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 & not ( p = s & A \/ B = [.r,q.[ ) 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 & not ( p = s & A \/ B = [.r,q.[ ) 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: ( ( p = s & A \/ B = [.r,q.[ ) or ( q = r & A \/ B = [.p,s.[ ) )
A5: ( p < q & r < s ) by A1, A2, XXREAL_1:27;
then A6: ( inf A = p & sup A = q & inf B = r & sup B = s ) by A1, A2, MEASURE6:11, MEASURE6:15;

A7: now :: thesis: not q < r

assume A8: q < r ; :: thesis: contradiction
then consider x being R_eal such that
A9: ( q < x & x < r & x in REAL ) by MEASURE5:2;
( not x in A & not x in B ) by A1, A2, A9, XXREAL_1:3;
then A10: not x in A \/ B by XBOOLE_0:def 3;
A11: ( inf A < inf B & sup A < sup B ) by A6, A8, A1, A2, XXREAL_1:27, XXREAL_0:2;
( 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 A & sup (A \/ B) = sup B ) by A11, XXREAL_0:def 9, XXREAL_0:def 10;
then ( inf (A \/ B) < x & x < sup (A \/ B) ) by A6, A9, A1, A2, XXREAL_1:27, XXREAL_0:2;
hence contradiction by A10, A4, XXREAL_2:83; :: thesis: verum

end;

A12: now :: thesis: not s < p

assume A13: s < p ; :: thesis: contradiction
then consider x being R_eal such that
A14: ( s < x & x < p & x in REAL ) by MEASURE5:2;
( not x in A & not x in B ) by A1, A2, A14, XXREAL_1:3;
then A15: not x in A \/ B by XBOOLE_0:def 3;
A16: ( inf B < inf A & sup B < sup A ) by A6, A13, A1, A2, XXREAL_1:27, XXREAL_0:2;
( 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 A16, XXREAL_0:def 9, XXREAL_0:def 10;
then ( inf (A \/ B) < x & x < sup (A \/ B) ) by A6, A14, A1, A2, XXREAL_1:27, XXREAL_0:2;
hence contradiction by A15, A4, XXREAL_2:83; :: thesis: verum

end;

( q <= r or s <= p ) by A1, A2, A3, Th8;
then ( q = r or s = p ) by A7, A12, XXREAL_0:1;
hence ( ( p = s & A \/ B = [.r,q.[ ) or ( q = r & A \/ B = [.p,s.[ ) ) by A1, A2, A5, XXREAL_1:168; :: thesis: verum