set X = NAT ;
set a = (INT --> 0) +* (id NAT);
A1: dom (id NAT) = NAT by RELAT_1:45;
A2: INT \/ NAT = INT by NUMBERS:17, XBOOLE_1:12;
dom (INT --> 0) = INT ;
then A3: dom ((INT --> 0) +* (id NAT)) = INT by A1, A2, FUNCT_4:def 1;
rng ((INT --> 0) +* (id NAT)) c= NAT ;
then reconsider a = (INT --> 0) +* (id NAT) as INT-Array of NAT by A3, FUNCT_2:2;
let f be INT-Exec ; :: thesis:
set S = ECIW-signature ;
set G = INT-ElemIns ;
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
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 Th16; :: according to AOFA_I00:def 22 :: thesis:
thus for i being Integer holds . (i,NAT) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th18; :: 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 Th18; :: thesis:
thus for x being Element of NAT holds ^ x is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th17; :: thesis:

hereby :: thesis:

take a = a; :: thesis:
dom (id NAT) = NAT by RELAT_1:45;
hence a | (card NAT) is one-to-one by FUNCT_4:23; :: 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 Th17; :: 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 Th18; :: thesis: