for S being non empty non void ManySortedSign
for I being non empty finite set
for A being non-empty MSAlgebra over S
for F being MSAlgebra-Family of I,S st ( for i being Element of I ex C being strict non-empty finitely-generated MSSubAlgebra of A st C = F . i ) holds
ex B being strict non-empty finitely-generated MSSubAlgebra of A st
for i being Element of I holds F . i is MSSubAlgebra of B