 reserve x,y,X,Y for set;

theorem Th3:
  for f being BinOp of {x} holds f = (x,x) .--> x & f is
  having_a_unity & f is commutative & f is associative & f is idempotent & f is
  invertible cancelable uniquely-decomposable
proof
  let f be BinOp of {x};
  reconsider a = x as Element of {x} by TARSKI:def 1;
A1: [:{x},{x}:] = {[x,x]} by ZFMISC_1:29;
A2: now
    let y be object;
    assume
A3: y in {[x,x]};
    then reconsider b = y as Element of [:{x},{x}:] by ZFMISC_1:29;
    thus f.y = f.b .= x by TARSKI:def 1
      .= ((x,x) .--> x).y by A3,FUNCOP_1:7;
  end;
  dom f = [:{x},{x}:] & dom ((x,x) .--> x) = {[x,x]} by FUNCT_2:def 1;
  hence f = (x,x) .--> x by A1,A2,FUNCT_1:2;
A4: now
    let a,b be Element of {x};
    a = x by TARSKI:def 1;
    hence a = b by TARSKI:def 1;
  end;
  then for b being Element of {x} holds f.(a,b) = b & f.(b,a) = b;
  then
A5: a is_a_unity_wrt f by BINOP_1:3;
  hence ex a being Element of {x} st a is_a_unity_wrt f;
  thus for a,b being Element of {x} holds f.(a,b) = f.(b,a) by A4;
  thus for a,b,c being Element of {x} holds f.(a,f.(b,c)) = f.(f.(a,b),c) by A4
;
  thus for a being Element of {x} holds f.(a,a) = a by A4;
  thus for a,b being Element of {x} ex r,l being Element of {x} st f.(a,r) = b
  & f.(l,a) = b
  proof
    let a,b be Element of {x};
    take a,a;
    thus thesis by A4;
  end;
  thus for a,b,c being Element of {x} st f.(a,b) = f.(a,c) or f.(b,a) = f.(c,a
  ) holds b = c by A4;
  thus ex a being Element of {x} st a is_a_unity_wrt f by A5;
  let a,b be Element of {x};
  thus thesis by A4;
end;
