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 . (d,e) = e holds
G . (d,(F \$\$ (B,f))) = F \$\$ (B,(G [;] (d,f)))

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 . (d,e) = e holds
G . (d,(F \$\$ (B,f))) = F \$\$ (B,(G [;] (d,f)))

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 . (d,e) = e holds
G . (d,(F \$\$ (B,f))) = F \$\$ (B,(G [;] (d,f)))

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 . (d,e) = e holds
G . (d,(F \$\$ (B,f))) = F \$\$ (B,(G [;] (d,f)))

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 . (d,e) = e implies G . (d,(F \$\$ (B,f))) = F \$\$ (B,(G [;] (d,f))) )
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 . (d,e) = e or G . (d,(F \$\$ (B,f))) = F \$\$ (B,(G [;] (d,f))) )
defpred S1[ Element of Fin C] means G . (d,(F \$\$ (\$1,f))) = F \$\$ (\$1,(G [;] (d,f)));
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 . (d,(F \$\$ (B9,f))) = F \$\$ (B9,(G [;] (d,f))) and
A6: not c in B9 ; :: thesis: S1[B9 \/ {.c.}]
thus G . (d,(F \$\$ ((B9 \/ {.c.}),f))) = G . (d,(F . ((F \$\$ (B9,f)),(f . c)))) by A1, A6, Th2
.= F . ((G . (d,(F \$\$ (B9,f)))),(G . (d,(f . c)))) by
.= F . ((F \$\$ (B9,(G [;] (d,f)))),((G [;] (d,f)) . c)) by
.= F \$\$ ((B9 \/ {.c.}),(G [;] (d,f))) by A1, A6, Th2 ; :: thesis: verum
end;
assume G . (d,e) = e ; :: thesis: G . (d,(F \$\$ (B,f))) = F \$\$ (B,(G [;] (d,f)))
then G . (d,(F \$\$ (({}. C),f))) = e by
.= F \$\$ (({}. C),(G [;] (d,f))) by ;
then A7: S1[ {}. C] ;
for B being Element of Fin C holds S1[B] from hence G . (d,(F \$\$ (B,f))) = F \$\$ (B,(G [;] (d,f))) ; :: thesis: verum