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
  J is isomorphism iff J is additive &
  (for x,y being Scalar of K holds J.(x*y) = J.x*J.y) & J.(1_K) = 1_K &
  J is one-to-one onto
proof
  thus J is isomorphism implies J is additive &
  (for x,y being Scalar of K holds J.(x*y) = J.x*J.y) & J.(1_K) = 1_K &
  J is one-to-one onto by GROUP_1:def 13,GROUP_6:def 6;
  assume (J is additive & for x,y
    being Scalar of K holds J.(x*y) = J.x*J.y )& J.(1_K) = 1_K;
  then J is additive multiplicative unity-preserving;
  hence thesis;
end;
