let a be Real; :: thesis: [.a,+infty.[ = {a} \/ ].a,+infty.[
thus [.a,+infty.[ c= {a} \/ ].a,+infty.[ :: according to XBOOLE_0:def 10 :: thesis: {a} \/ ].a,+infty.[ c= [.a,+infty.[

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in [.a,+infty.[ or x in {a} \/ ].a,+infty.[ )
assume A1: x in [.a,+infty.[ ; :: thesis: x in {a} \/ ].a,+infty.[
then reconsider x = x as Real ;
A2: x >= a by A1, XXREAL_1:236;

per cases ( x = a or x > a ) by A2, XXREAL_0:1;

suppose x = a ; :: thesis: x in {a} \/ ].a,+infty.[

then x in {a} by TARSKI:def 1;
hence x in {a} \/ ].a,+infty.[ by XBOOLE_0:def 3; :: thesis: verum

end;

suppose x > a ; :: thesis: x in {a} \/ ].a,+infty.[

then x in ].a,+infty.[ by XXREAL_1:235;
hence x in {a} \/ ].a,+infty.[ by XBOOLE_0:def 3; :: thesis: verum

end;

end;

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {a} \/ ].a,+infty.[ or x in [.a,+infty.[ )
assume A3: x in {a} \/ ].a,+infty.[ ; :: thesis: x in [.a,+infty.[
then reconsider x = x as Real ;
( x in {a} or x in ].a,+infty.[ ) by A3, XBOOLE_0:def 3;
then ( x = a or x > a ) by TARSKI:def 1, XXREAL_1:235;
hence x in [.a,+infty.[ by XXREAL_1:236; :: thesis: verum