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

theorem Th11:
  for a,b,c being Element of R holds
    c <> 0.R implies
    (c divides (a * b) & c divides a implies
       (a * b) / c = (a / c) * b) &
    (c divides (a * b) & c divides b implies
      (a * b) / c = a * (b / c))
proof
  let A,B,C be Element of R;
  assume
A1: C <> 0.R;
A2: now
    set A2 = B / C;
    set A1 = (A * B) / C;
    assume C divides (A * B) & C divides B;
    then A1 * C = A * B & A2 * C = B by A1,Def4;
    then A1 * C = (A * A2) * C by GROUP_1:def 3;
    hence C divides (A * B) & C divides B implies (A * B) / C = A * (B / C) by
A1,Th1;
  end;
  now
    set A2 = A / C;
    set A1 = (A * B) / C;
    assume C divides (A * B) & C divides A;
    then A1 * C = A * B & A2 * C = A by A1,Def4;
    then A1 * C = (A2 * B) * C by GROUP_1:def 3;
    hence C divides (A * B) & C divides A implies (A * B) / C = (A / C) * B by
A1,Th1;
  end;
  hence thesis by A2;
end;
