set A = the non empty set ;
set a = the Element of the non empty set ;
reconsider w = (<*> the non empty set ) .--> the Element of the non empty set as Element of PFuncs (( the non empty set *), the non empty set ) by MARGREL1:19;
set U0 = UAStr(# the non empty set ,<*w*> #);
A1: ( the charact of UAStr(# the non empty set ,<*w*> #) is quasi_total & the charact of UAStr(# the non empty set ,<*w*> #) is homogeneous ) by MARGREL1:20;
the charact of UAStr(# the non empty set ,<*w*> #) is non-empty by MARGREL1:20;
then reconsider U0 = UAStr(# the non empty set ,<*w*> #) as Universal_Algebra by ;
take U0 ; :: thesis: ( U0 is with_const_op & U0 is strict )
( dom <*w*> = {1} & 1 in {1} ) by ;
then reconsider o = the charact of U0 . 1 as operation of U0 by FUNCT_1:def 3;
o = w by FINSEQ_1:40;
then A2: dom o = {(<*> the non empty set )} by FUNCOP_1:13;
A3: now :: thesis: for x being FinSequence st x in dom o holds
len x = 0
let x be FinSequence; :: thesis: ( x in dom o implies len x = 0 )
assume x in dom o ; :: thesis:
then x = <*> the non empty set by ;
hence len x = 0 ; :: thesis: verum
end;
<*> the non empty set in {(<*> the non empty set )} by TARSKI:def 1;
then arity o = 0 by ;
hence ( U0 is with_const_op & U0 is strict ) ; :: thesis: verum