reserve x,y for set;

theorem Th21:
  for F being Field-like Abelian distributive add-associative
  right_zeroed right_complementable non degenerated doubleLoopStr, a being
  Element of F holds a*1.F = a & 1.F*a = a
proof
  let F be Field-like Abelian distributive add-associative right_zeroed
right_complementable non degenerated doubleLoopStr, a be Element of suppf1(F);
  per cases by ZFMISC_1:56;
  suppose
A1: a = 0.F;
    hence a*1.F = a;
    thus thesis by A1;
  end;
  suppose
    a is Element of NonZero F;
    hence thesis by Th6;
  end;
end;
