let I, J be Subset of REAL; :: thesis: ( I is open_interval & J is closed_interval & I meets J implies ex K, L being Interval st
( K misses L & I \ J = K \/ L ) )
assume A1: ( I is open_interval & J is closed_interval & I meets J ) ; :: thesis: ex K, L being Interval st
( K misses L & I \ J = K \/ L )
then consider p, q being R_eal such that
A2: I = ].p,q.[ ;
consider r, s being Real such that
A3: J = [.r,s.] by A1;
( I <> {} & J <> {} ) by A1;
then A4: ( p < q & r <= s & p < s & r < q ) by A1, A2, A3, XXREAL_1:28, XXREAL_1:29, XXREAL_1:89, XXREAL_1:93;
reconsider K = ].p,r.[, L = ].s,q.[ as Subset of REAL ;
( r is R_eal & s is R_eal ) by XXREAL_0:def 1;
then ( K is open_interval & L is open_interval ) ;
then reconsider K = K, L = L as Interval ;
take K ; :: thesis: ex L being Interval st
( K misses L & I \ J = K \/ L )
take L ; :: thesis: ( K misses L & I \ J = K \/ L )
thus ( K misses L & I \ J = K \/ L ) by A4, A2, A3, XXREAL_1:309, XXREAL_1:275; :: thesis: verum