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 ^)
A1: (A ``1) /\ ((A ``2) `) c= {}
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (A ``1) /\ ((A ``2) `) or x in {} )
A2: A ``1 c= A ``2 by Th16;
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 A2, XBOOLE_0:def 5; :: thesis: verum
end;
A3: A ^ = Inter (((A ``2) `),((A ``1) `)) by Th45;
A ^ = Inter (((A ^) ``1),((A ^) ``2)) by Th15;
then ( (A ^) ``1 = (A ``2) ` & (A ^) ``2 = (A ``1) ` ) by Th6, A3;
then A4: A _/\_ (A ^) = Inter (((A ``1) /\ ((A ``2) `)),((A ``2) /\ ((A ``1) `))) by Th18;
( (A ``1) /\ ((A ``2) `) c= {} & {} c= (A ``2) /\ ((A ``1) `) ) by A1;
hence {} in A _/\_ (A ^) by A4, Th1; :: thesis: verum