reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;
reserve z for Complex;

theorem Th34:
  a divides m|^k + 1 implies a divides (a*n+m)|^k + 1
  proof
    assume
A1: a divides m|^k + 1;
    then
A2: a <> 0;
    then
A3: (m|^k+1) mod a = 0 by A1,INT_1:62;
    (a*n+m) mod a = m mod a by NAT_D:21;
    then ((a*n+m)|^k) mod a = (m|^k) mod a by INT_4:8;
    then 0 = ((a*n+m)|^k mod a + (1 mod a)) mod a by A3,NAT_D:66
    .= ((a*n+m)|^k + 1) mod a by NAT_D:66;
    hence thesis by A2,INT_1:62;
  end;
