let f1, f2 be non empty NAT * -defined to-naturals homogeneous Function; :: thesis: ( f1 is len-total & arity f1 = 0 implies (primrec (f1,f2,1)) . <*0*> = f1 . {} )
assume that
A1: f1 is len-total and
A2: arity f1 = 0 ; :: thesis: (primrec (f1,f2,1)) . <*0*> = f1 . {}
reconsider p = <*0*> as Element of ((arity f1) + 1) -tuples_on NAT by A2, FINSEQ_2:131;
len p = 1 by FINSEQ_1:39;
then A3: 1 in dom p by FINSEQ_3:25;
p +* (1,0) = p by FUNCT_7:95;
hence (primrec (f1,f2,1)) . <*0*> = f1 . (Del (p,1)) by A1, A3, Th59
.= f1 . {} by WSIERP_1:19 ;
:: thesis: verum