reserve m,n for Nat,
  I for Program of SCM+FSA,
  s,s1,s2 for State of SCM+FSA,
  a for Int-Location,
  f for FinSeq-Location,
  p,p1,p2 for Instruction-Sequence of SCM+FSA;

theorem Th16:
  for I being keepInt0_1 InitHalting really-closed Program of SCM+FSA,
      J being InitHalting really-closed Program of SCM+FSA holds
   LifeSpan(p +* (I ";" J),Initialized s)
     = LifeSpan(p+*I,Initialized s) + 1 +
       LifeSpan(p +* I +* J,
           Result(p+*I,Initialized s)
            +*Initialize ((intloc 0) .--> 1))
proof
  let I be keepInt0_1 InitHalting really-closed Program of SCM+FSA;
  let J be InitHalting really-closed Program of SCM+FSA;
  set inI=iS;
  set inIJ=iS;
  set inJ=iS;
A1: inJ c= Result(p+* (I ";" J)+*I,s +* inIJ) +* inJ & inJ c=
Result(p +* I,s +* inI) +* inJ
  by FUNCT_4:25;
A2: J c= p +* (I ";" J) +* I +* J & J c= p +* I +* J by FUNCT_4:25;
A3: inI c= s +* inI & inI c= s +* inIJ by FUNCT_4:25;
A4: I c= p +* I & I c= p +* (I ";" J) +* I by FUNCT_4:25;
  then
A5: (Result(p+* (I ";" J)+*I,s +* inIJ) +* inJ) = (Result(
p +* I,s +* inI) +* inJ) by Th6,A3;
A6:  I ";" J c= p +* (I ";" J) by FUNCT_4:25;
  inIJ c= s +* inIJ by FUNCT_4:25;
  then
A7: LifeSpan(p +* (I ";" J),s +* inIJ)
    = LifeSpan(p+* (I ";" J)+*I,s +* inIJ) + 1 +
    LifeSpan(p+* (I ";" J)+*I+*J,
    Result(p+* (I ";" J)+*I,s +* inIJ) +* inJ) by Th13,A6;
  LifeSpan(p +* I,s +* inI) = LifeSpan(p+* (I ";" J)+*I,
  s +* inIJ) by A3,Th6,A4;
  hence thesis by A7,A1,A5,Th6,A2;
end;
