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 Th34: :::
  for x1,x2,p1,p2 be Nat st (2*x1+1) / (2|^p1) = x2 / (2|^p2)
    holds p1 <= p2
proof
  let x1,x2,p1,p2 be Nat such that
A1: (2*x1+1) / (2|^p1) = x2 / (2|^p2);
  assume p2 < p1;
  then p2+1 <= p1 by NAT_1:13;
  then reconsider P = p1-(p2+1)as Nat by NAT_1:21;
  p1 = P+(p2+1);
  then 2|^p1 = (2|^P) * (2|^(p2+1)) & 2|^(p2+1) = 2*2|^p2 by NEWTON:6,8;
  then (2*x1+1) * (2|^p2) = x2 * ((2|^P)*(2*2|^p2)) = x2 * (2|^P)*2*2|^p2
  by A1,XCMPLX_1:95;
  hence thesis by XCMPLX_1:5;
end;
