reserve Q,Q1,Q2 for multLoop;
reserve x,y,z,w,u,v for Element of Q;

theorem Th12:
  x in Nucl Q iff x in Nucl_l Q & x in Nucl_m Q & x in Nucl_r Q
proof
  thus x in Nucl Q implies x in Nucl_l Q & x in Nucl_m Q & x in Nucl_r Q
  proof
    assume A1: x in Nucl Q;
    then x in Nucl_l Q /\ Nucl_m Q by XBOOLE_0:def 4;
    hence thesis by XBOOLE_0:def 4, A1;
  end;
  assume that A2: x in Nucl_l Q & x in Nucl_m Q
  and A3: x in Nucl_r Q;
  x in Nucl_l Q /\ Nucl_m Q by XBOOLE_0:def 4, A2;
  hence x in Nucl Q by XBOOLE_0:def 4, A3;
end;
