
theorem Th6:
  for G being Group, E being non empty set, x,y being Element of E,
T being LeftOperation of G, E holds the_orbit_of(x,T) misses the_orbit_of(y,T)
  or the_orbit_of(x,T)=the_orbit_of(y,T)
proof
  let G be Group;
  let E be non empty set;
  let x,y be Element of E;
  let T be LeftOperation of G, E;
  assume not the_orbit_of(x,T) misses the_orbit_of(y,T);
  then the_orbit_of(x,T) /\ the_orbit_of(y,T) <> {} by XBOOLE_0:def 7;
  then consider z1 be object such that
A1: z1 in the_orbit_of(x,T) /\ the_orbit_of(y,T) by XBOOLE_0:def 1;
  z1 in the_orbit_of(y,T) by A1,XBOOLE_0:def 4;
  then consider z199 be Element of E such that
A2: z1=z199 and
A3: y,z199 are_conjugated_under T;
  z1 in the_orbit_of(x,T) by A1,XBOOLE_0:def 4;
  then consider z19 be Element of E such that
A4: z1=z19 and
A5: x,z19 are_conjugated_under T;
  now
    let z2 be object;
    hereby
      assume z2 in the_orbit_of(x,T);
      then consider z29 be Element of E such that
A6:   z2=z29 and
A7:   x,z29 are_conjugated_under T;
      z29,x are_conjugated_under T by A7,Th4;
      then z29,z199 are_conjugated_under T by A4,A5,A2,Th5;
      then z199,z29 are_conjugated_under T by Th4;
      then y,z29 are_conjugated_under T by A3,Th5;
      hence z2 in the_orbit_of(y,T) by A6;
    end;
    assume z2 in the_orbit_of(y,T);
    then consider z29 be Element of E such that
A8: z2=z29 and
A9: y,z29 are_conjugated_under T;
    z29,y are_conjugated_under T by A9,Th4;
    then z29,z19 are_conjugated_under T by A4,A2,A3,Th5;
    then z19,z29 are_conjugated_under T by Th4;
    then x,z29 are_conjugated_under T by A5,Th5;
    hence z2 in the_orbit_of(x,T) by A8;
  end;
  hence thesis by TARSKI:2;
end;
