theorem Th11:
  for I being Program of SCMPDS holds I is paraclosed iff
  for s being State of SCMPDS, P holds I is_closed_on s,P
proof
  let I be Program of SCMPDS;
  thus I is paraclosed implies for s be State of SCMPDS,P holds
  I is_closed_on s,P by FUNCT_4:25;
  assume
A1: for s being State of SCMPDS,P holds I is_closed_on s,P;
    let s be 0-started State of SCMPDS;
    let k be Nat;
    let P;
A2:  Initialize s = s by MEMSTR_0:44;
    assume stop I c= P;
    then
A3: P = P +* stop I by FUNCT_4:98;
    I is_closed_on s,P by A1;
    hence IC Comput(P,s,k) in dom stop I by A2,A3;
end;
