let D be non empty set ; :: thesis: for I, J being non empty set
for F, G being BinOp of D
for f being Function of I,D
for g being Function of J,D
for X being Element of Fin I
for Y being Element of Fin J st F is having_a_unity & F is commutative & F is associative & F is having_an_inverseOp & G is_distributive_wrt F holds
F $$ [:X,Y:],(G * f,g) = F $$ X,(G [:] f,(F $$ Y,g))
let I, J be non empty set ; :: thesis: for F, G being BinOp of D
for f being Function of I,D
for g being Function of J,D
for X being Element of Fin I
for Y being Element of Fin J st F is having_a_unity & F is commutative & F is associative & F is having_an_inverseOp & G is_distributive_wrt F holds
F $$ [:X,Y:],(G * f,g) = F $$ X,(G [:] f,(F $$ Y,g))
let F, G be BinOp of D; :: thesis: for f being Function of I,D
for g being Function of J,D
for X being Element of Fin I
for Y being Element of Fin J st F is having_a_unity & F is commutative & F is associative & F is having_an_inverseOp & G is_distributive_wrt F holds
F $$ [:X,Y:],(G * f,g) = F $$ X,(G [:] f,(F $$ Y,g))
let f be Function of I,D; :: thesis: for g being Function of J,D
for X being Element of Fin I
for Y being Element of Fin J st F is having_a_unity & F is commutative & F is associative & F is having_an_inverseOp & G is_distributive_wrt F holds
F $$ [:X,Y:],(G * f,g) = F $$ X,(G [:] f,(F $$ Y,g))
let g be Function of J,D; :: thesis: for X being Element of Fin I
for Y being Element of Fin J st F is having_a_unity & F is commutative & F is associative & F is having_an_inverseOp & G is_distributive_wrt F holds
F $$ [:X,Y:],(G * f,g) = F $$ X,(G [:] f,(F $$ Y,g))
let X be Element of Fin I; :: thesis: for Y being Element of Fin J st F is having_a_unity & F is commutative & F is associative & F is having_an_inverseOp & G is_distributive_wrt F holds
F $$ [:X,Y:],(G * f,g) = F $$ X,(G [:] f,(F $$ Y,g))
let Y be Element of Fin J; :: thesis: ( F is having_a_unity & F is commutative & F is associative & F is having_an_inverseOp & G is_distributive_wrt F implies F $$ [:X,Y:],(G * f,g) = F $$ X,(G [:] f,(F $$ Y,g)) )
assume that
A1: ( F is having_a_unity & F is commutative & F is associative ) and
A2: ( F is having_an_inverseOp & G is_distributive_wrt F ) ; :: thesis: F $$ [:X,Y:],(G * f,g) = F $$ X,(G [:] f,(F $$ Y,g))
defpred S₁[ Element of Fin I] means F $$ [:$1,Y:],(G * f,g) = F $$ $1,(G [:] f,(F $$ Y,g));
A3: S₁[ {}. I]

proof

reconsider T = {}. [:I,J:] as Element of Fin [:I,J:] ;
T = [:({}. I),Y:] by ZFMISC_1:113;
then F $$ [:({}. I),Y:],(G * f,g) = the_unity_wrt F by A1, SETWISEO:40;
hence S₁[ {}. I] by A1, SETWISEO:40; :: thesis: verum

end;

A4: for X1 being Element of Fin I
for x being Element of I st S₁[X1] holds
S₁[X1 \/ {.x.}]

proof

let X1 be Element of Fin I; :: thesis: for x being Element of I st S₁[X1] holds
S₁[X1 \/ {.x.}]
let x be Element of I; :: thesis: ( S₁[X1] implies S₁[X1 \/ {.x.}] )
assume A5: F $$ [:X1,Y:],(G * f,g) = F $$ X1,(G [:] f,(F $$ Y,g)) ; :: thesis: S₁[X1 \/ {.x.}]
reconsider s = {.x.} as Element of Fin I ;

now

per cases ( x in X1 or not x in X1 ) ;

case x in X1 ; :: thesis: S₁[X1 \/ {.x.}]

then X1 \/ {x} = X1 by ZFMISC_1:46;
hence S₁[X1 \/ {.x.}] by A5; :: thesis: verum

end;

case not x in X1 ; :: thesis: F $$ [:(X1 \/ {.x.}),Y:],(G * f,g) = F $$ (X1 \/ {.x.}),(G [:] f,(F $$ Y,g))

then A6: X1 misses {x} by ZFMISC_1:56;
then A7: [:X1,Y:] misses [:s,Y:] by ZFMISC_1:127;
thus F $$ [:(X1 \/ {.x.}),Y:],(G * f,g) = F $$ ([:X1,Y:] \/ [:s,Y:]),(G * f,g) by ZFMISC_1:120
.= F . (F $$ [:X1,Y:],(G * f,g)),(F $$ [:s,Y:],(G * f,g)) by A1, A7, SETWOP_2:6
.= F . (F $$ X1,(G [:] f,(F $$ Y,g))),(F $$ s,(G [:] f,(F $$ Y,g))) by A1, A2, A5, Th27
.= F $$ (X1 \/ {.x.}),(G [:] f,(F $$ Y,g)) by A1, A6, SETWOP_2:6 ; :: thesis: verum

end;

end;

end;

hence S₁[X1 \/ {.x.}] ; :: thesis: verum

end;

for X1 being Element of Fin I holds S₁[X1] from SETWISEO:sch 4(A3, A4);
hence F $$ [:X,Y:],(G * f,g) = F $$ X,(G [:] f,(F $$ Y,g)) ; :: thesis: verum