
theorem Th20:
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_1:27,28; then
A6: inf A = p & sup A = q & inf B = r & sup B = s
      by A1,A2,MEASURE6:8,11,12,15;
    now assume A7: q <= r; then
     not q in A & not q in B by A1,A2,XXREAL_1:3,4; then
A8:  not q in A \/ B by XBOOLE_0:def 3;
A9: inf A < inf B & sup A < sup B by A6,A7,A1,A2,XXREAL_1:27,28,XXREAL_0:2;
     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) < q & q < sup(A \/ B) by A5,A6,A9,XXREAL_0:def 9,def 10;
     hence contradiction by A8,A4,XXREAL_2:83;
    end; then
A10:s <= p by A1,A2,A3,Th10;
    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_1:3,4; 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,A12,A1,A2,XXREAL_1:27,28,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_1:173;
end;
