let A be non empty set ; :: thesis: for b being Element of A ex o being Element of LinPreorders A st
for a being Element of A st a <> b holds
a <_ o,b
let b be Element of A; :: thesis: ex o being Element of LinPreorders A st
for a being Element of A st a <> b holds
a <_ o,b
defpred S₁[ set , set ] means ( $1 <> b or $2 = b );
consider R being Relation of A such that
A1: for x, y being Element of A holds
( [x,y] in R iff S₁[x,y] ) from RELSET_1:sch 2();

A2: now :: thesis: for x, y being Element of A holds
( [x,y] in R or [y,x] in R )

let x, y be Element of A; :: thesis: ( [x,y] in R or [y,x] in R )
( S₁[x,y] or S₁[y,x] ) ;
hence ( [x,y] in R or [y,x] in R ) by A1; :: thesis: verum

end;

now :: thesis: for x, y, z being Element of A st [x,y] in R & [y,z] in R holds
[x,z] in R

let x, y, z be Element of A; :: thesis: ( [x,y] in R & [y,z] in R implies [x,z] in R )
assume that
A3: [x,y] in R and
A4: [y,z] in R ; :: thesis: [x,z] in R
S₁[x,y] by A1, A3;
hence [x,z] in R by A1, A4; :: thesis: verum

end;

then reconsider o = R as Element of LinPreorders A by A2, Def1;
take o ; :: thesis: for a being Element of A st a <> b holds
a <_ o,b
let a be Element of A; :: thesis: ( a <> b implies a <_ o,b )
assume a <> b ; :: thesis: a <_ o,b
then not [b,a] in R by A1;
hence a <_ o,b ; :: thesis: verum