reserve G for strict Group,
  a,b,x,y,z for Element of G,
  H,K for strict Subgroup of G,
  p for Element of NAT,
  A for Subset of G;

theorem Th15:
  for G being Group, x,y,z being Element of G for A being Subset of G holds
  z in x * A * y iff ex a being Element of G st z = x * a * y & a in A
proof
  let G be Group;
  let x,y,z be Element of G;
  let A be Subset of G;
  thus z in x * A * y implies ex a being Element of G st z = x * a * y & a in A
  proof
    assume z in x * A * y;
    then consider b being Element of G such that
A1: z = b * y and
A2: b in x * A by GROUP_2:28;
    consider u being Element of G such that
A3: b = x * u and
A4: u in A by A2,GROUP_2:27;
    take u;
    thus thesis by A1,A3,A4;
  end;
  given a being Element of G such that
A5: z = x * a * y and
A6: a in A;
  ex h being Element of G st z = h * y & h in x * A by A5,A6,GROUP_2:27;
  hence thesis by GROUP_2:28;
end;
