for I being non empty set
for S being non empty non void ManySortedSign
for i being Element of I
for A being MSAlgebra-Family of I,S
for s being SortSymbol of S
for U1 being non-empty MSAlgebra over S
for F being ManySortedFunction of I st ( for i being Element of I ex F1 being ManySortedFunction of U1,(A . i) st
( F1 = F . i & F1 is_homomorphism U1,A . i ) ) holds
for F9 being ManySortedFunction of U1,(A . i) st F9 = F . i holds
for x being set st x in the Sorts of U1 . s holds
for f being Function st f = (commute ((commute F) . s)) . x holds
f . i = (F9 . s) . x