let S be non empty non void ManySortedSign ; :: thesis: for A being non-empty MSAlgebra over S
for s1, s2, s3 being SortSymbol of S st TranslationRel S reduces s2,s3 holds
for f being Function st f is_e.translation_of A,s1,s2 holds
for t being Translation of A,s2,s3 holds t * f is Translation of A,s1,s3
let A be non-empty MSAlgebra over S; :: thesis: for s1, s2, s3 being SortSymbol of S st TranslationRel S reduces s2,s3 holds
for f being Function st f is_e.translation_of A,s1,s2 holds
for t being Translation of A,s2,s3 holds t * f is Translation of A,s1,s3
let s1, s2, s3 be SortSymbol of S; :: thesis: ( TranslationRel S reduces s2,s3 implies for f being Function st f is_e.translation_of A,s1,s2 holds
for t being Translation of A,s2,s3 holds t * f is Translation of A,s1,s3 )
assume A1: TranslationRel S reduces s2,s3 ; :: thesis: for f being Function st f is_e.translation_of A,s1,s2 holds
for t being Translation of A,s2,s3 holds t * f is Translation of A,s1,s3
let f be Function; :: thesis: ( f is_e.translation_of A,s1,s2 implies for t being Translation of A,s2,s3 holds t * f is Translation of A,s1,s3 )
assume A2: f is_e.translation_of A,s1,s2 ; :: thesis: for t being Translation of A,s2,s3 holds t * f is Translation of A,s1,s3
then A3: f is Translation of A,s1,s2 by Th17;
TranslationRel S reduces s1,s2 by A2, Th17;
hence for t being Translation of A,s2,s3 holds t * f is Translation of A,s1,s3 by A1, A3, Th18; :: thesis: verum