reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;
reserve r,r1,r2 for Real;

theorem Th51:
  uReal.r1 < uReal.r2 iff r1 < r2
proof
  uReal.r1 = Unique_No sReal.r1 & uReal.r2 = Unique_No sReal.r2 by Def7;
  then
A1:uReal.r1 == sReal.r1 & uReal.r2 == sReal.r2 by SURREALO:def 10;
  thus uReal.r1 < uReal.r2 implies r1 < r2
  proof
    assume uReal.r1 < uReal.r2;
    then uReal.r1 < sReal.r2 by A1,SURREALO:4;
    then sReal.r1 < sReal.r2 by A1,SURREALO:4;
    hence thesis by Th50;
  end;
  assume r1 < r2;
  then uReal.r1 < sReal.r2 by A1,SURREALO:4,Th50;
  hence thesis by A1,SURREALO:4;
end;
