A1: [:A,A:] c= [:A,A:] ;
reconsider R = [:A,A:] as Relation of A by A1;
A2: ( ( for a, b being Element of A holds
( [a,b] in R or [b,a] in R ) ) & ( for a, b, c being Element of A st [a,b] in R & [b,c] in R holds
[a,c] in R ) ) by ZFMISC_1:106;
thus not LinPreorders A is empty by A2, Def1; :: thesis: verum