let A be Interval; :: thesis: for x being Real holds
( A is open_interval iff x ++ A is open_interval )
let x be Real; :: thesis: ( A is open_interval iff x ++ A is open_interval )
reconsider y = - x as Element of REAL by XREAL_0:def 1;
A1: for B being Interval
for y being Real st B is open_interval holds
y ++ B is open_interval

proof

let B be Interval; :: thesis: for y being Real st B is open_interval holds
y ++ B is open_interval
let y be Real; :: thesis: ( B is open_interval implies y ++ B is 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 open_interval ; :: thesis: y ++ B is open_interval
then consider a, b being R_eal such that
A2: B = ].a,b.[ by MEASURE5:def 2;
reconsider s = z + a, t = z + b as R_eal ;
y ++ B = ].s,t.[

proof

thus y ++ B c= ].s,t.[ :: according to XBOOLE_0:def 10 :: thesis: ].s,t.[ c= y ++ B

proof

let c be object ; :: according to TARSKI:def 3 :: thesis: ( not c in y ++ B or c in ].s,t.[ )
assume A3: c in y ++ B ; :: thesis: c in ].s,t.[
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, MEASURE5:def 1;
then A6: s < z + d1 by XXREAL_3:43;
d1 < b by A2, A4, MEASURE5:def 1;
then A7: z + d1 < t by XXREAL_3:43;
z + d1 = c by A5, SUPINF_2:1;
hence c in ].s,t.[ by A6, A7; :: thesis: verum

end;

let c be object ; :: according to TARSKI:def 3 :: thesis: ( not c in ].s,t.[ or c in y ++ B )
assume A8: c in ].s,t.[ ; :: thesis: c in y ++ B
then reconsider c = c as Element of REAL ;
reconsider c1 = c as R_eal by XXREAL_0:def 1;
A9: c = y + (c - y) ;
c1 < z + b by A8, MEASURE5:def 1;
then c1 - z < (b + z) - z by XXREAL_3:43;
then A10: c1 - z < b by XXREAL_3:22;
z + a < c1 by A8, MEASURE5:def 1;
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 open_interval by MEASURE5:def 2; :: thesis: verum

end;

hence ( A is open_interval implies x ++ A is open_interval ) ; :: thesis: ( x ++ A is open_interval implies A is open_interval )
assume A12: x ++ A is open_interval ; :: thesis: A is open_interval
then reconsider B = x ++ A as Interval ;
y ++ B = A by Th23;
hence A is open_interval by A1, A12; :: thesis: verum