let D be non empty set ; :: thesis: for F being FinSequence of D
for g being BinOp of D st len F = 0 & g is having_a_unity & g is associative & g is commutative holds
g "**" F = g $$ (findom F),((NAT --> (the_unity_wrt g)) +* F)
let F be FinSequence of D; :: thesis: for g being BinOp of D st len F = 0 & g is having_a_unity & g is associative & g is commutative holds
g "**" F = g $$ (findom F),((NAT --> (the_unity_wrt g)) +* F)
let g be BinOp of D; :: thesis: ( len F = 0 & g is having_a_unity & g is associative & g is commutative implies g "**" F = g $$ (findom F),((NAT --> (the_unity_wrt g)) +* F) )
assume A1: ( len F = 0 & g is having_a_unity & g is associative & g is commutative ) ; :: thesis: g "**" F = g $$ (findom F),((NAT --> (the_unity_wrt g)) +* F)
then F = {} ;
then A2: dom F = {}. NAT by RELAT_1:60;
thus g "**" F = the_unity_wrt g by A1, Def1
.= g $$ (findom F),((NAT --> (the_unity_wrt g)) +* F) by A1, A2, SETWISEO:40 ; :: thesis: verum