
theorem Th1:
  for X being real-membered set ex R being strict RelStr st the
  carrier of R = X & R is real
proof
  let X be real-membered set;
  per cases;
  suppose
A1: X is empty;
    set E = the empty strict RelStr;
    take E;
    thus thesis by A1;
  end;
  suppose
    X is non empty;
    then reconsider Y = X as non empty set;
    defpred X[set,set] means ex x,y being Real st $1=x & $2=y & x<=y;
    consider L being non empty strict RelStr such that
A2: the carrier of L = Y and
A3: for a,b being Element of L holds a <= b iff X[a,b] from YELLOW_0:
    sch 1;
    take L;
    thus the carrier of L = X by A2;
    thus the carrier of L c= REAL by A2,MEMBERED:3;
    let x,y be Real;
    assume x in the carrier of L & y in the carrier of L;
    then reconsider a = x, b = y as Element of L;
    a <= b iff [x,y] in the InternalRel of L by ORDERS_2:def 5;
    then
    [x,y] in the InternalRel of L iff ex x,y being Real st a=x & b=
    y & x<=y by A3;
    hence thesis;
  end;
end;
