
theorem Th9:
for A,B be non empty set, p,q,r,s be R_eal st
 A = [.p,q.[ & B = ].r,s.] & A misses B holds q <= r or s < p
proof
    let A,B be non empty set, p,q,r,s be R_eal;
    assume that
A1:  A = [.p,q.[ and
A2:  B = ].r,s.] and
A3:  A misses B;

    assume
A4:  q > r & s >= p;
    per cases by A3,A1,A2,XXREAL_1:40,44,XBOOLE_1:69;
    suppose r < p & s < q; then
     A /\ B = [.p,s.] by A1,A2,XXREAL_1:152; then
     ex x be object st x in A /\ B by A4,XXREAL_1:30,XBOOLE_0:def 1;
     hence contradiction by A3,XBOOLE_0:4;
    end;
    suppose r >= p & s >= q; then
     A /\ B = ].r,q.[ by A1,A2,XXREAL_1:153; then
     ex x be object st x in A /\ B by A4,XXREAL_1:33,XBOOLE_0:def 1;
     hence contradiction by A3,XBOOLE_0:4;
    end;
end;
