let A be Interval; :: thesis: for x being Real st 0 < x & A is right_open_interval holds
x ** A is right_open_interval
let x be Real; :: thesis: ( 0 < x & A is right_open_interval implies x ** A is right_open_interval )
assume A1: 0 < x ; :: thesis: ( not A is right_open_interval or x ** A is right_open_interval )
assume A is right_open_interval ; :: thesis: x ** A is right_open_interval
then consider a being real number , b being R_eal such that
A2: a <= b and
A3: A = [.a,b.[ by MEASURE5:def 7;
reconsider a = a as R_eal by XXREAL_0:def 1;
now
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 A2, Th6;
case A4: ( a in REAL & b in REAL ) ; :: thesis: x ** A is right_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 by A4;
A8: x * s <= x * r by A1, A2, A7, A5, XREAL_1:66;
x * s is R_eal by XXREAL_0:def 1;
then consider d being R_eal such that
A9: d = x * s ;
A10: [.d,g.[ c= x ** A
proof
let q be set ; :: according to TARSKI:def 3 :: thesis: ( not q in [.d,g.[ or q in x ** A )
assume A11: q in [.d,g.[ ; :: thesis: q in x ** A
then reconsider q = q as Real by A9, XREAL_0:def 1;
set q2 = q / x;
q is R_eal by XXREAL_0:def 1;
then consider q1 being R_eal such that
A12: q1 = q ;
A13: q / x in A
proof
q / x is R_eal by XXREAL_0:def 1;
then consider q3 being R_eal such that
A14: q3 = q / x ;
A15: q3 < b
proof
q1 <= g by A11, A12, XXREAL_1:3;
then consider p, o being Real such that
A16: ( p = q1 & o = g ) and
p <= o by A6, A12;
p < o by A11, A12, A16, XXREAL_1:3;
then p / x < o / x by A1, XREAL_1:76;
hence q3 < b by A1, A5, A6, A12, A14, A16, XCMPLX_1:90; :: thesis: verum
end;
a <= q3
proof
( d <= q1 & x * (q / x) = q ) by A1, A11, A12, XCMPLX_1:88, XXREAL_1:3;
hence a <= q3 by A1, A7, A9, A12, A14, XREAL_1:70; :: thesis: verum
end;
hence q / x in A by A3, A14, A15, XXREAL_1:3; :: thesis: verum
end;
q = x * (q / x) by A1, XCMPLX_1:88;
hence q in x ** A by A13, INTEGRA2:def 2; :: thesis: verum
end;
x ** A c= [.d,g.[
proof
let q be set ; :: according to TARSKI:def 3 :: thesis: ( not q in x ** A or q in [.d,g.[ )
assume A17: q in x ** A ; :: thesis: q in [.d,g.[
then reconsider q = q as Real ;
consider z2 being Real such that
A18: z2 in A and
A19: q = x * z2 by A17, INTEGRA2:def 2;
reconsider z2 = z2 as R_eal by XXREAL_0:def 1;
z2 <= b by A3, 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 A3, A18, A20, XXREAL_1:3;
then A22: 2o1 < 2r1 by A1, A21, XREAL_1:70;
a <= z2 by A3, A18, XXREAL_1:3;
then consider 1o, 1ra being Real such that
A23: ( 1o = a & 1ra = z2 ) and
A24: 1o <= 1ra by A4;
x * 1o <= x * 1ra by A1, A24, XREAL_1:66;
hence q in [.d,g.[ by A7, A5, A9, A6, A19, A23, A20, A21, A22, XXREAL_1:3; :: thesis: verum
end;
then x ** A = [.d,g.[ by A10, XBOOLE_0:def 10;
hence x ** A is right_open_interval by A9, A6, A8, MEASURE5:def 7; :: thesis: verum
end;
case A25: ( a in REAL & b = +infty ) ; :: thesis: x ** A is right_open_interval
then 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: [.d,c.[ c= x ** A
proof
let q be set ; :: according to TARSKI:def 3 :: thesis: ( not q in [.d,c.[ or q in x ** A )
assume A30: q in [.d,c.[ ; :: thesis: q in x ** A
then reconsider q = q as Real by A27, XREAL_0:def 1;
set q2 = q / x;
q is R_eal by XXREAL_0:def 1;
then consider q1 being R_eal such that
A31: q1 = q ;
A32: q / x in A
proof
q / x is R_eal by XXREAL_0:def 1;
then consider q3 being R_eal such that
A33: q3 = q / x ;
A34: a <= q3
proof
( d <= q1 & x * (q / x) = q ) by A1, A30, A31, XCMPLX_1:88, XXREAL_1:3;
hence a <= q3 by A1, A26, A27, A31, A33, XREAL_1:70; :: thesis: verum
end;
q3 < b by A25, A33, XXREAL_0:9;
hence q / x in A by A3, A33, A34, XXREAL_1:3; :: thesis: verum
end;
q = x * (q / x) by A1, XCMPLX_1:88;
hence q in x ** A by A32, INTEGRA2:def 2; :: thesis: verum
end;
A35: d <= c by A28, XXREAL_0:4;
x ** A c= [.d,c.[
proof
let q be set ; :: according to TARSKI:def 3 :: thesis: ( not q in x ** A or q in [.d,c.[ )
assume A36: q in x ** A ; :: thesis: q in [.d,c.[
then reconsider q = q as Real ;
consider z2 being Real such that
A37: z2 in A and
A38: q = x * z2 by A36, INTEGRA2:def 2;
reconsider q = q as R_eal by XXREAL_0:def 1;
A39: q < +infty by XXREAL_0:9;
reconsider z2 = z2 as R_eal by XXREAL_0:def 1;
a <= z2 by A3, A37, XXREAL_1:3;
then consider o, r being Real such that
A40: ( o = a & r = z2 ) and
A41: o <= r by A25;
x * o <= x * r by A1, A41, XREAL_1:66;
hence q in [.d,c.[ by A26, A28, A27, A38, A40, A39, XXREAL_1:3; :: thesis: verum
end;
then x ** A = [.d,c.[ by A29, XBOOLE_0:def 10;
hence x ** A is right_open_interval by A27, A35, MEASURE5:def 7; :: thesis: verum
end;
end;
end;
hence x ** A is right_open_interval ; :: thesis: verum