reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th30:
  10 divides a|^10 + 1 iff
  ex r,k being Nat st
   a = 10*k+r & 10 divides r|^10+1 & (r = 0 or ... or r = 9)
  proof
    thus 10 divides a|^10 + 1 implies
    ex r,k being Nat st
    a = 10*k+r & 10 divides r|^10+1 & (r = 0 or ... or r = 9)
    proof
      assume
A1:   10 divides a|^10 + 1;
      take r = a mod 10;
      take k = a div 10;
      thus a = 10*k+r by NAT_D:2;
      thus 10 divides r|^10+1 by A1,Th29;
      thus r = 0 or ... or r = 9 by NUMPOLY1:8;
    end;
    given r,k being Nat such that
A2: a = 10*k+r and
A3: 10 divides r|^10+1 and
A4: r = 0 or ... or r = 9;
    r = a mod 10 by A2,A4,NAT_D:def 2;
    hence thesis by A3,Th29;
  end;
