let A, B be non empty Interval; :: thesis: for p, q, r, s being R_eal st A = [.p,q.] & B = ].r,s.] & A misses B & A \/ B is Interval holds
( q = r & A \/ B = [.p,s.] )
let p, q, r, s be R_eal; :: thesis: ( A = [.p,q.] & B = ].r,s.] & A misses B & A \/ B is Interval implies ( q = r & A \/ B = [.p,s.] ) )
assume that
A1: A = [.p,q.] and
A2: B = ].r,s.] and
A3: A misses B and
A4: A \/ B is Interval ; :: thesis: ( q = r & A \/ B = [.p,s.] )
A5: ( p <= q & r < s ) by A1, A2, XXREAL_1:26, XXREAL_1:29;
then A6: ( inf A = p & sup A = q & inf B = r & sup B = s ) by A1, A2, MEASURE6:10, MEASURE6:14, MEASURE6:9, MEASURE6:13;

now :: thesis: not s < p

assume A7: s < p ; :: thesis: contradiction
then consider x being R_eal such that
A8: ( s < x & x < p & x in REAL ) by MEASURE5:2;
( not x in A & not x in B ) by A1, A2, A8, XXREAL_1:1, XXREAL_1:2;
then A9: not x in A \/ B by XBOOLE_0:def 3;
( inf (A \/ B) = min ((inf A),(inf B)) & sup (A \/ B) = max ((sup A),(sup B)) ) by XXREAL_2:9, XXREAL_2:10;
then ( inf (A \/ B) = inf B & sup (A \/ B) = sup A ) by A5, A6, A7, XXREAL_0:2, XXREAL_0:def 9, XXREAL_0:def 10;
then ( inf (A \/ B) < x & x < sup (A \/ B) ) by A5, A6, A8, XXREAL_0:2;
hence contradiction by A9, A4, XXREAL_2:83; :: thesis: verum

end;

then A10: q <= r by A1, A2, A3, Th6;

now :: thesis: not q < r

assume A11: q < r ; :: thesis: contradiction
then consider x being R_eal such that
A12: ( q < x & x < r & x in REAL ) by MEASURE5:2;
( not x in A & not x in B ) by A1, A2, A12, XXREAL_1:1, XXREAL_1:2;
then A13: not x in A \/ B by XBOOLE_0:def 3;
( min ((inf A),(inf B)) = inf A & max ((sup A),(sup B)) = sup B ) by A11, A6, A5, XXREAL_0:2, XXREAL_0:def 9, XXREAL_0:def 10;
then ( inf (A \/ B) = inf A & sup (A \/ B) = sup B ) by XXREAL_2:9, XXREAL_2:10;
then ( inf (A \/ B) < x & x < sup (A \/ B) ) by A6, A5, A12, XXREAL_0:2;
hence contradiction by A13, A4, XXREAL_2:83; :: thesis: verum

end;

hence q = r by A10, XXREAL_0:1; :: thesis: A \/ B = [.p,s.]
hence A \/ B = [.p,s.] by A1, A2, A5, XXREAL_1:167; :: thesis: verum