let S, X be non empty set ; :: thesis: for R being Relation of X st R is asymmetric holds

R is_asymmetric_in S

let R be Relation of X; :: thesis: ( R is asymmetric implies R is_asymmetric_in S )

assume R is asymmetric ; :: thesis: R is_asymmetric_in S

then A1: R is_asymmetric_in field R ;

let x, y be object ; :: according to RELAT_2:def 5 :: thesis: ( not x in S or not y in S or not [x,y] in R or not [y,x] in R )

assume ( x in S & y in S ) ; :: thesis: ( not [x,y] in R or not [y,x] in R )

assume A2: [x,y] in R ; :: thesis: not [y,x] in R

then ( x in field R & y in field R ) by RELAT_1:15;

hence not [y,x] in R by A1, A2; :: thesis: verum

R is_asymmetric_in S

let R be Relation of X; :: thesis: ( R is asymmetric implies R is_asymmetric_in S )

assume R is asymmetric ; :: thesis: R is_asymmetric_in S

then A1: R is_asymmetric_in field R ;

let x, y be object ; :: according to RELAT_2:def 5 :: thesis: ( not x in S or not y in S or not [x,y] in R or not [y,x] in R )

assume ( x in S & y in S ) ; :: thesis: ( not [x,y] in R or not [y,x] in R )

assume A2: [x,y] in R ; :: thesis: not [y,x] in R

then ( x in field R & y in field R ) by RELAT_1:15;

hence not [y,x] in R by A1, A2; :: thesis: verum