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

theorem Th35:
  not 7 divides n
  iff
  ex k st n = 7*k+1 or n = 7*k+2 or n = 7*k+3 or n = 7*k+4 or n = 7*k+5 or
   n = 7*k+6
  proof
    consider K being Nat such that
A1: n = 7*K or n = 7*K+1 or n = 7*K+2 or n = 7*K+3 or n = 7*K+4 or n = 7*K+5 or
    n = 7*K+6 by Th27;
    thus not 7 divides n implies
    ex k st n = 7*k+1 or n = 7*k+2 or n = 7*k+3 or n = 7*k+4 or n = 7*k+5 or
     n = 7*k+6 by A1;
    given k such that
A2: n = 7*k+1 or n = 7*k+2 or n = 7*k+3 or n = 7*k+4 or n = 7*k+5 or n = 7*k+6;
    given t being Nat such that
A3: n = 7*t;
    per cases by A2;
    suppose n = 7*k+1;
      then 1/7 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 7*k+2;
      then 2/7 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 7*k+3;
      then 3/7 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 7*k+4;
      then 4/7 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 7*k+5;
      then 5/7 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 7*k+6;
      then 6/7 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
  end;
