reserve m for Nat;
reserve P for Instruction-Sequence of SCM+FSA;

theorem
  for I being Program of SCM+FSA
   holds I is parahalting iff
  for s being State of SCM+FSA
  for P being Instruction-Sequence of SCM+FSA
   holds I is_halting_on s,P
proof
set SAt = Start-At(0,SCM+FSA);
  let I be Program of SCM+FSA;
  thus I is parahalting implies
   for s being State of SCM+FSA
  for P being Instruction-Sequence of SCM+FSA
   holds I is_halting_on s,P
  by FUNCT_4:25;
  assume
A1: for s being State of SCM+FSA
  for P being Instruction-Sequence of SCM+FSA
   holds I is_halting_on s,P;
  let s be 0-started State of SCM+FSA;
A2: Start-At(0,SCM+FSA) c= s by MEMSTR_0:29;
  let P be Instruction-Sequence of SCM+FSA such that
A3: I c= P;
A4: P = P+*I by A3,FUNCT_4:98;
A5: Initialize s = s by A2,FUNCT_4:98;
   I is_halting_on s,P by A1;
  hence P halts_on s by A4,A5;
end;
