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 Th21:
  for s being 0-started State of SCMPDS
  for I being parahalting halt-free Program of SCMPDS,
      J being parahalting shiftable Program of SCMPDS holds
  LifeSpan(P+*stop(I ';' J), s)
   = LifeSpan(P +* stop I,s) +
     LifeSpan(P +* stop I +* stop J,Initialize Result(P +* stop I,s))
proof
  let s be 0-started State of SCMPDS;
  let I be parahalting halt-free Program of SCMPDS,J be parahalting
  shiftable Program of SCMPDS;
  set sI=stop I, sIJ = stop (I ';' J),
  s1= s, s2= s,
  P1 = P +* sIJ, P2 = P +* stop I;
A1: sIJ c= P1 by FUNCT_4:25;
  set
  s3=Initialize Result(P1 +* stop I,s1),
  P3 = P1 +* stop I +* stop J,
  s4=Initialize Result(P2,s2),
  P4 = P2 +* stop J;
A2: stop I c= P2 by FUNCT_4:25;
A3: stop J c= P4 by FUNCT_4:25;
A4: stop J c= P3 by FUNCT_4:25;
A5: stop I c= P1 +* stop I by FUNCT_4:25;
  then s4 =  s3 by A2,Th6;
  then
A6: LifeSpan(P3,s3) = LifeSpan(P4,s4) by A4,A3,Th6;
  LifeSpan(P1 +* stop I,s1) =
  LifeSpan(P2,s2) by A5,A2,Th6;
  hence thesis by A1,A6,Lm4;
end;
