reserve m, n for Nat,
  x for set,
  i for Instruction of SCM+FSA,
  I for Program of SCM+FSA,
  a for Int-Location,
  f for FinSeq-Location,
  l, l1 for Nat,
  s,s1,s2 for State of SCM+FSA,
  P,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem Th17:
  for I being keeping_0 parahalting really-closed Program of SCM+FSA,
      J being parahalting really-closed Program of SCM+FSA holds
   LifeSpan(P+*(I ";" J),s +* Initialize((intloc 0).-->1)) =
  LifeSpan(P+*I,s +* Initialize((intloc 0).-->1)) + 1 + LifeSpan(P+*I+*J,
   Result(P+*I,s +* Initialize((intloc 0).-->1)) +*
    Initialize((intloc 0).-->1))
proof
  let I be keeping_0 parahalting really-closed Program of SCM+FSA;
  let J be parahalting really-closed Program of SCM+FSA;
A1: I ";" J c= P +* (I ";" J) by FUNCT_4:25;
 Initialize((intloc 0).-->1) c= s +* Initialize((intloc 0).-->1) by FUNCT_4:25;
  then
A2: LifeSpan(P+*(I ";" J),s +* Initialize((intloc 0).-->1)) =
    LifeSpan(P+*(I ";" J)+*I,s +* Initialize((intloc 0).-->1)) + 1 +
    LifeSpan(P+*(I ";" J)+*I +* J,
       Result(P+*(I ";" J)+*I,
              s +* Initialize((intloc 0).-->1)) +* Initialize((intloc 0).-->1))
               by Lm2,A1;
A3: J c= P+*(I ";" J) +* I +* J by FUNCT_4:25;
A4: J c= P+*I+*J by FUNCT_4:25;
A5: I c= P +* (I ";" J) +* I by FUNCT_4:25;
A6: I c= P+* I by FUNCT_4:25;
A7: (Result(P+*(I ";" J)+*I,s +* Initialize((intloc 0).-->1))
       +* Initialize((intloc 0).-->1))
   = (Result(P+*I,s +* Initialize((intloc 0).-->1))
    +* Initialize((intloc 0).-->1)) by Th8,A5,A6;
A8:   LifeSpan(P+*I,s +* Initialize((intloc 0).-->1)) =
    LifeSpan(P+*(I ";" J)+*I,s +* Initialize((intloc 0).-->1))
            by Th8,A5,A6;
  thus thesis by A7,Th8,A3,A4,A2,A8;
end;
