now

let x, y be set ; :: thesis: ( [x,y] in R ~ implies [x,y] in [:Y,X:] )
assume [x,y] in R ~ ; :: thesis: [x,y] in [:Y,X:]
then [y,x] in R by RELAT_1:def 7;
hence [x,y] in [:Y,X:] by ZFMISC_1:107; :: thesis: verum

end;

hence R ~ is Relation of Y,X by RELAT_1:def 3; :: thesis: verum