
theorem
  for P being RelStr, D being Subset-Family of P
  for A, B being set holds [A,B] in pcs-general-power-IR(P,D) iff
  A in D & B in D &
  for a being Element of P st a in A ex b being Element of P st b in B & a <= b
proof
  let P be RelStr, D be Subset-Family of P;
  let A, B be set;
  thus [A,B] in pcs-general-power-IR(P,D) implies A in D & B in D &
  for a being Element of P st a in A ex b being Element of P st b in B &
  a <= b
  proof
    assume
A1: [A,B] in pcs-general-power-IR(P,D);
    hence
A2: A in D & B in D by Def45;
    let a be Element of P;
    assume a in A;
    then consider b being set such that
A3: b in B and
A4: [a,b] in the InternalRel of P by A1,Def45;
    reconsider b as Element of P by A2,A3;
    take b;
    thus thesis by A3,A4;
  end;
  assume that
A5: A in D and
A6: B in D and
A7: for a being Element of P st a in A ex b being Element of P st b in B &
  a <= b;
  for a being set st a in A ex b being set st b in B &
  [a,b] in the InternalRel of P
  proof
    let a be set;
    assume
A8: a in A;
    then reconsider a as Element of P by A5;
    consider b being Element of P such that
A9: b in B and
A10: a <= b by A7,A8;
    take b;
    thus thesis by A9,A10;
  end;
  hence thesis by A5,A6,Def45;
end;
