let I, J be Interval; :: thesis: ( I is open_interval & J is right_open_interval & I meets J implies I /\ J is Interval )
assume A1: ( I is open_interval & J is right_open_interval & I meets J ) ; :: thesis: I /\ J is Interval
then consider p, q being R_eal such that
A2: I = ].p,q.[ ;
consider r being Real, s being R_eal such that
A3: J = [.r,s.[ by A1;

per cases ( ( p < r & q <= s ) or ( p < r & s <= q ) or ( p >= r & q >= s ) or ( r <= p & q < s ) ) ;

suppose ( p < r & q <= s ) ; :: thesis: I /\ J is Interval

then I /\ J = [.r,q.[ by A2, A3, XXREAL_1:154;
then I /\ J is right_open_interval ;
hence I /\ J is Interval ; :: thesis: verum

end;

suppose ( p < r & s <= q ) ; :: thesis: I /\ J is Interval

hence I /\ J is Interval by A2, A3, XXREAL_1:48, XBOOLE_1:28; :: thesis: verum

end;

suppose ( p >= r & q >= s ) ; :: thesis: I /\ J is Interval

then I /\ J = ].p,s.[ by A2, A3, XXREAL_1:155;
then I /\ J is open_interval ;
hence I /\ J is Interval ; :: thesis: verum

end;

suppose ( r <= p & q < s ) ; :: thesis: I /\ J is Interval

hence I /\ J is Interval by A2, A3, XXREAL_1:45, XBOOLE_1:28; :: thesis: verum

end;

end;