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
b <_ o,a
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
b <_ o,a
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

let x, y be Element of A; :: thesis: ( [x,y] in R or [y,x] in R )
A3: ( S₁[x,y] or S₁[y,x] ) ;
thus ( [x,y] in R or [y,x] in R ) by A1, A3; :: thesis: verum

end;

A4: now

let x, y, z be Element of A; :: thesis: ( [x,y] in R & [y,z] in R implies [x,z] in R )
assume that
A5: [x,y] in R and
A6: [y,z] in R ; :: thesis: [x,z] in R
A7: S₁[y,z] by A1, A6;
thus [x,z] in R by A1, A5, A7; :: thesis: verum

end;

reconsider o = R as Element of LinPreorders A by A2, A4, Def1;
take o ; :: thesis: for a being Element of A st a <> b holds
b <_ o,a
let a be Element of A; :: thesis: ( a <> b implies b <_ o,a )
assume A8: a <> b ; :: thesis: b <_ o,a
A9: not [a,b] in R by A1, A8;
thus b <_ o,a by A9, Def4; :: thesis: verum