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_1:28;
then A6: ( inf A = p & sup A = q & inf B = r & sup B = s ) by A1, A2, MEASURE6:8, MEASURE6:9, MEASURE6:13, MEASURE6:12;
now :: thesis: not s <= p
assume A7: s <= p ; :: thesis: contradiction
then ( not s in A & not s in B ) by A1, A2, XXREAL_1:2, XXREAL_1:4;
then A8: not s in A \/ B by XBOOLE_0:def 3;
A9: ( inf B < inf A & sup B < sup A ) by A6, A7, A1, A2, XXREAL_1:26, XXREAL_1:28, 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) < s & s < sup (A \/ B) ) by A5, A6, A9, XXREAL_0:def 9, XXREAL_0:def 10;
hence contradiction by A8, A4, XXREAL_2:83; :: thesis: verum
end;
then A10: q <= r by A1, A2, A3, Th12;
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_1:2, XXREAL_1:4;
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, A12, A1, A2, XXREAL_1:26, XXREAL_1:28, 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_1:171; :: thesis: verum