reserve A,B,C for non empty set,
  f for Function of [:A,B:],C;
reserve K for non empty doubleLoopStr;
reserve V for non empty ModuleStr over K;
reserve W for non empty RightModStr over K;
reserve J for Function of K,K;
reserve K,L for Ring;
reserve J for Function of K,L;
reserve x,y for Scalar of K;

theorem
  for K being Ring holds id K is antiisomorphism iff K is comRing
proof
  let K be Ring;
  set J = id K;
A1: now
    assume
A2: K is comRing;
A3: for x,y being Scalar of K holds J.(x*y) = J.y*J.x
    by A2,Lm5;
    J.(1_K) = 1_K;
    hence J is antiisomorphism by A3,Th22;
  end;
  now
    assume
A4: J is antiisomorphism;
    for x,y being Element of K holds x*y = y*x
    proof
      let x,y be Element of K;
A5:   J.y = y;
      J.(x*y) = x*y & J.x = x;
      hence thesis by A4,A5,Th22;
    end;
    hence K is comRing by GROUP_1:def 12;
  end;
  hence thesis by A1;
end;
