reserve x,y for set;

theorem Th5:
  for F being Field-like Abelian distributive add-associative
  right_zeroed right_complementable non degenerated doubleLoopStr, a,b being
  Element of NonZero F holds a*b = b*a
proof
  let F be Field-like Abelian distributive add-associative right_zeroed
right_complementable non degenerated doubleLoopStr, a,b 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,b as Element of D;
  reconsider x = a, y = b as Element of F;
  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 x*y = a+b .= b+a
    .= y*x by A3;
  hence thesis;
end;
