let r, s, p, q be ext-real number ; :: thesis: ( ].r,s.] meets ].p,q.] implies ].r,s.] \/ ].p,q.] = ].(min r,p),(max s,q).] )
assume ].r,s.] meets ].p,q.] ; :: thesis: ].r,s.] \/ ].p,q.] = ].(min r,p),(max s,q).]
then consider u being ext-real number such that
A1: u in ].r,s.] and
A2: u in ].p,q.] by MEMBERED:def 20;
let t be ext-real number ; :: according to MEMBERED:def 14 :: thesis: ( ( not t in ].r,s.] \/ ].p,q.] or t in ].(min r,p),(max s,q).] ) & ( not t in ].(min r,p),(max s,q).] or t in ].r,s.] \/ ].p,q.] ) )
thus ( t in ].r,s.] \/ ].p,q.] implies t in ].(min r,p),(max s,q).] ) :: thesis: ( not t in ].(min r,p),(max s,q).] or t in ].r,s.] \/ ].p,q.] )

proof

assume t in ].r,s.] \/ ].p,q.] ; :: thesis: t in ].(min r,p),(max s,q).]
then ( t in ].r,s.] or t in ].p,q.] ) by XBOOLE_0:def 3;
then A3: ( ( r < t & t <= s ) or ( p < t & t <= q ) ) by Th2;
then A4: min r,p < t by XXREAL_0:22;
t <= max s,q by A3, XXREAL_0:31;
hence t in ].(min r,p),(max s,q).] by A4, Th2; :: thesis: verum

end;

A5: r < u by A1, Th2;
A6: u <= s by A1, Th2;
A7: p < u by A2, Th2;
A8: u <= q by A2, Th2;
assume A9: t in ].(min r,p),(max s,q).] ; :: thesis: t in ].r,s.] \/ ].p,q.]
then A10: min r,p < t by Th2;
A11: t <= max s,q by A9, Th2;

per cases ( ( r < t & t <= s ) or ( p < t & t <= q ) or ( p < t & t <= s ) or ( r < t & t <= q ) ) by A10, A11, XXREAL_0:20, XXREAL_0:29;

suppose ( ( r < t & t <= s ) or ( p < t & t <= q ) ) ; :: thesis: t in ].r,s.] \/ ].p,q.]

then ( t in ].r,s.] or t in ].p,q.] ) by Th2;
hence t in ].r,s.] \/ ].p,q.] by XBOOLE_0:def 3; :: thesis: verum

end;

suppose that A12: p < t and
A13: t <= s ; :: thesis: t in ].r,s.] \/ ].p,q.]

( u <= t or t <= u ) ;
then ( r < t or t <= q ) by A5, A8, XXREAL_0:2;
then ( t in ].r,s.] or t in ].p,q.] ) by A12, A13, Th2;
hence t in ].r,s.] \/ ].p,q.] by XBOOLE_0:def 3; :: thesis: verum

end;

suppose that A14: r < t and
A15: t <= q ; :: thesis: t in ].r,s.] \/ ].p,q.]

( u <= t or t <= u ) ;
then ( t <= s or p < t ) by A6, A7, XXREAL_0:2;
then ( t in ].r,s.] or t in ].p,q.] ) by A14, A15, Th2;
hence t in ].r,s.] \/ ].p,q.] by XBOOLE_0:def 3; :: thesis: verum

end;

end;