let f be INT-Exec ; :: thesis:
set X = NAT ;
set S = ECIW-signature ;
set G = INT-ElemIns ;
set A = FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ;
set T = (Funcs NAT ,INT ) \ 0 ,0 ;
thus for v being INT-Variable of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f
for t being INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f holds v,t form_assignment_wrt f by II1; :: according to AOFA_I00:def 22 :: thesis:
thus for i being integer number holds . i,NAT is INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f by II3; :: thesis:
thus for v being INT-Variable of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f holds . v is INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f by II3; :: thesis:
thus for x being Element of NAT holds ^ x is INT-Variable of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f by II2; :: thesis:
set a = (INT --> 0 ) +* (id NAT );
{0 } c= NAT by ZFMISC_1:37;
then ( dom (INT --> 0 ) = INT & rng (INT --> 0 ) = {0 } & dom (id NAT ) = NAT & rng (id NAT ) = NAT & INT \/ NAT = INT & {0 } \/ NAT = NAT ) by FUNCOP_1:14, FUNCOP_1:19, NUMBERS:17, RELAT_1:71, XBOOLE_1:12;
then ( dom ((INT --> 0 ) +* (id NAT )) = INT & rng ((INT --> 0 ) +* (id NAT )) c= NAT ) by FUNCT_4:18, FUNCT_4:def 1;
then reconsider a = (INT --> 0 ) +* (id NAT ) as INT-Array of NAT by FUNCT_2:4;

hereby :: thesis:

take a = a; :: thesis:
( card NAT = NAT & dom (id NAT ) = NAT ) by CARD_1:84, RELAT_1:71;
hence a | (card NAT ) is one-to-one by FUNCT_4:24; :: thesis:
thus for t being INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f holds a * t is INT-Variable of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f by II2; :: thesis:

end;

thus ( ( for t being INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f holds - t is INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f ) & ( for t1, t2 being INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f holds
( t1 (#) t2 is INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f & t1 + t2 is INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f & t1 div t2 is INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f & t1 mod t2 is INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f & leq t1,t2 is INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f & gt t1,t2 is INT-Expression of FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ,f ) ) ) by II3; :: thesis: