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

proof

let B be Interval; :: thesis: for y being real number st B is open_interval holds
y ++ B is open_interval
let y be real number ; :: thesis: ( B is open_interval implies y ++ B is open_interval )
reconsider y = y as 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: ( a <= b & B = ].a,b.[ ) by MEASURE5:def 5;
reconsider s = z + a, t = z + b as R_eal ;
A3: s <= t by A2, XXREAL_3:38;
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 set ; :: according to TARSKI:def 3 :: thesis: ( not c in y ++ B or c in ].s,t.[ )
assume A4: c in y ++ B ; :: thesis: c in ].s,t.[
then reconsider c = c as Real ;
consider d being Real such that
A5: ( d in B & c = y + d ) by A4, Def6;
reconsider d1 = d as R_eal by XXREAL_0:def 1;
( a < d1 & d1 < b & d1 in REAL ) by A2, A5, MEASURE5:def 2;
then A6: ( s < z + d1 & z + d1 < t ) by XXREAL_3:47;
z + d1 = c by A5, SUPINF_2:1;
hence c in ].s,t.[ by A6, MEASURE5:def 2; :: thesis: verum

end;

let c be set ; :: according to TARSKI:def 3 :: thesis: ( not c in ].s,t.[ or c in y ++ B )
assume A7: c in ].s,t.[ ; :: thesis: c in y ++ B
then reconsider c = c as Real ;
reconsider c1 = c as R_eal by XXREAL_0:def 1;
( z + a < c1 & c1 < z + b ) by A7, MEASURE5:def 2;
then ( (a + z) - z < c1 - z & c1 - z < (b + z) - z ) by XXREAL_3:47;
then A8: ( a < c1 - z & c1 - z < b ) by XXREAL_3:22;
c1 - z = c - y by SUPINF_2:5;
then ( c - y in B & c = y + (c - y) ) by A2, A8, MEASURE5:def 2;
hence c in y ++ B by Def6; :: thesis: verum

end;

hence y ++ B is open_interval by A3, MEASURE5:def 5; :: 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 A9: x ++ A is open_interval ; :: thesis: A is open_interval
then reconsider B = x ++ A as Interval by MEASURE5:def 9;
reconsider y = - x as Real by XREAL_0:def 1;
y ++ B = A by Th59;
hence A is open_interval by A1, A9; :: thesis: verum