
theorem Th31:
  for x,y,n be Integer st
  x,n are_coprime & x,y are_congruent_mod n holds y,n are_coprime
proof
  let x,y,n be Integer;
  assume that
A1: x,n are_coprime and
A2: x,y are_congruent_mod n and
A3: not y,n are_coprime;
  consider z be Integer such that
A4: n*z = x-y by A2,INT_1:def 5;
  set gcdyn = y gcd n;
A5: gcdyn divides y by INT_2:21;
A6: gcdyn divides n by INT_2:21;
  gcdyn divides n*z by INT_2:2,21;
  then gcdyn divides n*z+y by A5,WSIERP_1:4;
  then gcdyn divides (x gcd n) by A4,A6,INT_2:22;
  then gcdyn divides 1 by A1,INT_2:def 3;
  then
A7: gcdyn = 1 or gcdyn = -1 by INT_2:13;
  0 <= gcdyn;
  hence contradiction by A3,A7,INT_2:def 3;
end;
