
theorem Th15:
for A,B be non empty Interval, p,q,r,s be R_eal st
 A = [.p,q.] & B = [.r,s.[ & A misses B & A \/ B is Interval
  holds p = s & A \/ B = [.r,q.]
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 and
A4:  A \/ B is Interval;

A5: p <= q & r < s by A1,A2,XXREAL_1:27,29; then
A6: inf A = p & sup A = q & inf B = r & sup B = s
      by A1,A2,MEASURE6:10,14,11,15;

    now assume A7: q < r; then
     consider x be R_eal such that
A8:   q < x & x < r & x in REAL by MEASURE5:2;
     not x in A & not x in B by A1,A2,A8,XXREAL_1:1,3; then
A9:  not x in A \/ B by XBOOLE_0:def 3;

     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 A5,A6,A7,XXREAL_0:2,def 9,def 10; then
     inf(A \/ B) < x & x < sup(A \/ B) by A5,A6,A8,XXREAL_0:2;
     hence contradiction by A9,A4,XXREAL_2:83;
    end; then
A10:s <= p by A1,A2,A3,Th5;

    now assume 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_1:1,3; then
A13: not x in A \/ B by XBOOLE_0:def 3;

     min(inf A,inf B) = inf B & max(sup A,sup B) = sup A
       by A11,A6,A5,XXREAL_0:2,def 9,def 10; then
     inf(A \/ B) = inf B & sup(A \/ B) = sup A by XXREAL_2:9,10; then
     inf(A \/ B) < x & x < sup(A \/ B) by A6,A5,A12,XXREAL_0:2;
     hence contradiction by A13,A4,XXREAL_2:83;
    end;
    hence p = s by A10,XXREAL_0:1;
    hence A \/ B = [.r,q.] by A1,A2,A5,XXREAL_1:166;
end;
