let X be ARS ; :: thesis: for x, y being Element of X st x >><< y holds

x <=*=> y

let x, y be Element of X; :: thesis: ( x >><< y implies x <=*=> y )

given u being Element of X such that A2: ( x =*=> u & u <=*= y ) ; :: according to ABSRED_0:def 21 :: thesis: x <=*=> y

A3: ( x <=*=> u & u <=*=> y ) by A2, LemZ;

thus x <=*=> y by A3, Th7; :: thesis: verum

x <=*=> y

let x, y be Element of X; :: thesis: ( x >><< y implies x <=*=> y )

given u being Element of X such that A2: ( x =*=> u & u <=*= y ) ; :: according to ABSRED_0:def 21 :: thesis: x <=*=> y

A3: ( x <=*=> u & u <=*=> y ) by A2, LemZ;

thus x <=*=> y by A3, Th7; :: thesis: verum