reserve k for Nat;
reserve N for with_zero set,
   S for IC-recognized
    halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N;
reserve
   S for IC-recognized CurIns-recognized
    halting IC-Ins-separated
  non empty with_non-empty_values AMI-Struct over N;

theorem
 for q be non halt-free finite
  (the InstructionsF of S)-valued NAT-defined Function
 for p being q-autonomic non empty FinPartState of S,
     s1,s2 being State of S st  p c= s1 &  p c= s2
 for P1,P2 being Instruction-Sequence of S
    st q c= P1 & q c= P2
 for i being Nat
 holds IC Comput(P1,s1,i) = IC Comput(P2,s2,i) &
   CurInstr(P1,Comput(P1,s1,i)) = CurInstr(P2,Comput(P2,s2,i))
proof
 let q be non halt-free finite
  (the InstructionsF of S)-valued NAT-defined Function;
  let p be q-autonomic non empty FinPartState of S,
      s1, s2 be State of S such that
A1:  p c= s1 and
A2:  p c= s2;
 let P1,P2 be Instruction-Sequence of S such that
A3: q c= P1 and
A4: q c= P2;
A5: dom q c= dom P1 by A3,RELAT_1:11;
A6: dom q c= dom P2 by A4,RELAT_1:11;
  let i be Nat;
  set Cs2i = Comput(P2,s2,i);
  set Cs1i = Comput(P1,s1,i);
A7: IC Cs1i in dom q by A3,Def4,A1;
A8: IC Cs2i in dom q by A4,Def4,A2;
  thus
A9: IC Cs1i = IC Cs2i
  proof
    assume
A10: IC Comput(P1,s1,i) <> IC Comput(P2,s2,i);
    (Cs1i|dom  p).IC S = Cs1i.IC S & (Cs2i|dom  p).IC S = Cs2i.IC S
    by Th6,FUNCT_1:49;
    hence contradiction by A10,A3,A4,A1,A2,EXTPRO_1:def 10;
  end;
  thus CurInstr(P1,Comput(P1,s1,i))
   = P1.IC Cs1i by A7,A5,PARTFUN1:def 6
  .= q.IC Cs1i by A7,A3,GRFUNC_1:2
  .= P2.IC Cs2i by A8,A4,A9,GRFUNC_1:2
  .= CurInstr(P2,Comput(P2,s2,i)) by A8,A6,PARTFUN1:def 6;
end;
