
theorem
  for a,b be Integer,n,x be Nat st
  a,b are_congruent_mod n & n <> 0 holds a|^x,b|^x are_congruent_mod n
proof
  let a,b be Integer;
  let n,x be Nat;
  assume that
A1: a,b are_congruent_mod n and
A2: n <> 0;
  consider an be Nat such that
A3: an,a are_congruent_mod n by A2,Lm8;
  b mod n >= 0 by A2,NAT_D:62;
  then b mod n is Element of NAT by INT_1:3;
  then consider nb be Nat such that
A4: nb = b mod n;
  reconsider ai = an as Integer;
A5: ai,b are_congruent_mod n by A1,A3,INT_1:15;
  reconsider bi = nb as Integer;
  b mod n = (b mod n) mod n by NAT_D:65;
  then b,bi are_congruent_mod n by A2,A4,NAT_D:64;
  then an,nb are_congruent_mod n by A5,INT_1:15;
  then an|^x,nb|^x are_congruent_mod n by A2,Lm9;
  then
A6: (an|^x mod n) = (nb|^x mod n) by NAT_D:64;
A7: b|^x mod n = (b mod n)|^x mod n by Th30;
A8: a|^x mod n = (a mod n)|^x mod n by Th30;
  an mod n = a mod n by A3,NAT_D:64;
  then (a mod n)|^x mod n = (b mod n)|^x mod n by A4,A6,PEPIN:12;
  hence thesis by A2,A8,A7,NAT_D:64;
end;
