
theorem
  for a,b,c,d be Real st a < b & b < c & c < d holds
    [. a,d .] \ [. b,c .] c= [. a,b .] \/ [. c,d .]
proof
 let a,b,c,d be Real;
 assume A1: a < b & b < c & c < d; then
 a < c by XXREAL_0:2; then
 B1:[. a,d .] = [' a,d '] by INTEGRA5:def 3,XXREAL_0:2,A1;
 B2:[. b,c .] = [' b,c '] by INTEGRA5:def 3,A1;
 B3:[. a,b .] = [' a,b '] by INTEGRA5:def 3,A1;
 [. c,d .] = [' c,d '] by INTEGRA5:def 3,A1;
 hence thesis by B1,B2,B3,Th6,A1;
end;
