 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem Th22:
  x is positive implies born ||.x.|| c= born x
proof
A1:0_No <= 0_No;
  assume
A2: x is positive;
  then
A3: {} in born x by A1,SURREAL0:37,ORDINAL3:8;
A4: born 0_No={} by SURREAL0:37;
  set Nx=||.x.||;
  for o be object st o in L_Nx \/ R_Nx
    ex O st O in born x & o in Day O
  proof
    let o;
    assume
A5: o in L_Nx \/ R_Nx;
    then reconsider o as Surreal by SURREAL0:def 16;
    per cases;
    suppose o = 0_No;
      then o in Day {} by SURREAL0:def 18,A4;
      hence thesis by A3;
    end;
    suppose o <> 0_No;
      then
A6:   o in (L_Nx \/R_Nx)\{0_No} by A5,ZFMISC_1:56;
      (L_Nx \/R_Nx)\{0_No} c= L_x \/R_x by A2,Th20;
      then born o in born x & o in Day born o
      by A6,SURREALO:1,SURREAL0:def 18;
      hence thesis;
    end;
  end;
  then Nx=[L_Nx,R_Nx] in Day born x by SURREAL0:45,46;
  hence thesis by SURREAL0:def 18;
end;
