 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem Th57:
  for a,b being Element of product F
  for i
  holds ([. a, b .]).i = [. a/.i, b/.i .]
proof
  let a,b be Element of product F;
  let i;
  thus [. a/.i, b/.i .] = (((a/.i)")*((b/.i)"))*((a/.i)*(b/.i)) by GROUP_5:16
      .= (((a")/.i) * ((b/.i)")) * ((a/.i) * (b/.i)) by GROUP_7:8
      .= (((a")/.i) * ((b")/.i)) * ((a/.i) * (b/.i)) by GROUP_7:8
      .= (((a")/.i) * ((b")/.i)) * ((a * b)/.i) by GROUP_7:1
      .= (((a") * (b"))/.i) * ((a * b)/.i) by GROUP_7:1
      .= (((a") * (b")) * (a * b))/.i by GROUP_7:1
      .= ([. a, b .]).i by GROUP_5:16;
end;
