
theorem
  for P1,P2 being RelStr st the RelStr of P1 = the RelStr of P2 & P1 is
  complete holds P2 is complete
proof
  let P1,P2 be RelStr such that
A1: the RelStr of P1 = the RelStr of P2 and
A2: for X being set ex a being Element of P1 st X is_<=_than a & for b
  being Element of P1 st X is_<=_than b holds a <= b;
  let X be set;
  consider a being Element of P1 such that
A3: X is_<=_than a and
A4: for b being Element of P1 st X is_<=_than b holds a <= b by A2;
  reconsider a9 = a as Element of P2 by A1;
  take a9;
  thus X is_<=_than a9 by A1,A3,Th2;
  let b9 be Element of P2;
  reconsider b = b9 as Element of P1 by A1;
  assume X is_<=_than b9;
  then a <= b by A1,A4,Th2;
  hence thesis by A1;
end;
