reserve x,y for set;

theorem Th7:
  for F being Field-like Abelian distributive add-associative
  right_zeroed right_complementable non degenerated doubleLoopStr, a being
  Element of NonZero F ex b being Element of NonZero F st a*b = 1.F & b*a = 1.F
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;
  addLoopStr(#B,P,e#) is AbGroup by A1,Def4;
  then consider D being strict AbGroup such that
A2: D = addLoopStr(#B,P,e#);
  reconsider a as Element of D by A2;
  consider b being Element of D such that
A3: a+b = 0.D and
A4: b+a = 0.D by Th3;
  reconsider b as Element of NonZero F by A2;
  take b;
  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
A5: dom(omf(F)||(suppf1(F)\{0.F})) = [:suppf1(F)\{0.F},suppf1(F)\{0.F}:] by
FUNCT_2:def 1;
  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,A5,FUNCT_1:47;
  end;
  hence thesis by A2,A3,A4;
end;
