theorem ex u st u=I.l.{} & (l,u) ReassignIn I = I
proof
set O=OwnSymbolsOf S; O\(dom I)={}; then
A1: O c= dom I & {{}}={{}} by XBOOLE_1:37;
reconsider lo=l as Element of O by FOMODEL1:def 19; reconsider i=I.l as
Interpreter of l, U;
i is Function of 0-tuples_on U, U & 0-tuples_on U={{}}
by FOMODEL0:10, Def2; then reconsider ii=i as Function of {{}}, U;
reconsider e={} as Element of {{}} by TARSKI:def 1;
reconsider u=ii.e as Element of U; take u; thus u=I.l.{};
set h={}.-->u, H=l.-->h, J=(l,u) ReassignIn I; h= {{}} --> u; then
reconsider hh=h as Function of {{}}, U;
A2: dom H={lo}; then
A3: dom H c= dom I by A1;
for z be Element of {{}} holds ii.z=hh.z; then
A4: ii=hh by FUNCT_2:63;
now
let z be object; assume
A5: z in dom H;
thus H.z=h by FUNCOP_1:7, A5
.= I.z by A4, A5, TARSKI:def 1;
end; then H tolerates I by PARTFUN1:53, A3;
then J=H+*I by FUNCT_4:34 .= I
by A2, A1, XBOOLE_1:1, FUNCT_4:19; hence thesis;
end;
