reserve N for with_zero set;

theorem Th10:
  for A being IC-Ins-separated non empty
with_non-empty_values AMI-Struct over N, I being Instruction of A
 holds I is halting iff Output I
  is empty
proof
  let A be IC-Ins-separated non empty with_non-empty_values AMI-Struct over
  N, I be Instruction of A;
  thus I is halting implies Output I is empty
  proof
    assume
A1: for s being State of A holds Exec(I,s) = s;
    assume not thesis;
    then consider o being Object of A such that
A2: o in Output I;
    ex s being State of A st s.o <> Exec(I,s).o by A2,Def3;
    hence thesis by A1;
  end;
  assume
A3: Output I is empty;
  let s be State of A;
  assume
A4: Exec(I,s) <> s;
  dom s = the carrier of A & dom Exec(I,s) = the carrier of A
  by PARTFUN1:def 2;
  then ex x being object st x in the carrier of A & Exec(I,s).x <> s.x by A4,
FUNCT_1:2;
  hence contradiction by A3,Def3;
end;
