
theorem Th14:
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;

A4: p <= q & r <= s by A1,A2,XXREAL_1:29;

A5: inf A = p & sup A = q & inf B = r & sup B = s
      by A1,A2,XXREAL_1:29,MEASURE6:10,14;

    per cases by A1,A2,A3,Th4;
    suppose A6: q < r; then
     consider x be R_eal such that
A7:   q < x & x < r & x in REAL by MEASURE5:2;
     not x in A & not x in B by A1,A2,A7,XXREAL_1:1; then
A8:  not x in A \/ B by XBOOLE_0:def 3;

A9: inf A < x & x < sup B by A7,A4,A5,XXREAL_0:2;
     now assume
A10:   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,A4,A5,XXREAL_0:2,def 9,def 10;
      hence contradiction by A8,A9,A10,XXREAL_2:83;
     end;
     hence not A \/ B is Interval;
    end;
    suppose A11: s < p; then
     consider x be R_eal such that
A12:   s < x & x < p & x in REAL by MEASURE5:2;
     not x in A & not x in B by A1,A2,A12,XXREAL_1:1; then
A13:  not x in A \/ B by XBOOLE_0:def 3;

A14: inf B < x & x < sup A by A12,A4,A5,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,A4,A5,XXREAL_0:2,XXREAL_0:def 9,def 10;
      hence contradiction by A13,A14,A15,XXREAL_2:83;
     end;
     hence not A \/ B is Interval;
    end;
end;
