reserve X,Y for set;
reserve R for domRing-like commutative Ring;
reserve c for Element of R;

theorem Th12:
  for a,b,c being Element of R holds
  c <> 0.R & c divides a & c divides b & c divides (a + b)
    implies (a/c) + (b/c) = (a + b)/c
proof
  let a,b,c be Element of R;
  assume
A1: c <> 0.R;
  set e = b/c;
  set d = a/c;
  assume that
A2: c divides a & c divides b and
A3: c divides (a + b);
  d * c = a & e * c = b by A1,A2,Def4;
  then a + b = (d + e) * c by VECTSP_1:def 3;
  then (a + b)/c = (d + e) * (c/c) by A1,A3,Th11
    .= (d + e) * 1.R by A1,Th9
    .= d + e;
  hence thesis;
end;
