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;
