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 Th11:
  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)) = a
=0_goto loc & loc <> (IC Comput(P1,s1,n)) + 1 holds Comput(
P1,s1,n).a = 0.
  R iff Comput(P2,s2,n).a = 0.R
proof
  set Cs2i1 = Comput(P2,s2,n+1);
  set Cs1i1 = Comput(P1,s1,n+1);
  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: I = CurInstr(P2,Comput(P2,s2,n))
        by A1,A2,AMISTD_5:7;
  set Cs2i = Comput(P2,s2,n);
  set Cs1i = Comput(P1,s1,n);
A4: Cs1i1 = Following(P1,Cs1i) by EXTPRO_1:3
    .= Exec (CurInstr(P1,Cs1i), Cs1i);
A5: Cs2i1 = Following(P2,Cs2i) by EXTPRO_1:3
    .= Exec (CurInstr(P2,Cs2i), Cs2i);
   IC SCM R in dom p by AMISTD_5:6;
   then
A6: (Cs1i1|dom  p).IC SCM R = Cs1i1.IC SCM R &
   (Cs2i1|dom  p).IC SCM R = Cs2i1.IC SCM R by FUNCT_1:49;
  assume that
A7: I = a=0_goto loc and
A8: loc <> (IC Comput(P1,s1,n)) + 1;
A9: IC Cs1i = IC Cs2i by A1,A2,AMISTD_5:7;
  hereby
    assume
    Comput(P1,s1,n).a = 0.R & Comput(P2,s2,n).a <> 0.
R;
    then Cs1i1.IC SCM R = loc & Cs2i1.IC SCM R = IC Cs2i + 1 by A3,A4,A5,A7,
SCMRING2:16;
    hence contradiction by A1,A9,A6,A8,A2,EXTPRO_1:def 10;
  end;
  assume that
A10: Comput(P2,s2,n).a = 0.R and
A11: Comput(P1,s1,n).a <> 0.R;
A12: Cs1i1.IC SCM R = IC Cs1i + 1 by A4,A7,A11,SCMRING2:16;
  Cs2i1.IC SCM R = loc by A3,A5,A7,A10,SCMRING2:16;
  hence contradiction by A1,A6,A8,A12,A2,EXTPRO_1:def 10;
end;
