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;

theorem Th7:
  (n * m) mod 4 = 3 implies
  (n mod 4 = 1 & m mod 4 = 3) or (n mod 4 = 3 & m mod 4 = 1)
  proof
    assume
A1: (n * m) mod 4 = 3;
A2: m mod (3+1) = 0 or ... or m mod (3+1) = 3 by NUMBER03:11;
    n mod (3+1) = 0 or ... or n mod (3+1) = 3 by NUMBER03:11;
    then per cases by A1,Th5;
    suppose
A3:   n mod (3+1) = 1;
      m mod (3+1) = 3
      proof
        assume m mod (3+1) <> 3;
        then (n*m) mod 4 = (1*n) mod 4 by A2,A1,Th5,RADIX_2:3;
        hence thesis by A1,A3;
      end;
      hence thesis by A3;
    end;
    suppose
A4:   n mod (3+1) = 3;
      m mod (3+1) = 1
      proof
A5:     3*3 = 4*2+1;
        assume m mod (3+1) <> 1;
        then (n*m) mod 4 = (3*n) mod 4 by A2,A1,Th5,RADIX_2:3;
        then (n*m) mod 4 = (3*3) mod 4 by A4,RADIX_2:3;
        hence thesis by A1,A5,NUMBER02:16;
      end;
      hence thesis by A4;
    end;
  end;
