let R be non empty RelStr ; :: thesis: ( R is transitive & R is serial implies R is positive_alliance )
assume AA: ( R is transitive & R is serial ) ; :: thesis: R is positive_alliance
set X = the carrier of R;
set I = the InternalRel of R;
for x, y being object st x in the carrier of R & y in the carrier of R & not [x,y] in the InternalRel of R holds
ex z being object st
( z in the carrier of R & [x,z] in the InternalRel of R & not [z,y] in the InternalRel of R )

proof

let x, y be object ; :: thesis: ( x in the carrier of R & y in the carrier of R & not [x,y] in the InternalRel of R implies ex z being object st
( z in the carrier of R & [x,z] in the InternalRel of R & not [z,y] in the InternalRel of R ) )
assume A0: ( x in the carrier of R & y in the carrier of R & not [x,y] in the InternalRel of R ) ; :: thesis: ex z being object st
( z in the carrier of R & [x,z] in the InternalRel of R & not [z,y] in the InternalRel of R )
then consider z being object such that
A1: ( z in the carrier of R & [x,z] in the InternalRel of R ) by ROUGHS_2:def 1, AA, ROUGHS_2:def 3;
take z ; :: thesis: ( z in the carrier of R & [x,z] in the InternalRel of R & not [z,y] in the InternalRel of R )
thus ( z in the carrier of R & [x,z] in the InternalRel of R & not [z,y] in the InternalRel of R ) by A0, A1, AA, ORDERS_2:def 3, RELAT_2:def 8; :: thesis: verum

end;

then the InternalRel of R is_positive_alliance_in the carrier of R ;
hence R is positive_alliance ; :: thesis: verum