
theorem Th2: :: Nat0:
for x, y, z being Nat holds x in Segm y \ Segm z iff z <= x & x < y
proof
 let x, y, z be Nat;
 hereby assume
 A1: x in Segm y \ Segm z;
    not x in Segm z by A1,XBOOLE_0:def 5;
   hence z <= x by NAT_1:44;
    x in Segm y by A1,XBOOLE_0:def 5;
   hence x < y by NAT_1:44;
 end;
 assume z <= x & x < y;
  then x in Segm y & not x in Segm z by NAT_1:44;
 hence x in Segm y \ Segm z by XBOOLE_0:def 5;
end;
