let X, S be non empty set ; :: thesis: for R being Relation of X st R is antisymmetric holds
R is_antisymmetric_in S
let R be Relation of X; :: thesis: ( R is antisymmetric implies R is_antisymmetric_in S )
assume R is antisymmetric ; :: thesis: R is_antisymmetric_in S
then A1: R is_antisymmetric_in field R by RELAT_2:def 12;
let x, y be set ; :: according to RELAT_2:def 4 :: thesis: ( not x in S or not y in S or not [x,y] in R or not [y,x] in R or x = y )
assume ( x in S & y in S ) ; :: thesis: ( not [x,y] in R or not [y,x] in R or x = y )
assume A2: [x,y] in R ; :: thesis: ( not [y,x] in R or x = y )
then ( x in field R & y in field R ) by RELAT_1:15;
hence ( not [y,x] in R or x = y ) by A1, A2, RELAT_2:def 4; :: thesis: verum