
theorem Th19:
for A,B be non empty Interval, p,q,r,s be R_eal st
 A = [.p,q.[ & B = ].r,s.] & A misses B holds not A \/ B is Interval
proof
    let A,B be non empty Interval, p,q,r,s be R_eal;
    assume that
A1:  A = [.p,q.[ and
A2:  B = ].r,s.] and
A3:  A misses B;

    p < q & r < s by A1,A2,XXREAL_1:26,27; then
A4: inf A = p & sup A = q & inf B = r & sup B = s
      by A1,A2,MEASURE6:11,15,9,13;

    per cases by A1,A2,A3,Th9;
    suppose A5: q <= r; then
A6:  inf A < inf B & sup A < sup B by A4,A1,A2,XXREAL_1:26,27,XXREAL_0:2;
     not q in A & not q in B by A1,A2,A5,XXREAL_1:2,3; then
A7:  not q in A \/ B by XBOOLE_0:def 3;
A8: inf A < q & q < sup B by A4,A5,A1,A2,XXREAL_1:26,27,XXREAL_0:2;
     now assume
A9:   A \/ B is Interval;
      inf(A \/ B) = min(inf A,inf B)
    & sup(A \/ B) = max(sup A,sup B) by XXREAL_2:9,10; then
      inf(A \/ B) = inf A & sup(A \/ B) = sup B
        by A6,XXREAL_0:def 9,def 10;
      hence contradiction by A7,A8,A9,XXREAL_2:83;
     end;
     hence not A \/ B is Interval;
    end;
    suppose A10: s < p; then
A11:  inf B < inf A & sup B < sup A by A4,A1,A2,XXREAL_1:26,27,XXREAL_0:2;
     consider x be R_eal such that
A12:   s < x & x < p & x in REAL by A10,MEASURE5:2;
     not x in A & not x in B by A1,A2,A12,XXREAL_1:2,3; then
A13:  not x in A \/ B by XBOOLE_0:def 3;
A14: inf B < x & x < sup A by A12,A4,A1,A2,XXREAL_1:26,27,XXREAL_0:2;
     now assume
A15:   A \/ B is Interval;
      inf(A \/ B) = min(inf A,inf B)
    & sup(A \/ B) = max(sup A,sup B) by XXREAL_2:9,10; then
      inf(A \/ B) = inf B & sup(A \/ B) = sup A
        by A11,XXREAL_0:def 9,def 10;
      hence contradiction by A13,A14,A15,XXREAL_2:83;
     end;
     hence not A \/ B is Interval;
    end;
end;
