reserve x,y for set;

theorem Th6:
  for F being Field-like Abelian distributive add-associative
  right_zeroed right_complementable non degenerated doubleLoopStr, a being
  Element of NonZero 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 NonZero F;
  set B = suppf1(F)\{0.F};
  set P = omf(F)!(suppf1(F),0.F);
A1: B = NonZero F;
  then reconsider e = 1.F as Element of B by STRUCT_0:2;
  reconsider D = addLoopStr(#B,P,e#) as strict AbGroup by A1,Def4;
  reconsider a as Element of D;
  omf(F)||(suppf1(F)\{0.F}) is Function of [:suppf1(F)\{0.F},suppf1(F)\{0.
  F}:], (suppf1(F)\{0.F}) by REALSET1:7;
  then
A2: dom(omf(F)||(suppf1(F)\{0.F})) = [:suppf1(F)\{0.F},suppf1(F)\{0.F}:] by
FUNCT_2:def 1;
A3: for x,y being Element of suppf1(F)\{0.F} holds omf(F).(x,y) = (the addF
  of D).(x,y)
  proof
    let x,y be Element of suppf1(F)\{0.F};
    [x,y] in [:suppf1(F)\{0.F},suppf1(F)\{0.F}:];
    hence thesis by A2,FUNCT_1:47;
  end;
  then
A4: omf(F).(1.F,a) = 0.D + a .= a;
  omf(F).(a,1.F) = a + 0.D by A3
    .= a;
  hence thesis by A4;
end;
