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 S1[ 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 S1[B9] & not b in B9 holds
S1[B9 \/ {.b.}]
proof
let B9 be Element of Fin C; :: thesis: for b being Element of C st S1[B9] & not b in B9 holds
S1[B9 \/ {.b.}]

let c be Element of C; :: thesis: ( S1[B9] & not c in B9 implies S1[B9 \/ {.c.}] )
assume that
A5: G . ((F \$\$ (B9,f)),d) = F \$\$ (B9,(G [:] (f,d))) and
A6: not c in B9 ; :: thesis: S1[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
.= F . ((F \$\$ (B9,(G [:] (f,d)))),((G [:] (f,d)) . c)) by
.= F \$\$ ((B9 \/ {.c.}),(G [:] (f,d))) by A1, A6, Th2 ; :: thesis: verum
end;
assume G . (e,d) = e ; :: thesis: G . ((F \$\$ (B,f)),d) = F \$\$ (B,(G [:] (f,d)))
then G . ((F \$\$ (({}. C),f)),d) = e by
.= F \$\$ (({}. C),(G [:] (f,d))) by ;
then A7: S1[ {}. C] ;
for B being Element of Fin C holds S1[B] from hence G . ((F \$\$ (B,f)),d) = F \$\$ (B,(G [:] (f,d))) ; :: thesis: verum