let I be non empty set ; :: thesis: for S being non empty non void ManySortedSign
for A being MSAlgebra-Family of I,S
for o being OperSymbol of S st the_arity_of o <> {} holds
for y being Element of Args o,(product A)
for i9 being Element of I
for g being Function st g = (Den o,(product A)) . y holds
g . i9 = (Den o,(A . i9)) . ((commute y) . i9)
let S be non empty non void ManySortedSign ; :: thesis: for A being MSAlgebra-Family of I,S
for o being OperSymbol of S st the_arity_of o <> {} holds
for y being Element of Args o,(product A)
for i9 being Element of I
for g being Function st g = (Den o,(product A)) . y holds
g . i9 = (Den o,(A . i9)) . ((commute y) . i9)
let A be MSAlgebra-Family of I,S; :: thesis: for o being OperSymbol of S st the_arity_of o <> {} holds
for y being Element of Args o,(product A)
for i9 being Element of I
for g being Function st g = (Den o,(product A)) . y holds
g . i9 = (Den o,(A . i9)) . ((commute y) . i9)
let o be OperSymbol of S; :: thesis: ( the_arity_of o <> {} implies for y being Element of Args o,(product A)
for i9 being Element of I
for g being Function st g = (Den o,(product A)) . y holds
g . i9 = (Den o,(A . i9)) . ((commute y) . i9) )
assume A1: the_arity_of o <> {} ; :: thesis: for y being Element of Args o,(product A)
for i9 being Element of I
for g being Function st g = (Den o,(product A)) . y holds
g . i9 = (Den o,(A . i9)) . ((commute y) . i9)
let y be Element of Args o,(product A); :: thesis: for i9 being Element of I
for g being Function st g = (Den o,(product A)) . y holds
g . i9 = (Den o,(A . i9)) . ((commute y) . i9)
let i9 be Element of I; :: thesis: for g being Function st g = (Den o,(product A)) . y holds
g . i9 = (Den o,(A . i9)) . ((commute y) . i9)
A2: y in dom (Commute (Frege (A ?. o))) by A1, Th19;
A3: commute y in product (doms (A ?. o)) by A1, Th18;
A4: Den o,(product A) = (OPS A) . o by MSUALG_1:def 11
.= IFEQ (the_arity_of o),{} ,(commute (A ?. o)),(Commute (Frege (A ?. o))) by PRALG_2:def 20
.= Commute (Frege (A ?. o)) by A1, FUNCOP_1:def 8 ;
A5: dom (A ?. o) = I by PARTFUN1:def 4;
let g be Function; :: thesis: ( g = (Den o,(product A)) . y implies g . i9 = (Den o,(A . i9)) . ((commute y) . i9) )
assume g = (Den o,(product A)) . y ; :: thesis: g . i9 = (Den o,(A . i9)) . ((commute y) . i9)
then g = (Frege (A ?. o)) . (commute y) by A4, A2, PRALG_2:def 6
.= (A ?. o) .. (commute y) by A3, PRALG_2:def 8 ;
then g . i9 = ((A ?. o) . i9) . ((commute y) . i9) by A5, PRALG_1:def 17
.= (Den o,(A . i9)) . ((commute y) . i9) by PRALG_2:14 ;
hence g . i9 = (Den o,(A . i9)) . ((commute y) . i9) ; :: thesis: verum