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
for t being INT-Expression of X holds v,t form_assignment_wrt 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
for t being INT-Expression of X holds v,t form_assignment_wrt f
let c be Enumeration of X; :: thesis: for f being INT-Exec of c,T
for v being INT-Variable of X
for t being INT-Expression of X holds v,t form_assignment_wrt f
A0: ( rng c c= NAT & dom c = X ) by ThNum1, ThNum5;
then reconsider c' = c as Function of X,NAT by FUNCT_2:4;
set S = ECIW-signature ;
set G = INT-ElemIns ;
set A = FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ;
let f be INT-Exec of c,T; :: thesis: for v being INT-Variable of X
for t being INT-Expression of X holds v,t form_assignment_wrt f
let v be INT-Variable of X; :: thesis: for t being INT-Expression of X holds v,t form_assignment_wrt f
let t be INT-Expression of X; :: thesis: v,t form_assignment_wrt f
reconsider v' = v as Element of Funcs (Funcs X,INT ),X by FUNCT_2:11;
reconsider t' = t as Element of Funcs (Funcs X,INT ),INT by FUNCT_2:11;
reconsider cv = c' * v as Element of Funcs (Funcs X,INT ),NAT by FUNCT_2:11;
set v1 = cv ** c',NAT ;
set t1 = t' ** c',NAT ;
AA: ElementaryInstructions (FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ) = FreeGenSetNSG ECIW-signature ,INT-ElemIns by AOFA_000:70;
BB: Terminals (DTConUA ECIW-signature ,INT-ElemIns ) = INT-ElemIns by FREEALG:3;
CC: [(cv ** c',NAT ),(t' ** c',NAT )] in INT-ElemIns by ZFMISC_1:106;
then root-tree [(cv ** c',NAT ),(t' ** c',NAT )] in ElementaryInstructions (FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ) by AA, BB;
then reconsider I = root-tree [(cv ** c',NAT ),(t' ** c',NAT )] as Element of (FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ) ;
take I ; :: according to AOFA_I00:def 15 :: thesis: ( I in ElementaryInstructions (FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ) & ( for s being Element of Funcs X,INT holds f . s,I = s +* (v . s),(t . s) ) )
for s being Element of Funcs X,INT holds f . s,(root-tree [((c * v') ** c,NAT ),(t' ** c,NAT )]) = s +* (v' . s),(t' . s) by A0, INTiwaEXECc;
hence ( I in ElementaryInstructions (FreeUnivAlgNSG ECIW-signature ,INT-ElemIns ) & ( for s being Element of Funcs X,INT holds f . s,I = s +* (v . s),(t . s) ) ) by AA, BB, CC; :: thesis: verum