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

theorem
  not 10 divides n
  iff
  ex k st n = 10*k+1 or n = 10*k+2 or n = 10*k+3 or n = 10*k+4 or n = 10*k+5 or
   n = 10*k+6 or n = 10*k+7 or n = 10*k+8 or n = 10*k+9
  proof
    consider K being Nat such that
A1: n = 10*K or n = 10*K+1 or n = 10*K+2 or n = 10*K+3 or n = 10*K+4 or
    n = 10*K+5 or n = 10*K+6 or n = 10*K+7 or n = 10*K+8 or n = 10*K+9
    by Th30;
    thus not 10 divides n implies
    ex k st n = 10*k+1 or n = 10*k+2 or n = 10*k+3 or n = 10*k+4 or
    n = 10*k+5 or n = 10*k+6 or n = 10*k+7 or n = 10*k+8 or n = 10*k+9 by A1;
    given k such that
A2: n = 10*k+1 or n = 10*k+2 or n = 10*k+3 or n = 10*k+4 or n = 10*k+5 or
    n = 10*k+6 or n = 10*k+7 or n = 10*k+8 or n = 10*k+9;
    given t being Nat such that
A3: n = 10*t;
    per cases by A2;
    suppose n = 10*k+1;
      then 1/10 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 10*k+2;
      then 2/10 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 10*k+3;
      then 3/10 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 10*k+4;
      then 4/10 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 10*k+5;
      then 5/10 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 10*k+6;
      then 6/10 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 10*k+7;
      then 7/10 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 10*k+8;
      then 8/10 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 10*k+9;
      then 9/10 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
  end;
