reserve a,b,c,k,k9,m,n,n9,p,p9 for Nat;
reserve i,i9 for Integer;

theorem Th4:
  i is odd implies i^2 mod 4 = 1
proof
  assume i is odd;
  then consider i9 such that
A1: i = 2*i9 + 1 by ABIAN:1;
  i^2 = 4*(i9^2 + i9) + 1 by A1;
  hence i^2 mod 4 = 1 mod 4 by NAT_D:61
    .= 1 by NAT_D:24;
end;
