let K be non empty doubleLoopStr ; :: thesis: for J being Function of K,K holds
( J is antiautomorphism iff ( ( for x, y being Scalar of K holds J . (x + y) = (J . x) + (J . y) ) & ( 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 & rng J = the carrier of K ) )
let J be Function of K,K; :: thesis: ( J is antiautomorphism iff ( ( for x, y being Scalar of K holds J . (x + y) = (J . x) + (J . y) ) & ( 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 & rng J = the carrier of K ) )

A1: now

assume J is antiautomorphism ; :: thesis: ( ( for x, y being Scalar of K holds
( J . (x + y) = (J . x) + (J . y) & ( 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 & rng J = the carrier of K )
then A2: J is antiisomorphism by Def17;
then A3: J is antimonomorphism by Def13;
then J is antilinear by Def9;
then ( J is additive & J is antimultiplicative & J is unity-preserving ) ;
hence for x, y being Scalar of K holds
( J . (x + y) = (J . x) + (J . y) & ( for x, y being Scalar of K holds
( J . (x * y) = (J . y) * (J . x) & J . (1_ K) = 1_ K ) ) ) by Def6, GRCAT_1:def 13, GROUP_1:def 17; :: thesis: ( J is one-to-one & rng J = the carrier of K )
thus J is one-to-one by A3, Def9; :: thesis: rng J = the carrier of K
thus rng J = the carrier of K by A2, Def13; :: thesis: verum

end;

now

assume that
A4: ( ( for x, y being Scalar of K holds J . (x + y) = (J . x) + (J . y) ) & ( for x, y being Scalar of K holds J . (x * y) = (J . y) * (J . x) ) & J . (1_ K) = 1_ K ) and
A5: J is one-to-one and
A6: rng J = the carrier of K ; :: thesis: J is antiautomorphism
( J is additive & J is antimultiplicative & J is unity-preserving ) by A4, Def6, GRCAT_1:def 13, GROUP_1:def 17;
then J is antilinear ;
then J is antimonomorphism by A5, Def9;
then J is antiisomorphism by A6, Def13;
hence J is antiautomorphism by Def17; :: thesis: verum

end;

hence ( J is antiautomorphism iff ( ( for x, y being Scalar of K holds J . (x + y) = (J . x) + (J . y) ) & ( 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 & rng J = the carrier of K ) ) by A1; :: thesis: verum