reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;
reserve N1,N2 for Subgroup of G;

theorem
  for N,N1,N2 be Subgroup of G st N = N1 /\ N2 holds N ~ H c= N1 ~ H /\ N2 ~ H
proof
  let N,N1,N2 be Subgroup of G;
  assume N = N1 /\ N2; then
A1:N is Subgroup of N1 & N is Subgroup of N2 by GROUP_2:88;
  let x be object;
  assume
A2: x in N ~ H;
  N ~ H c= N1 ~ H & N ~ H c= N2 ~ H by A1,Th57;
  hence thesis by A2,XBOOLE_0:def 4;
end;
