set t = the INT-Expression of NAT;
let f be INT-Exec ; :: thesis: for v being INT-Variable of NAT holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
set S = ECIW-signature ;
set G = INT-ElemIns ;
set X = NAT ;
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
let v be INT-Variable of NAT; :: thesis: v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
A1: Terminals (DTConUA (ECIW-signature,INT-ElemIns)) = INT-ElemIns by FREEALG:3;
reconsider t9 = the INT-Expression of NAT as Element of Funcs ((Funcs (NAT,INT)),INT) by FUNCT_2:8;
reconsider v9 = v as Element of Funcs ((Funcs (NAT,INT)),NAT) by FUNCT_2:8;
A2: [v9,t9] in INT-ElemIns by ZFMISC_1:87;
A3: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) = FreeGenSetNSG (ECIW-signature,INT-ElemIns) by AOFA_000:70;
then root-tree [v9,t9] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A2;
then reconsider I = root-tree [v9,t9] as Element of (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) ;

hereby :: according to AOFA_I00:def 16 :: thesis: ex t being INT-Expression of NAT st v,t form_assignment_wrt f

take I = I; :: thesis: I is_assignment_wrt FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), NAT ,f
thus I is_assignment_wrt FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), NAT ,f :: thesis: verum

proof

thus I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A3, A1, A2; :: according to AOFA_I00:def 14 :: thesis: ex v being INT-Variable of NAT ex t being INT-Expression of NAT st
for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),(t . s))
take v ; :: thesis: ex t being INT-Expression of NAT st
for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),(t . s))
take the INT-Expression of NAT ; :: thesis: for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),( the INT-Expression of NAT . s))
thus for s being Element of Funcs (NAT,INT) holds f . (s,I) = s +* ((v . s),( the INT-Expression of NAT . s)) by Def25; :: thesis: verum

end;

end;

take the INT-Expression of NAT ; :: thesis: v, the INT-Expression of NAT form_assignment_wrt f
thus v, the INT-Expression of NAT form_assignment_wrt f by Th16; :: thesis: verum