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 Th6:
  for I being InitHalting really-closed Program of SCM+FSA
   st Initialize ((intloc 0) .--> 1) c= s1 &
     Initialize ((intloc 0) .--> 1) c= s2 & I c= p1 & I c= p2 &
    s1 =  s2
 holds LifeSpan(p1,s1) = LifeSpan(p2,
  s2) &  Result(p1,s1) =  Result(p2,s2)
proof
  let I be InitHalting really-closed Program of SCM+FSA;
  assume that
A1: iS c= s1 and
A2: iS c= s2 and
A3: I c= p1 and
A4: I c= p2 and
A5:  s1 =  s2;
A6: p2 halts_on s2 by A2,Def1,A4;
A7: p1 halts_on s1 by A1,Def1,A3;
A8: now
    let l be Nat;
    assume
A9: CurInstr(p2,Comput(p2,s2,l)) = halt SCM+FSA;
    CurInstr(p1,Comput(p1,s1,l)) = CurInstr(p2,Comput(p2,s2,l))
     by A1,A5,Th5,A3,A4;
    hence LifeSpan(p1,s1) <= l by A7,A9,EXTPRO_1:def 15;
  end;
  CurInstr(p2,Comput(p2,s2,LifeSpan(p1,s1)))
    = CurInstr(p1,Comput(p1,s1,LifeSpan(p1,s1))) by A1,A5,Th5,A3,A4
    .= halt SCM+FSA by A7,EXTPRO_1:def 15;
  hence
A10:  LifeSpan(p1,s1) = LifeSpan(p2,s2) by A8,A6,EXTPRO_1:def 15;
     p2 halts_on s2 by A2,Def1,A4;
     then
A11: Result(p2,s2) = Comput(p2,s2,LifeSpan(p1,s1))
     by A10,EXTPRO_1:23;
   p1 halts_on s1 by A1,Def1,A3;
   then
  Result(p1,s1) = Comput(p1,s1,LifeSpan(p1,s1)) by EXTPRO_1:23;

  hence thesis by A1,A5,A11,Th5,A3,A4;
end;
