let A be non empty Interval; :: thesis: for x being Real st 0 < x & A is left_open_interval holds
x ** A is left_open_interval
let x be Real; :: thesis: ( 0 < x & A is left_open_interval implies x ** A is left_open_interval )
assume A1: 0 < x ; :: thesis: ( not A is left_open_interval or x ** A is left_open_interval )
assume A is left_open_interval ; :: thesis: x ** 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 = b 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 = -infty & b in REAL ) ; :: thesis: x ** A is left_open_interval
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 ; :: according to TARSKI:def 3 :: thesis: ( not q in ].c,d.] or q in x ** A )
assume A9: q in ].c,d.] ; :: thesis: q in x ** A
then reconsider q = 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 q3 <= b by A1, A5, A6, XREAL_1:68; :: thesis: verum
end;
a < q3 by A4, XXREAL_0:12;
hence q2 in A by A2, A12, XXREAL_1:2; :: thesis: verum
end;
q = x * (q / x) by A1, XCMPLX_1:87;
hence q in x ** A by A10, 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 A13: q in x ** A ; :: thesis: q in ].c,d.]
then reconsider q = 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 = q as R_eal by XXREAL_0:def 1;
A16: -infty < q by XXREAL_0:12;
reconsider z2 = 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 q in ].c,d.] by A5, A7, A6, A15, A17, A16, XXREAL_1:2; :: thesis: verum
end;
then x ** A = ].c,d.] by A8;
hence x ** A is left_open_interval by A6, MEASURE5:def 5; :: thesis: verum
end;
case A19: ( a in REAL & b in REAL ) ; :: thesis: x ** A is left_open_interval
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 ; :: according to TARSKI:def 3 :: thesis: ( not q in ].d,g.] or q in x ** A )
assume A24: q in ].d,g.] ; :: thesis: q in x ** A
then reconsider q = q as Real by A22;
set q2 = q / x;
reconsider q1 = q as R_eal by XXREAL_0:def 1;
A25: q / x in A
proof
reconsider q3 = q / x 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 q3 <= b by A1, A21, A22, A28, XCMPLX_1:89; :: thesis: verum
end;
( d < q1 & x * (q / x) = q ) by A1, A24, XCMPLX_1:87, XXREAL_1:2;
then a < q3 by A1, A20, XREAL_1:64;
hence q / x in A by A2, A27, XXREAL_1:2; :: thesis: verum
end;
q = x * (q / x) by A1, XCMPLX_1:87;
hence q in x ** A by A25, MEMBER_1:193; :: thesis: verum
end;
x ** A c= ].d,g.]
proof
let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in x ** A or q in ].d,g.] )
assume A30: q in x ** A ; :: thesis: q in ].d,g.]
then reconsider q = q as Real ;
consider z2 being Real such that
A31: z2 in A and
A32: q = x * z2 by A30, INTEGRA2:39;
reconsider z2 = 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 q in ].d,g.] by A21, A20, A22, A32, A33, A35, A34, A37, XXREAL_1:2; :: thesis: verum
end;
then x ** A = ].d,g.] by A23;
hence x ** A is left_open_interval by A22, MEASURE5:def 5; :: thesis: verum
end;
end;
end;
hence x ** A is left_open_interval ; :: thesis: verum