reserve F for total
  NAT-defined (the InstructionsF of SCM)-valued Function;

theorem
  for F being total
   NAT-defined (the InstructionsF of SCM)-valued Function
    st <%MultBy(dl.0,dl.1)%>^<%halt SCM%> c= F
  for i1, i2 being Integer,
     s being 0-started State-consisting of <%i1,i2%>
  holds F halts_on s & LifeSpan(F,s) = 1 & (Result(F,s)).dl.0 = i1 * i2 &
   for d being Data-Location st d<>dl.0 holds (Result(F,s)).d = s.d
proof
  let F being total
   NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%MultBy(dl.0,dl.1)%>^<%halt SCM%> c= F;
  let i1, i2 be Integer,
      s be 0-started State-consisting of <%i1,i2%>;
  set s0 = Comput(F,s,0);
  set s1 = Comput(F,s,0+1);
A2: s = s0 by EXTPRO_1:2;
A3: s.dl.0 = i1 & s.dl.1 = i2 by Th2;
A4: IC s = 0 by MEMSTR_0:def 11;
A5: F.0 = MultBy(dl.0,dl.1) by A1,Th3;
  then
A6: IC s1 = (0+1) by A4,A2,Th7;
A7: F.1 = halt SCM by A1,Th3;
  hence F halts_on s by A6,EXTPRO_1:30;
  thus LifeSpan(F,s) = 1 by A4,A7,A2,A6,EXTPRO_1:33;
  s1.dl.0 = s0.dl.0 * s0.dl.1 by A4,A5,A2,Th7;
  hence (Result(F,s)).dl.0 = i1 * i2 by A7,A3,A2,A6,EXTPRO_1:7;
  let d be Data-Location;
  assume
A8: d<>dl.0;
  thus (Result(F,s)).d = s1.d by A7,A6,EXTPRO_1:7
    .= s.d by A4,A5,A2,A8,Th7;
end;
