reserve x,O for set,
  o for Element of O,
  G,H,I for GroupWithOperators of O,
  A, B for Subset of G,
  N for normal StableSubgroup of G,
  H1,H2,H3 for StableSubgroup of G,
  g1,g2 for Element of G,
  h1,h2 for Element of H1,
  h for Homomorphism of G,H;

theorem Th37:
  for H3 being strict StableSubgroup of G holds H1 is
  StableSubgroup of H3 & H2 is StableSubgroup of H3 implies H1 "\/" H2 is
  StableSubgroup of H3
proof
  let H3 be strict StableSubgroup of G;
  assume that
A1: H1 is StableSubgroup of H3 and
A2: H2 is StableSubgroup of H3;
  H2 is Subgroup of H3 by A2,Def7;
  then
A3: carr H2 c= carr H3 by GROUP_2:def 5;
  H1 is Subgroup of H3 by A1,Def7;
  then carr H1 c= carr H3 by GROUP_2:def 5;
  then the_stable_subgroup_of(carr H1 \/ carr H2) is StableSubgroup of
  the_stable_subgroup_of carr H3 by A3,Th26,XBOOLE_1:8;
  hence thesis by Th25;
end;
