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 Th32:
  for x1,x2,y1,y2,p1,p2 be Nat st
    x1 + y1 / (2|^p1) = x2 + y2 / (2|^p2) & y1 < 2|^p1 & y2 < 2|^p2
  holds x1=x2
proof
  let x1,x2,y1,y2,p1,p2 be Nat such that
A1: x1 + y1 / (2|^p1) = x2 + y2 / (2|^p2) & y1 < 2|^p1 & y2 < 2|^p2;
  0<= y1 / (2|^p1) < 1 & 0<= y2 / (2|^p2) < 1 by A1,XREAL_1:189;
  then x1+0<= x1+y1 / (2|^p1) < x1+1 & x2+0<= x2+y2 / (2|^p2) < x2+1
  by XREAL_1:6;
  then x1 <x2+1 & x2 < x1+1 by A1,XXREAL_0:2;
  then x1 <= x2 <= x1 by NAT_1:13;
  hence thesis by XXREAL_0:1;
end;
