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, Th18;
A3: commute y in product (doms (A ?. o)) by A1, Th17;
A4: Den (o,(product A)) = (OPS A) . o by MSUALG_1:def 6
.= IFEQ ((the_arity_of o),{},(commute (A ?. o)),(Commute (Frege (A ?. o)))) by PRALG_2:def 13
.= Commute (Frege (A ?. o)) by A1, FUNCOP_1:def 8 ;
set C = union { ( the Sorts of (A . i9) . s9) where i9 is Element of I, s9 is Element of the carrier of S : verum } ;
y in Funcs ((dom (the_arity_of o)),(Funcs (I,(union { ( the Sorts of (A . i9) . s9) where i9 is Element of I, s9 is Element of the carrier of S : verum } )))) by Th14;
then commute y in Funcs (I,(Funcs ((dom (the_arity_of o)),(union { ( the Sorts of (A . i9) . s9) where i9 is Element of I, s9 is Element of the carrier of S : verum } )))) by A1, FUNCT_6:55;
then WW: dom (commute y) = I by FUNCT_2:92;
dom (A ?. o) = I by PARTFUN1:def 2;
then a5: dom ((A ?. o) .. (commute y)) = I /\ (dom (commute y)) by PRALG_1:def 19
.= I by WW ;
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 1
.= (A ?. o) .. (commute y) by A3, PRALG_2:def 2 ;
then g . i9 = ((A ?. o) . i9) . ((commute y) . i9) by a5, PRALG_1:def 19
.= (Den (o,(A . i9))) . ((commute y) . i9) by PRALG_2:7 ;
hence g . i9 = (Den (o,(A . i9))) . ((commute y) . i9) ; :: thesis: verum