
theorem Th10: :: Br2:
  for G being Group, a being Element of G, x, y being Element of con_class a
  st x<>y holds (a-con_map"{x}) misses (a-con_map"{y})
proof
  let G be Group, a be Element of G, x,y be Element of con_class a such that
A1: x <> y;
  now
    assume ex g being object st g in (a-con_map"{x}) /\ (a-con_map"{y });
    then consider g being set such that
A2: g in (a-con_map"{x}) /\ (a-con_map"{y});
A3: g in a-con_map"{x} by A2,XBOOLE_0:def 4;
A4: g in a-con_map"{y} by A2,XBOOLE_0:def 4;
    a-con_map.g in {x} by A3,FUNCT_1:def 7;
    then
A5: a-con_map.g = x by TARSKI:def 1;
    a-con_map.g in {y} by A4,FUNCT_1:def 7;
    hence contradiction by A1,A5,TARSKI:def 1;
  end;
  then (a-con_map"{x}) /\ (a-con_map"{y}) = {} by XBOOLE_0:def 1;
  hence thesis by XBOOLE_0:def 7;
end;
