
theorem Th13:
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_1:46,XBOOLE_1:69;
    suppose r <= p & s <= q; then
     A /\ B = ].p,s.[ by A1,A2,XXREAL_1:160; 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;
    suppose r > p & s > q; then
     A /\ B = ].r,q.[ by A1,A2,XXREAL_1:160; 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;
