let C, D be non empty set ; :: thesis: for B being Element of Fin C

for d, e being Element of D

for F, G being BinOp of D

for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . (e,d) = e holds

G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d)))

let B be Element of Fin C; :: thesis: for d, e being Element of D

for F, G being BinOp of D

for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . (e,d) = e holds

G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d)))

let d, e be Element of D; :: thesis: for F, G being BinOp of D

for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . (e,d) = e holds

G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d)))

let F, G be BinOp of D; :: thesis: for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . (e,d) = e holds

G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d)))

let f be Function of C,D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . (e,d) = e implies G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d))) )

assume that

A1: ( F is commutative & F is associative & F is having_a_unity ) and

A2: e = the_unity_wrt F and

A3: G is_distributive_wrt F ; :: thesis: ( not G . (e,d) = e or G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d))) )

defpred S_{1}[ Element of Fin C] means G . ((F $$ ($1,f)),d) = F $$ ($1,(G [:] (f,d)));

A4: for B9 being Element of Fin C

for b being Element of C st S_{1}[B9] & not b in B9 holds

S_{1}[B9 \/ {.b.}]

then G . ((F $$ (({}. C),f)),d) = e by A1, A2, SETWISEO:31

.= F $$ (({}. C),(G [:] (f,d))) by A1, A2, SETWISEO:31 ;

then A7: S_{1}[ {}. C]
;

for B being Element of Fin C holds S_{1}[B]
from SETWISEO:sch 2(A7, A4);

hence G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d))) ; :: thesis: verum

for d, e being Element of D

for F, G being BinOp of D

for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . (e,d) = e holds

G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d)))

let B be Element of Fin C; :: thesis: for d, e being Element of D

for F, G being BinOp of D

for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . (e,d) = e holds

G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d)))

let d, e be Element of D; :: thesis: for F, G being BinOp of D

for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . (e,d) = e holds

G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d)))

let F, G be BinOp of D; :: thesis: for f being Function of C,D st F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . (e,d) = e holds

G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d)))

let f be Function of C,D; :: thesis: ( F is commutative & F is associative & F is having_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . (e,d) = e implies G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d))) )

assume that

A1: ( F is commutative & F is associative & F is having_a_unity ) and

A2: e = the_unity_wrt F and

A3: G is_distributive_wrt F ; :: thesis: ( not G . (e,d) = e or G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d))) )

defpred S

A4: for B9 being Element of Fin C

for b being Element of C st S

S

proof

assume
G . (e,d) = e
; :: thesis: G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d)))
let B9 be Element of Fin C; :: thesis: for b being Element of C st S_{1}[B9] & not b in B9 holds

S_{1}[B9 \/ {.b.}]

let c be Element of C; :: thesis: ( S_{1}[B9] & not c in B9 implies S_{1}[B9 \/ {.c.}] )

assume that

A5: G . ((F $$ (B9,f)),d) = F $$ (B9,(G [:] (f,d))) and

A6: not c in B9 ; :: thesis: S_{1}[B9 \/ {.c.}]

thus G . ((F $$ ((B9 \/ {.c.}),f)),d) = G . ((F . ((F $$ (B9,f)),(f . c))),d) by A1, A6, Th2

.= F . ((G . ((F $$ (B9,f)),d)),(G . ((f . c),d))) by A3, BINOP_1:11

.= F . ((F $$ (B9,(G [:] (f,d)))),((G [:] (f,d)) . c)) by A5, FUNCOP_1:48

.= F $$ ((B9 \/ {.c.}),(G [:] (f,d))) by A1, A6, Th2 ; :: thesis: verum

end;S

let c be Element of C; :: thesis: ( S

assume that

A5: G . ((F $$ (B9,f)),d) = F $$ (B9,(G [:] (f,d))) and

A6: not c in B9 ; :: thesis: S

thus G . ((F $$ ((B9 \/ {.c.}),f)),d) = G . ((F . ((F $$ (B9,f)),(f . c))),d) by A1, A6, Th2

.= F . ((G . ((F $$ (B9,f)),d)),(G . ((f . c),d))) by A3, BINOP_1:11

.= F . ((F $$ (B9,(G [:] (f,d)))),((G [:] (f,d)) . c)) by A5, FUNCOP_1:48

.= F $$ ((B9 \/ {.c.}),(G [:] (f,d))) by A1, A6, Th2 ; :: thesis: verum

then G . ((F $$ (({}. C),f)),d) = e by A1, A2, SETWISEO:31

.= F $$ (({}. C),(G [:] (f,d))) by A1, A2, SETWISEO:31 ;

then A7: S

for B being Element of Fin C holds S

hence G . ((F $$ (B,f)),d) = F $$ (B,(G [:] (f,d))) ; :: thesis: verum