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
  for G being Group, H being Subgroup of G, a1,a2 being Element of H
  for b1,b2 being Element of G st a1 = b1 & a2 = b2 holds a1*a2 = b1*b2
proof
  let G be Group, H be Subgroup of G;
  let a1,a2 be Element of H;
A1: the carrier of H c= the carrier of G by GROUP_2:def 5;
  let b1,b2 be Element of G such that
A2: a1 = b1 and
A3: a2 = b2;
  dom the multF of G = [:the carrier of G, the carrier of G:] by FUNCT_2:def 1;
  then
  A4: the
 multF of G is ManySortedSet of [:the carrier of G, the carrier of G :]
  by PARTFUN1:def 2;
  the multF of H = (the multF of G)||the carrier of H by GROUP_2:def 5;
  hence thesis by A1,A2,A3,A4,ALTCAT_1:5;
end;
