reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;

theorem Th11:
  L_x << {x} & {x} << R_x
proof
  thus L_x << {x}
  proof
    let xl,X be Surreal such that
    A1: xl in L_x & X in {x} & xl >= X;
    xl >= x by A1,TARSKI:def 1;
    then L_x << {xl} & xl in {xl} by SURREAL0:43,TARSKI:def 1;
    then not xl <= xl by A1;
    hence thesis;
  end;
  let X,xr be Surreal such that
  A2: X in {x} & xr in R_x & X >= xr;
  xr <= x by A2,TARSKI:def 1;
  then {xr} << R_x & xr in {xr} by SURREAL0:43,TARSKI:def 1;
  then not xr <= xr by A2;
  hence thesis;
end;
