let A be Interval; :: thesis: for x being Real holds
( A is left_open_interval iff x ++ A is left_open_interval )
let x be Real; :: thesis: ( A is left_open_interval iff x ++ A is left_open_interval )
A1: for B being Interval
for y being Real st B is left_open_interval holds
y ++ B is left_open_interval

proof

let B be Interval; :: thesis: for y being Real st B is left_open_interval holds
y ++ B is left_open_interval
let y be Real; :: thesis: ( B is left_open_interval implies y ++ B is left_open_interval )
reconsider y = y as Element of REAL by XREAL_0:def 1;
reconsider z = y as R_eal by XXREAL_0:def 1;
assume B is left_open_interval ; :: thesis: y ++ B is left_open_interval
then consider a being R_eal, b1 being Real such that
A2: B = ].a,b1.] by MEASURE5:def 5;
reconsider b = b1 as R_eal by XXREAL_0:def 1;
set s = z + a;
set t = z + b;
y ++ B = ].(z + a),(z + b).]

proof

thus y ++ B c= ].(z + a),(z + b).] :: according to XBOOLE_0:def 10 :: thesis: ].(z + a),(z + b).] c= y ++ B

proof

let c be object ; :: according to TARSKI:def 3 :: thesis: ( not c in y ++ B or c in ].(z + a),(z + b).] )
assume A3: c in y ++ B ; :: thesis: c in ].(z + a),(z + b).]
then reconsider c = c as Element of REAL ;
consider d being Real such that
A4: d in B and
A5: c = y + d by A3, Lm1;
reconsider d1 = d as R_eal by XXREAL_0:def 1;
a < d1 by A2, A4, XXREAL_1:2;
then A6: z + a < z + d1 by XXREAL_3:43;
d1 <= b by A2, A4, XXREAL_1:2;
then A7: z + d1 <= z + b by XXREAL_3:36;
z + d1 = c by A5, SUPINF_2:1;
hence c in ].(z + a),(z + b).] by A6, A7, XXREAL_1:2; :: thesis: verum

end;

let c be object ; :: according to TARSKI:def 3 :: thesis: ( not c in ].(z + a),(z + b).] or c in y ++ B )
assume A8: c in ].(z + a),(z + b).] ; :: thesis: c in y ++ B
then reconsider c = c as Real ;
reconsider c1 = c as R_eal by XXREAL_0:def 1;
A9: c = y + (c - y) ;
c1 <= z + b by A8, XXREAL_1:2;
then c1 - z <= (b + z) - z by XXREAL_3:36;
then A10: c1 - z <= b by XXREAL_3:22;
z + a < c1 by A8, XXREAL_1:2;
then (a + z) - z < c1 - z by XXREAL_3:43;
then A11: a < c1 - z by XXREAL_3:22;
c1 - z = c - y by SUPINF_2:3;
then c - y in B by A2, A11, A10;
hence c in y ++ B by A9, Lm1; :: thesis: verum

end;

hence y ++ B is left_open_interval by MEASURE5:def 5; :: thesis: verum

end;

( x ++ A is left_open_interval implies A is left_open_interval )

proof

reconsider y = - x as Real ;
assume A12: x ++ A is left_open_interval ; :: thesis: A is left_open_interval
then reconsider B = x ++ A as Interval ;
y ++ B = A by Th23;
hence A is left_open_interval by A1, A12; :: thesis: verum

end;

hence ( A is left_open_interval iff x ++ A is left_open_interval ) by A1; :: thesis: verum