let U be non empty set ; :: thesis: for A being non empty IntervalSet of U holds U in A _\/_ (A ^)
let A be non empty IntervalSet of U; :: thesis: U in A _\/_ (A ^)
A1: U c= (A ``2) \/ ((A ``1) `)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in U or x in (A ``2) \/ ((A ``1) `) )
assume A2: x in U ; :: thesis: x in (A ``2) \/ ((A ``1) `)
A ``1 c= A ``2 by Th16;
then A3: (A ``2) ` c= (A ``1) ` by SUBSET_1:12;
( x in A ``2 or x in (A ``2) ` ) by A2, XBOOLE_0:def 5;
hence x in (A ``2) \/ ((A ``1) `) by A3, XBOOLE_0:def 3; :: thesis: verum
end;
A4: 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, A4;
then A _\/_ (A ^) = Inter (((A ``1) \/ ((A ``2) `)),((A ``2) \/ ((A ``1) `))) by Th17;
hence U in A _\/_ (A ^) by A1; :: thesis: verum