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 Th27:
  uDyadic.-d = - uDyadic.d
proof
  defpred P[Nat] means
    for d st d in DYADIC($1) holds uDyadic.-d = - uDyadic.d;
A1:P[0]
  proof
    let d;
    assume d in DYADIC 0;
    then reconsider i=d as Integer by Th21;
    uDyadic.-d = uInt.(-i) = -uInt.i by Def5,Th12;
    hence thesis by Def5;
  end;
A2: P[n] implies P[n+1]
  proof
    assume
A3: P[n];
    let d such that
A4: d in DYADIC(n+1);
    per cases;
    suppose d in DYADIC(n);
      hence thesis by A3;
    end;
    suppose not d in DYADIC(n);
      then d in DYADIC(n+1)\DYADIC(n) by A4,XBOOLE_0:def 5;
      then consider i be Integer such that
A5:    d = (2*i+1) / 2|^(n+1) by Th20;
      set j = -(i+1);
      2*j+1 = -(2*i+1);
      then
A6:   - d = (2*j+1) / 2|^(n+1) by A5,XCMPLX_1:187;
A7:   -(i/(2|^n)) = (-i) / (2|^n) by XCMPLX_1:187
      .= (j+1)/ (2|^n);
A8:   -((i+1)/(2|^n)) = j / (2|^n) by XCMPLX_1:187;
      i/(2|^n) in DYADIC(n) & (i+1)/(2|^n) in DYADIC(n) by Def4;
      then
A9:   -uDyadic.(i/(2|^n)) = uDyadic.((j+1)/(2|^n)) &
      -uDyadic.((i+1)/(2|^n)) = uDyadic.(j/(2|^n)) by A3,A7,A8;
      uDyadic.d = [{uDyadic.(i/(2|^n))},{uDyadic.((i+1)/(2|^n))}] by A5,Def5;
      then L_(uDyadic.d) ={uDyadic.(i/(2|^n))} &
      R_(uDyadic.d) = {uDyadic.((i+1)/(2|^n))};
      hence - (uDyadic.d) = [--{uDyadic.((i+1)/(2|^n))},--{uDyadic.(i/(2|^n))}]
      by SURREALR:7
      .= [{-uDyadic.((i+1)/(2|^n))},--{uDyadic.(i/(2|^n))}] by SURREALR:21
      .= [{-uDyadic.((i+1)/(2|^n))},{-uDyadic.(i/(2|^n))}] by SURREALR:21
      .= uDyadic.-d by A9,A6,Def5;
    end;
  end;
A10:P[n] from NAT_1:sch 2(A1,A2);
  consider i be Integer, n be Nat such that
A11: d = i / (2|^n) by Th18;
  d in DYADIC(n) by A11,Def4;
  hence thesis by A10;
end;
