 = a as R_eal by XXREAL_0:def 1;
now :: thesis: ( ( a = -infty & b = -infty & x ** A is left_open_interval ) or ( a = -infty & b in REAL & x ** A is left_open_interval ) or ( a = -infty & b = +infty & x ** A is left_open_interval ) or ( a in REAL & b in REAL & x ** A is left_open_interval ) or ( a in REAL & b = +infty & x ** A is left_open_interval ) or ( a = +infty & b = +infty & x ** A is left_open_interval ) )
per cases ( ( a = -infty & b = -infty ) or ( a = -infty & b in REAL ) or ( a = -infty & b = +infty ) or ( a in REAL & b in REAL ) or ( a in REAL & b = +infty ) or ( a = +infty & b = +infty ) ) by A3, Th5;
case A4: ( a in REAL & b in REAL ) ; :: thesis: x ** A is left_open_interval
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 ; :: according to TARSKI:def 3 :: thesis: ( not q in ].g,d.] or q in x ** A )
assume A10: q in ].g,d.] ; :: thesis: q in x ** A
then reconsider q = 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: q / x in A
proof
reconsider q3 = q / x 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 q3 < b by A1, A5, A6, A11, A16, XCMPLX_1:89; :: thesis: verum
end;
a <= q3
proof
( q1 <= d & x * (q / x) = q ) by A1, A10, A11, XCMPLX_1:87, XXREAL_1:2;
hence a <= q3 by A1, A7, A8, A11, XREAL_1:69; :: thesis: verum
end;
hence q / x in A by A2, A15, XXREAL_1:3; :: thesis: verum
end;
q = x * (q / x) by A1, XCMPLX_1:87;
hence q in x ** A by A13, MEMBER_1:193; :: thesis: verum
end;
x ** A c= ].g,d.]
proof
let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in x ** A or q in ].g,d.] )
assume A17: q in x ** A ; :: thesis: q in ].g,d.]
then reconsider q = q as Real ;
consider z2 being Real such that
A18: z2 in A and
A19: q = x * z2 by A17, INTEGRA2:39;
reconsider z2 = 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 q in ].g,d.] by A7, A5, A8, A6, A19, A23, A20, A21, A22, XXREAL_1:2; :: thesis: verum
end;
then x ** A = ].g,d.] by A9;
hence x ** A is left_open_interval by A8, MEASURE5:def 5; :: thesis: verum
end;
case A25: ( a in REAL & b = +infty ) ; :: thesis: x ** A is left_open_interval
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 ; :: according to TARSKI:def 3 :: thesis: ( not q in ].c,d.] or q in x ** A )
assume A30: q in ].c,d.] ; :: thesis: q in x ** A
then reconsider q = 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 a <= q3 by A1, A26, A27, XREAL_1:69; :: thesis: verum
end;
q3 < b by A25, XXREAL_0:9;
hence q2 in A by A2, A32, XXREAL_1:3; :: thesis: verum
end;
q = x * q2 by A1, XCMPLX_1:87;
hence q in x ** A by A31, MEMBER_1:193; :: thesis: verum
end;
x ** A c= ].c,d.]
proof
let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in x ** A or q in ].c,d.] )
assume A33: q in x ** A ; :: thesis: q in ].c,d.]
then reconsider q = 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 = q as R_eal by XXREAL_0:def 1;
A36: -infty < q by XXREAL_0:12;
reconsider z2 = 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 q in ].c,d.] by A26, A28, A27, A35, A37, A36, XXREAL_1:2; :: thesis: verum
end;
then x ** A = ].c,d.] by A29;
hence x ** A is left_open_interval by A27, MEASURE5:def 5; :: thesis: verum
end;
end;
end;
hence x ** A is left_open_interval ; :: thesis: verum