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

theorem Th46:
  not 7 divides a implies
  ex k st a^2 = 7*k+1 or a^2 = 7*k+2 or a^2 = 7*k+4
  proof
    assume not 7 divides a;
    then consider k such that
A1: a = 7*k+1 or a = 7*k+2 or a = 7*k+3 or a = 7*k+4 or a = 7*k+5 or
    a = 7*k+6 by Th35;
    a^2 = 7*(k*(7*k+2))+1 or
    a^2 = 7*(k*(7*k+4))+4 or
    a^2 = 7*(k*(7*k+6)+1)+2 or
    a^2 = 7*(k*(7*k+8)+2)+2 or
    a^2 = 7*(k*(7*k+10)+3)+4 or
    a^2 = 7*(k*(7*k+12)+5)+1 by A1;
    hence thesis;
  end;
