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;

theorem Th24:
  d1 < d2 iff uDyadic.d1 < uDyadic.d2
proof
  consider i1 be Integer, n1 be Nat such that
A1:d1 = i1 / (2|^n1) by Th18;
  consider i2 be Integer, n2 be Nat such that
A2:d2 = i2 / (2|^n2) by Th18;
  (2|^n1)*(2|^n2) = 2|^(n1+n2) by NEWTON:8;
  then
A3: d1 = (i1*(2|^n2)) / 2|^(n1+n2) &
  d2 = (i2*(2|^n1)) / 2|^(n1+n2) by A1,A2,XCMPLX_1:91;
  thus d1 < d2 implies uDyadic.d1 < uDyadic.d2
  proof
    assume d1 < d2;
    then i1*(2|^n2) < i2*(2|^n1) by A3,XREAL_1:72;
    hence thesis by A3,Lm5;
  end;
  assume uDyadic.d1 < uDyadic.d2;
  then i1*(2|^n2) < i2*(2|^n1) by A3,Lm5;
  hence thesis by A3,XREAL_1:74;
end;
