reserve m,n for Element of NAT,
  i,j for Instruction of SCMPDS,
  I for Program
  of SCMPDS,
  a for Int_position;
reserve Q,U,P for Instruction-Sequence of SCMPDS;

theorem
  for s being 0-started State of SCMPDS,n,
    p0 be Element of NAT, f be FinSequence
of INT st p0 >= 3 & f is_FinSequence_on s,p0 & len f=n
 holds IExec(sum(n,p0),P,s).intpos 1=Sum f & sum(n,p0) is parahalting
proof
  let s be 0-started State of SCMPDS,
      n,p0 be Element of NAT,f be FinSequence of INT;
  assume that
A1: p0 >= 3 and
A2: f is_FinSequence_on s,p0 & len f=n;
  thus IExec(sum(n,p0),P,s).a1=Sum f by A1,A2,Lm3;
  now
    let t be State of SCMPDS, Q;
    consider g being FinSequence of INT such that
A3:     len g=n & g is_FinSequence_on t,p0 by SCPISORT:2;
     g is_FinSequence_on Initialize t,p0
  proof let i be Nat;
    assume 1 <= i & i <= len g;
     then g.i=t.intpos(p0+i) by A3;
    hence thesis by SCMPDS_5:15;
   end;
     then sum(n,p0) is_halting_on Initialize t,Q by A1,Lm3,A3;
    hence sum(n,p0) is_halting_on t,Q by SCMPDS_6:126;
  end;
  hence thesis by SCMPDS_6:21;
end;
