reserve x,y for set;
reserve i, j, k for Nat;
reserve I,J,K for Element of Segm 9,
  a,a1 for Nat,
  b,b1,c for Element of Data-Locations SCM;
reserve a, b for Data-Location,
  loc for Nat;
reserve I,J,K for Element of Segm 9,
  a,a1 for Nat,
  b,b1,c for Element of Data-Locations SCM,
  da,db for Data-Location;

theorem
  for q being non halt-free finite
      (the InstructionsF of SCM)-valued NAT-defined Function
  for p being q-autonomic non empty FinPartState of SCM, s1, s2
  being State of SCM st  p c= s1 &  p c= s2
  for P1,P2 being Instruction-Sequence of SCM
      st q c= P1 & q c= P2
  for i being Nat, da, db
being Data-Location, I being Instruction of SCM st
 I = CurInstr(P1,Comput(P1,s1,i))
  holds I = Divide(da, db) & db in dom p implies Comput(P1,s1,i).
da mod
  Comput(P1,s1,i).db = Comput(P2,s2,i).da mod Comput(P2,s2,i).db
proof
  let q be non halt-free finite
      (the InstructionsF of SCM)-valued NAT-defined Function;
  let p be q-autonomic non empty FinPartState of SCM,
      s1, s2 be State of
  SCM such that
A1:  p c= s1 &  p c= s2;
  let P1,P2 be Instruction-Sequence of SCM
  such that
A2: q c= P1 & q c= P2;
  let i be Nat, da, db be Data-Location, I be Instruction of SCM
  such that
A3: I = CurInstr(P1,Comput(P1,s1,i));
  set Cs1i1 = Comput(P1,s1,i+1);
  set Cs1i = Comput(P1,s1,i);
  set Cs2i1 = Comput(P2,s2,i+1);
  set Cs2i = Comput(P2,s2,i);
A4: Cs2i1 = Following(P2,Cs2i) by EXTPRO_1:3
    .= Exec (CurInstr(P2,Cs2i), Cs2i);
  assume that
A5: I = Divide(da, db) and
A6: db in dom p and
A7: Comput(P1,s1,i).da mod Comput(P1,s1,i).db <>
Comput(P2,s2,i).
  da mod Comput(P2,s2,i).db;
A8: (Cs1i1|dom  p).db = Cs1i1.db &
    (Cs2i1|dom  p).db = Cs2i1.db by A6,FUNCT_1:49;
  I = CurInstr(P2,Comput(P2,s2,i)) by A3,A2,A1,AMISTD_5:7;
  then
A9: Cs2i1.db = Cs2i.da mod Cs2i.db by A4,A5,AMI_3:6;
  Cs1i1 = Following(P1,Cs1i) by EXTPRO_1:3
    .= Exec (CurInstr(P1,Cs1i), Cs1i);
  then Cs1i1.db = Cs1i.da mod Cs1i.db by A3,A5,AMI_3:6;
  hence contradiction by A7,A8,A9,A2,A1,EXTPRO_1:def 10;
end;
