set S = ECIW-signature ;
set G = INT-ElemIns ;
let X be non empty countable set ; :: thesis: for T being Subset of (Funcs (X,INT))
for c being Enumeration of X
for f being INT-Exec of c,T
for t being INT-Expression of X holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
let T be Subset of (Funcs (X,INT)); :: thesis: for c being Enumeration of X
for f being INT-Exec of c,T
for t being INT-Expression of X holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
let c be Enumeration of X; :: thesis: for f being INT-Exec of c,T
for t being INT-Expression of X holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);
let f be INT-Exec of c,T; :: thesis: for t being INT-Expression of X holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
set v = the INT-Variable of X;
let t be INT-Expression of X; :: thesis: t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
A1: ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) = FreeGenSetNSG (ECIW-signature,INT-ElemIns) by AOFA_000:70;
A2: rng c c= NAT by Th11;
dom c = X by Th6;
then reconsider c9 = c as Function of X,NAT by A2, FUNCT_2:2;
reconsider cv = c9 * the INT-Variable of X as Element of Funcs ((Funcs (X,INT)),NAT) by FUNCT_2:8;
reconsider v9 = the INT-Variable of X as Element of Funcs ((Funcs (X,INT)),X) by FUNCT_2:8;
reconsider t9 = t as Element of Funcs ((Funcs (X,INT)),INT) by FUNCT_2:8;
A3: Terminals (DTConUA (ECIW-signature,INT-ElemIns)) = INT-ElemIns by FREEALG:3;
set v1 = cv ** (c9,NAT);
set t1 = t9 ** (c9,NAT);
A4: [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] in INT-ElemIns by ZFMISC_1:87;
then root-tree [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A3;
then reconsider I = root-tree [(cv ** (c9,NAT)),(t9 ** (c9,NAT))] as Element of (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) ;

hereby :: according to AOFA_I00:def 17 :: thesis: ex v being INT-Variable of X st v,t form_assignment_wrt f

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

proof

thus I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,INT-ElemIns)) by A1, A3, A4; :: according to AOFA_I00:def 14 :: thesis: ex v being INT-Variable of X ex t being INT-Expression of X st
for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s))
take the INT-Variable of X ; :: thesis: ex t being INT-Expression of X st
for s being Element of Funcs (X,INT) holds f . (s,I) = s +* (( the INT-Variable of X . s),(t . s))
take t ; :: thesis: for s being Element of Funcs (X,INT) holds f . (s,I) = s +* (( the INT-Variable of X . s),(t . s))
for s being Element of Funcs (X,INT) holds f . (s,(root-tree [((c * v9) ** (c,NAT)),(t9 ** (c,NAT))])) = s +* ((v9 . s),(t9 . s)) by A2, Def28;
hence for s being Element of Funcs (X,INT) holds f . (s,I) = s +* (( the INT-Variable of X . s),(t . s)) ; :: thesis: verum

end;

end;

take the INT-Variable of X ; :: thesis: the INT-Variable of X,t form_assignment_wrt f
thus the INT-Variable of X,t form_assignment_wrt f by Th19; :: thesis: verum