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;

theorem Th22:
  J is antiisomorphism iff J is additive
  & (for x,y being Scalar of K holds J.(x*y) = J.y*J.x) & J.(1_K) =
  1_K & J is one-to-one & J is onto
proof
  thus J is antiisomorphism implies J is additive
  & (for x,y being Scalar of K holds J.(x*y) = J.y*J.x) & J.(1_K) =
  1_K & J is one-to-one & J is onto by Def6,GROUP_1:def 13;
    assume (J is additive & for x,y
    being Scalar of K holds J.(x*y) = J.y*J.x )& J.(1_K) = 1_K;
    then J is additive antimultiplicative unity-preserving;
    hence thesis;
end;
