 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;

theorem ThStrSubEq:
  for A,B being Subgroup-Family of F
  st (for i being Element of I holds A.i = B.i)
  holds A = B
proof
  let A,B be Subgroup-Family of F;
  assume for i being Element of I holds A.i = B.i;
  then for i being object st i in I holds A.i = B.i;
  hence A = B by PBOOLE:3;
end;
