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

theorem Th5:
  for x holds L_x <<= {x} <<= R_x
proof
 defpred P[Ordinal] means
     for x be Surreal st born x c= $1 holds L_x <<= {x} <<= R_x;
  A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
  proof
    let D be Ordinal such that A2:for C be Ordinal st C in D holds P[C];
    let x be Surreal such that
    A3:born x c= D;
    per cases by A3,ORDINAL1:11,XBOOLE_0:def 8;
    suppose born x in D;
      hence thesis by A2;
    end;
    suppose A4: born x = D;
      thus L_x <<= {x}
      proof
        given r,xl be Surreal such that
        A5: r in {x} & xl in L_x  & not xl <= r;
        not xl <= x by A5,TARSKI:def 1;
        then per cases by SURREAL0:43;
        suppose not L_xl << {x};
          then consider xll,X be Surreal such that
          A6: xll in L_xl & X in {x}  & X <= xll;
          A7: x <= xll by A6,TARSKI:def 1;
          A8:xl in {xl} by TARSKI:def 1;
          xl in L_x \/ R_x by A5,XBOOLE_0:def 3;
          then born xl in D by A4,Th1;
          then L_xl <<= {xl} by A2;
          then xll <= xl by A8,A6;
          then x <= xl by A7,Th4;
          then L_x << {xl} by SURREAL0:43;
          hence thesis by Th3,A5,A8;
        end;
        suppose not {xl} << R_x;
          then consider X,xr be Surreal such that
          A9: X in {xl} & xr in R_x  & xr <= X;
          A10: L_x << R_x by SURREAL0:45;
          xr <= xl by A9,TARSKI:def 1;
          hence thesis by A10,A9,A5;
        end;
      end;
      thus {x} <<= R_x
      proof
        given xr,l be Surreal such that
        A11: xr in R_x & l in {x} & not l <= xr;
        not x <= xr by A11,TARSKI:def 1;
        then per cases by SURREAL0:43;
        suppose not {x} << R_xr;
          then consider X,xrr be Surreal such that
          A12: X in {x} & xrr in R_xr  & xrr <= X;
          A13: xrr <= x by A12,TARSKI:def 1;
          A14:xr in {xr} by TARSKI:def 1;
          xr in L_x \/ R_x by A11,XBOOLE_0:def 3;
          then born xr in D by A4,Th1;
          then {xr} <<= R_xr by A2;
          then xr <= xrr by A14,A12;
          then xr <= x by A13,Th4;
          then {xr} << R_x by SURREAL0:43;
          hence thesis by A11,A14,Th3;
        end;
        suppose not L_x << {xr};
          then consider xl,X be Surreal such that
          A15: xl in L_x & X in {xr}  & X <= xl;
          A16: L_x << R_x by SURREAL0:45;
          xr <= xl by A15,TARSKI:def 1;
          hence thesis by A16,A15,A11;
        end;
      end;
    end;
  end;
  A17:for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
  let x be Surreal;
  P[born x] by A17;
  hence thesis;
end;
