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 v being INT-Variable of X holds v is INT-Variable 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 v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f

let c be Enumeration of X; :: thesis: for f being INT-Exec of c,T

for v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f

set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);

let f be INT-Exec of c,T; :: thesis: for v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f

let v be INT-Variable of X; :: thesis: v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f

set t = the INT-Expression of X;

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 * v as Element of Funcs ((Funcs (X,INT)),NAT) by FUNCT_2:8;

reconsider v9 = v as Element of Funcs ((Funcs (X,INT)),X) by FUNCT_2:8;

reconsider t9 = the INT-Expression of X 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)) ;

thus v, the INT-Expression of X form_assignment_wrt f by Th19; :: thesis: verum

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 v being INT-Variable of X holds v is INT-Variable 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 v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f

let c be Enumeration of X; :: thesis: for f being INT-Exec of c,T

for v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f

set A = FreeUnivAlgNSG (ECIW-signature,INT-ElemIns);

let f be INT-Exec of c,T; :: thesis: for v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f

let v be INT-Variable of X; :: thesis: v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f

set t = the INT-Expression of X;

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 * v as Element of Funcs ((Funcs (X,INT)),NAT) by FUNCT_2:8;

reconsider v9 = v as Element of Funcs ((Funcs (X,INT)),X) by FUNCT_2:8;

reconsider t9 = the INT-Expression of X 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 16 :: thesis: ex t being INT-Expression of X st v,t form_assignment_wrt f

take
the INT-Expression of X
; :: thesis: v, the INT-Expression of X form_assignment_wrt ftake 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

end;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 v ; :: thesis: 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-Expression of X ; :: thesis: for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),( the INT-Expression of X . 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 +* ((v . s),( the INT-Expression of X . s)) ; :: thesis: verum

end;for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s))

take v ; :: thesis: 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-Expression of X ; :: thesis: for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),( the INT-Expression of X . 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 +* ((v . s),( the INT-Expression of X . s)) ; :: thesis: verum

thus v, the INT-Expression of X form_assignment_wrt f by Th19; :: thesis: verum