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

theorem
  (a mod 7 = 1 or a mod 7 = 2 or a mod 7 = 4) &
  (b mod 7 = 1 or b mod 7 = 2 or b mod 7 = 4) implies
  a+b mod 7 = 1 or ... or a+b mod 7 = 6
  proof
A1: (a+b) mod 7 = ((a mod 7)+(b mod 7)) mod 7 by NAT_D:66;
    assume (a mod 7 = 1 or a mod 7 = 2 or a mod 7 = 4) &
    (b mod 7 = 1 or b mod 7 = 2 or b mod 7 = 4);
    then per cases;
    suppose
      a mod 7 = 1 & b mod 7 = 1 or
      a mod 7 = 1 & b mod 7 = 2 or
      a mod 7 = 1 & b mod 7 = 4 or
      a mod 7 = 2 & b mod 7 = 1 or
      a mod 7 = 2 & b mod 7 = 2 or
      a mod 7 = 2 & b mod 7 = 4 or
      a mod 7 = 4 & b mod 7 = 1 or
      a mod 7 = 4 & b mod 7 = 2;
      hence thesis by A1,NAT_D:24;
    end;
    suppose
A2:   a mod 7 = 4 & b mod 7 = 4;
      thus a+b mod 7 = 1 or ... or a+b mod 7 = 6
      proof
        take 1;
        (a+b) mod 7 = 7*1+1 mod 7 by A1,A2;
        hence thesis by Th16;
      end;
    end;
  end;
