reserve A,B,C,X,Y,Z,x,x1,x2,y,z for set, U,U1,U2,U3 for non empty set,
u,u1,u2 for (Element of U), P,Q,R for Relation, f,g for Function,
k,m,n for Nat, m1, n1 for non zero Nat, kk,mm,nn for (Element of NAT),
p, p1, p2 for FinSequence, q, q1, q2 for U-valued FinSequence;
reserve S, S1, S2 for Language, s,s1,s2 for Element of S,
l,l1,l2 for literal Element of S, a for ofAtomicFormula Element of S,
r for relational Element of S, w,w1,w2 for string of S,
t,t1,t2 for termal string of S;
reserve phi0 for 0wff string of S,
psi, psi1, psi2, phi,phi1,phi2 for wff string of S,
I for (S,U)-interpreter-like Function;
reserve tt,tt0,tt1,tt2 for Element of AllTermsOf S;

theorem for E being Equivalence_Relation of U,
i being E-respecting Element of U-InterpretersOf S holds
(l,E-class.u) ReassignIn (i quotient E) = ((l,u) ReassignIn i) quotient E
proof
set II=U-InterpretersOf S; let E be Equivalence_Relation of U;
let i be E-respecting Element of II; set x=u;
set O=OwnSymbolsOf S, UU=Class E, III=UU-InterpretersOf S;
reconsider X=(E-class).x as Element of UU;
reconsider I=i quotient E as Element of III;
reconsider j=(l,x) ReassignIn i as Element of II;
reconsider Jj=(l,X) ReassignIn (I qua Element of III) as Element of III;
reconsider jJ=j quotient E as Function;
A1: dom Jj=O & dom jJ=O by Def17, PARTFUN1:def 2;
set jJ=(j qua Element of II) quotient E, g=l .--> ({{}} --> x),
h={{}} --> x, G=l .--> ({{}} --> X); reconsider n=|.ar l.| as Nat;
A2: {{}}= (0-tuples_on U) & id {{}} is Equivalence_Relation of {{}} by
FOMODEL0:10; then reconsider Enn=id{{}} as Equivalence_Relation of
0-tuples_on U; set En=n-placesOf E, nE=n-tuple2Class E;
A3: dom g={l} & dom G={l} & l in dom g & l in dom G by TARSKI:def 1;
A4: En=Enn & dom (E-class)=U & dom ({{}} --> (E-class.x))={{}} &
dom h={{}} & (id {{}}) \+\ ({} .--> {}) = {} by Lm30, FUNCT_2:def 1; then
A5: En=Enn & x in dom (E-class) & {} in dom ({{}} --> (E-class.x)) &
id {{}} = {} .--> {} by TARSKI:def 1, FOMODEL0:29;
{} in dom h by  TARSKI:def 1; then
reconsider hh=h as (Enn,E)-respecting Function by Lm22;
reconsider hhh=hh as (En,E)-respecting Function of n-tuples_on U,U by A2,
A4;
now
let s be object; assume s in O;
then reconsider ss=s as own Element of S;
per cases;
suppose A6: s in dom G;
A7: s=l by A6, TARSKI:def 1; then
A8: jJ.s = (j.l) quotient E by Def18 .= (n-tuple2Class E)*((j.l) quotient
(n-placesOf E,E)) by Def15 .= nE*((g.l) quotient (En,E)) by
A3, FUNCT_4:13 .= nE*((h quotient (En,E)) qua Relation) by FUNCOP_1:72 .=
(n-placesOf (((E-class) qua Relation of U, Class E)~))*((E-class)*hhh)
by Lm21 .=
(id {{}} qua Relation)*((E-class)*hhh) by Lm30 .=
({{}} --> (E-class.x))*({{}} --> {}) by FUNCOP_1:17, A5 .=
{{}} --> (({} .--> (E-class.x)).{}) by FUNCOP_1:17, A5 .=
{{}} --> X by FUNCOP_1:72;
Jj.s = G.l by A6, A7, FUNCT_4:13 .= {{}} --> X by FUNCOP_1:72;
hence Jj.s=jJ.s by A8;
end;
suppose A9: not s in dom G; then Jj.s = I.s by FUNCT_4:11 .=
(i.ss) quotient E by Def18 .=
(j.ss) quotient E by A9, FUNCT_4:11 .= jJ.ss by Def18;
hence Jj.s=jJ.s;
end;
end;
hence thesis by A1, FUNCT_1:2;
end;
