A1: rng c c= NAT by Th11;
card X = rng c by Th6;
then A2: INT \/ (card X) = INT by A1, NUMBERS:17, XBOOLE_1:1, XBOOLE_1:12;
set x = (c ") . 0;
set a = (INT --> ((c ") . 0)) +* (c ");
A3: dom (INT --> ((c ") . 0)) = INT ;
A4: rng (INT --> ((c ") . 0)) = {((c ") . 0)} by FUNCOP_1:8;
(c ") . 0 in X by FUNCT_2:5, ORDINAL3:8;
then {((c ") . 0)} c= X by ZFMISC_1:31;
then A5: {((c ") . 0)} \/ X = X by XBOOLE_1:12;
rng (c ") = X by Th7;
then A6: rng ((INT --> ((c ") . 0)) +* (c ")) c= X by A4, A5, FUNCT_4:17;
dom (c ") = card X by Th7;
then dom ((INT --> ((c ") . 0)) +* (c ")) = INT by A3, A2, FUNCT_4:def 1;
then reconsider a = (INT --> ((c ") . 0)) +* (c ") as INT-Array of X by A6, FUNCT_2:2;
let f be INT-Exec of c,T; :: 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 Th19; :: according to AOFA_I00:def 22 :: thesis:
thus for i being Integer holds . (i,X) is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th21; :: 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 Th21; :: thesis:
thus for x being Element of X holds ^ x is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f by Th20; :: thesis:

hereby :: thesis:

take a = a; :: thesis:
dom (c ") = card X by Th7;
hence a | (card X) 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 Th20; :: 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 Th21; :: thesis: