
theorem Th21:
  for x,y,i,j being Integer st i,j are_coprime holds x,y
are_congruent_mod i & x,y are_congruent_mod j implies x,y are_congruent_mod i*j
proof
  let x,y,i,j be Integer;
  assume i,j are_coprime;
  then
A1: i * j divides i lcm j by Th20;
  assume that
A2: x,y are_congruent_mod i and
A3: x,y are_congruent_mod j;
A4: j divides (x - y) by A3,INT_2:15;
  i divides (x - y) by A2,INT_2:15;
  then i lcm j divides (x - y) by A4,INT_2:19;
  then i*j divides (x - y) by A1,INT_2:9;
  hence thesis by INT_2:15;
end;
