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 Th45: :::
  for i1,i2 be Integer, n1,n2 be Nat st i1/(2|^n1) < i2/(2|^n2)
    holds
       i1/(2|^n1) < (i1*(2|^n2)*2+1)/(2|^(n1+n2+1))
         <= (i2*(2|^n1)*2-1)/(2|^(n1+n2+1)) < i2/(2|^n2)
proof
  let i1,i2 be Integer, n1,n2 be Nat;
  assume i1/(2|^n1) < i2/(2|^n2);
  then i1*(2|^n2)+1 <= i2*(2|^n1) by INT_1:7,XREAL_1:102;
  then
A1: (i1*(2|^n2)+1)*2 <= i2*(2|^n1)*2 by XREAL_1:64;
  2|^(n1+n2+1) = 2|^(n1+(n2+1))
  .= (2|^n1)*(2|^(n2+1)) by NEWTON:8
  .= (2|^n1)*(2*2|^n2) by NEWTON:6;
  then
  (i1*(2|^n2)*2)/(2|^(n1+n2+1)) = (i1 * ((2|^n2)*2)) / ((2|^n1)*(2*2|^n2))
  .= i1 /(2|^n1) by XCMPLX_1:91;
  hence i1 /(2|^n1) < (i1*(2|^n2)*2+1)/(2|^(n1+n2+1))
  by XREAL_1:29,XREAL_1:74;
  i1*(2|^n2)*2+1 +1 <= i2*(2|^n1)*2 by A1;
  then i1*(2|^n2)*2+1 <= i2*(2|^n1)*2-1 by XREAL_1:19;
  hence (i1*(2|^n2)*2+1)/(2|^(n1+n2+1)) <= (i2*(2|^n1)*2-1)/(2|^(n1+n2+1))
  by XREAL_1:72;
A2: i2*(2|^n1)*2-1 < i2*(2|^n1)*2-1+1 by XREAL_1:29;
  2|^(n1+n2+1) = 2|^(n2+(n1+1))
  .= (2|^n2)*(2|^(n1+1)) by NEWTON:8
  .= (2|^n2)*(2*2|^n1) by NEWTON:6;
  then (i2*(2|^n1)*2)/(2|^(n1+n2+1)) = (i2*((2|^n1)*2))/((2|^n2)*(2*2|^n1))
  .= i2/(2|^n2) by XCMPLX_1:91;
  hence thesis by A2,XREAL_1:74;
end;
