let X be non empty set ; :: thesis: for r being Relation of X st r is symmetric holds
chi (r,[:X,X:]) is symmetric
let r be Relation of X; :: thesis: ( r is symmetric implies chi (r,[:X,X:]) is symmetric )
assume r is symmetric ; :: thesis: chi (r,[:X,X:]) is symmetric
then A1: r is_symmetric_in field r by RELAT_2:def 11;
let x, y be Element of X; :: according to LFUZZY_1:def 5 :: thesis: (chi (r,[:X,X:])) . (x,y) = (chi (r,[:X,X:])) . (y,x)
A2: ( x in field r & y in field r & [x,y] in r implies [y,x] in r ) by A1, RELAT_2:def 3;
A3: ( x in field r & y in field r & [y,x] in r implies [x,y] in r ) by A1, RELAT_2:def 3;

per cases ( [x,y] in r or not [x,y] in r ) ;

suppose A4: [x,y] in r ; :: thesis: (chi (r,[:X,X:])) . (x,y) = (chi (r,[:X,X:])) . (y,x)

then (chi (r,[:X,X:])) . [x,y] = 1 by FUNCT_3:def 3;
hence (chi (r,[:X,X:])) . (x,y) = (chi (r,[:X,X:])) . (y,x) by A2, A4, FUNCT_3:def 3, RELAT_1:15; :: thesis: verum

end;

suppose not [x,y] in r ; :: thesis: (chi (r,[:X,X:])) . (x,y) = (chi (r,[:X,X:])) . (y,x)

then ( not [y,x] in r & (chi (r,[:X,X:])) . [x,y] = 0 ) by A3, FUNCT_3:def 3, RELAT_1:15;
hence (chi (r,[:X,X:])) . (x,y) = (chi (r,[:X,X:])) . (y,x) by FUNCT_3:def 3; :: thesis: verum

end;

end;