reserve x for set,
  m,n for Nat,
  a,b,c for Int_position,
  i for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer,
  loc,l1 for Nat,
  I,J for Program of SCMPDS,
  N for with_non-empty_elements set;
reserve P,P1,P2,Q for Instruction-Sequence of SCMPDS;

theorem Th6:
  for s being 0-started State of SCMPDS
  for I being parahalting Program of SCMPDS
   st stop I c= P1 & stop I c= P2
  holds LifeSpan(P1,s) = LifeSpan(P2,s) &
   Result(P1,s) =  Result(P2,s)
proof
  let s be 0-started State of SCMPDS;
  let I be parahalting Program of SCMPDS;
  set SI=stop I;
  assume that
A1: SI c= P1 and
A2: SI c= P2;
A3: P2 halts_on s by A2,SCMPDS_4:def 7;
A4: P1 halts_on s by A1,SCMPDS_4:def 7;
A5: now
    let l be Nat;
    assume
A6: CurInstr(P2,Comput(P2,s,l)) = halt SCMPDS;
    CurInstr(P1,Comput(P1,s,l))
     = CurInstr(P2,Comput(P2,s,l))
     by A1,A2,Th5;
    hence LifeSpan(P1,s) <= l by A4,A6,EXTPRO_1:def 15;
  end;
  CurInstr(P2,Comput(P2,s,LifeSpan(P1,s)))
     = CurInstr(P1,Comput(P1,s,LifeSpan(P1,s))) by A1,A2,Th5
    .= halt SCMPDS by A4,EXTPRO_1:def 15;
  hence
A7:  LifeSpan(P1,s) = LifeSpan(P2,s) by A5,A3,EXTPRO_1:def 15;
   P2 halts_on s by A2,SCMPDS_4:def 7;
   then
A8: Result(P2,s) = Comput(P2,s,LifeSpan(P1,s))
     by A7,EXTPRO_1:23;
    P1 halts_on s by A1,SCMPDS_4:def 7;
    then
    Result(P1,s) = Comput(P1,s,LifeSpan(P1,s))
      by EXTPRO_1:23;
  hence thesis by A1,A2,A8,Th5;
end;
