reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem Th25:
  ex k st n = 5*k or n = 5*k+1 or n = 5*k+2 or n = 5*k+3 or n = 5*k+4
  proof
    consider k such that
A1: n = 5*k+(0 qua Nat) or ... or n = 5*k+(5-1) by Th22;
    consider i such that
A2: 0 <= i & i <= 4 and
A3: n = 5*k+i by A1;
    take k;
    i = 0 or ... or i = 4 by A2;
    hence thesis by A3;
  end;
