reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th16:
  a < n & b < n & n divides a-b implies a = b
  proof
    assume that
A1: a < n and
A2: b < n and
A3: n divides a-b;
A4: n divides -(a-b) by A3,INT_2:10;
    assume a <> b;
    then a > b or b > a by XXREAL_0:1;
    then a-b > 0 or b-a > 0 by XREAL_1:50;
    then n <= a-b or n <= b-a by A3,A4,INT_2:27;
    then a < a-b or b < b-a by A1,A2,XXREAL_0:2;
    hence thesis by XREAL_1:43;
  end;
