reserve i, j, k for Nat,
  n for Nat,
  IL for non empty set,
  N for with_non-empty_elements set;
reserve R for non trivial Ring,
  a, b for Data-Location of R,
  loc for Nat,
  I for Instruction of SCM R,
  p for FinPartState of SCM R,
  s, s1, s2 for State of SCM R,
  P,P1,P2 for Instruction-Sequence of SCM R,
  q for FinPartState of SCM;

theorem Th9:
  for q being non halt-free finite
   (the InstructionsF of SCM R)-valued NAT-defined Function
  for p being q-autonomic non empty
  FinPartState of SCM R st  p c= s1 &  p c= s2 &
  q c= P1 & q c= P2 &
  CurInstr(P1,Comput(P1,s1,n)) =
  SubFrom(a, b) & a in dom p holds Comput(P1,s1,n).a - Comput(
P1,s1,n).b =
  Comput(P2,s2,n).a - Comput(P2,s2,n).b
proof
  set Cs2i1 = Comput(P2,s2,n+1);
  set Cs1i1 = Comput(P1,s1,n+1);
  set Cs2i = Comput(P2,s2,n);
  set Cs1i = Comput(P1,s1,n);
  set I = CurInstr(P1,Comput(P1,s1,n));
  let q be non halt-free finite
   (the InstructionsF of SCM R)-valued NAT-defined Function;
  let p be q-autonomic non empty FinPartState of SCM R such that
A1:  p c= s1 &  p c= s2 and
A2: q c= P1 & q c= P2;
A3: a in dom  p
 implies (Cs1i1|dom  p).a = Cs1i1.a & (Cs2i1|dom  p).a = Cs2i1.a
  by FUNCT_1:49;
A4: Cs2i1 = Following(P2,Cs2i) by EXTPRO_1:3
    .= Exec (CurInstr(P2,Cs2i), Cs2i);
  assume that
A5: I = SubFrom(a,b) and
A6: a in dom p & Comput(P1,s1,n).a - Comput(P1,s1,n).
b <>
  Comput(P2,s2,n). a - Comput(P2,s2,n).b;
  Cs1i1 = Following(P1,Cs1i) by EXTPRO_1:3
    .= Exec (CurInstr(P1,Cs1i), Cs1i);
  then
A7: Cs1i1.a = Cs1i.a - Cs1i.b by A5,SCMRING2:13;
  I = CurInstr(P2,Comput(P2,s2,n))
   by A1,A2,AMISTD_5:7;
  then Cs2i1.a = Cs2i.a - Cs2i.b by A4,A5,SCMRING2:13;
  hence contradiction by A1,A3,A6,A7,A2,EXTPRO_1:def 10;
end;
