reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;
reserve f,g for complex-valued FinSequence;

theorem Th80:
  n is odd implies n mod 4 = 1 or n mod 4 = 3
  proof
    assume n is odd;
    then consider m being Nat such that
A1: n = 2*m+1 by ABIAN:9;
    (2*m+1) mod 4 = (((2*m) mod 4) + (1 mod 4)) mod 4 by NAT_D:66
    .= (((2*m) mod 4) + 1) mod 4 by NAT_D:24;
    then n mod 4 = (0+1) mod 4 or n mod 4 = (2+1) mod 4 by A1,Th78;
    hence thesis by NAT_D:24;
  end;
