let X be non empty set ; :: thesis: for r being Relation of X st r is antisymmetric holds
chi (r,[:X,X:]) is antisymmetric
let r be Relation of X; :: thesis: ( r is antisymmetric implies chi (r,[:X,X:]) is antisymmetric )
assume r is antisymmetric ; :: thesis: chi (r,[:X,X:]) is antisymmetric
then A1: r is_antisymmetric_in field r by RELAT_2:def 12;
for x, y being Element of X st (chi (r,[:X,X:])) . (x,y) <> 0 & (chi (r,[:X,X:])) . (y,x) <> 0 holds
x = y

proof

let x, y be Element of X; :: thesis: ( (chi (r,[:X,X:])) . (x,y) <> 0 & (chi (r,[:X,X:])) . (y,x) <> 0 implies x = y )
assume that
A2: (chi (r,[:X,X:])) . (x,y) <> 0 and
A3: (chi (r,[:X,X:])) . (y,x) <> 0 ; :: thesis: x = y
A4: ( x in field r & y in field r & [x,y] in r & [y,x] in r implies x = y ) by A1, RELAT_2:def 4;
[x,y] in r by A2, FUNCT_3:def 3;
hence x = y by A3, A4, FUNCT_3:def 3, RELAT_1:15; :: thesis: verum

end;

hence chi (r,[:X,X:]) is antisymmetric ; :: thesis: verum