let X1, X2 be set ; :: thesis: for S1 being semialgebra_of_sets of X1
for S2 being semialgebra_of_sets of X2 holds { [:A,B:] where A is Element of S1, B is Element of S2 : verum } is semialgebra_of_sets of [:X1,X2:]
let S1 be semialgebra_of_sets of X1; :: thesis: for S2 being semialgebra_of_sets of X2 holds { [:A,B:] where A is Element of S1, B is Element of S2 : verum } is semialgebra_of_sets of [:X1,X2:]
let S2 be semialgebra_of_sets of X2; :: thesis: { [:A,B:] where A is Element of S1, B is Element of S2 : verum } is semialgebra_of_sets of [:X1,X2:]
set S = { [:A,B:] where A is Element of S1, B is Element of S2 : verum } ;
A1: { [:A,B:] where A is Element of S1, B is Element of S2 : verum } is Semiring of [:X1,X2:] by Th08;
( X1 in S1 & X2 in S2 ) by SRINGS_3:def 6;
then [:X1,X2:] in { [:A,B:] where A is Element of S1, B is Element of S2 : verum } ;
hence { [:A,B:] where A is Element of S1, B is Element of S2 : verum } is semialgebra_of_sets of [:X1,X2:] by A1, SRINGS_3:def 6; :: thesis: verum