reserve i,j,n,n1,n2,m,k,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat;

theorem Th34:
  n1 <= 2*n & n2 <= 2*n & |.x.| = Py(a,n1) & |.y.| = Py(a,n2) &
    x,y are_congruent_mod Px(a,n)
  implies n1,n2 are_congruent_mod 2*n or n1,-n2 are_congruent_mod 2*n
proof
  assume that A1: n1 <= 2*n & n2 <= 2*n and
              A2: |.x.| = Py(a,n1) & |.y.| = Py(a,n2) &
  x,y are_congruent_mod Px(a,n);
  n1=n2 or n1>n2 or n1<n2 by XXREAL_0:1;
  then n1=n2 or n1 = 2*n -n2 or n2 = 2*n -n1 by Lm9,A2,INT_1:14,A1;
  then n1-n2 = 2*n*0 or n1 -- n2 = 2*n*1 or n1--n2 = 2*n*(-1);
  hence thesis  by INT_1:def 5;
end;
