let U be non empty set ; :: thesis: for A being non empty IntervalSet of U holds {} in A _/\_ (A ^)
let A be non empty IntervalSet of U; :: thesis: {} in A _/\_ (A ^)
a0: (A ``1) /\ ((A ``2) `) c= {}

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (A ``1) /\ ((A ``2) `) or x in {} )
c1: A ``1 c= A ``2 by wazne2;
assume x in (A ``1) /\ ((A ``2) `) ; :: thesis: x in {}
then ( x in A ``1 & x in (A ``2) ` ) by XBOOLE_0:def 4;
hence x in {} by c1, XBOOLE_0:def 5; :: thesis: verum

end;

e1: A ^ = Inter (((A ``2) `),((A ``1) `)) by ThDop0;
A ^ = Inter (((A ^) ``1),((A ^) ``2)) by Wazne33;
then ( (A ^) ``1 = (A ``2) ` & (A ^) ``2 = (A ``1) ` ) by Pik, e1;
then a1: A _/\_ (A ^) = Inter (((A ``1) /\ ((A ``2) `)),((A ``2) /\ ((A ``1) `))) by Int;
( (A ``1) /\ ((A ``2) `) c= {} & {} c= (A ``2) /\ ((A ``1) `) ) by a0, XBOOLE_1:2;
hence {} in A _/\_ (A ^) by a1, Lemacik; :: thesis: verum