reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  for H3 being strict Subgroup of G holds H1 is Subgroup of H3 & H2 is
  Subgroup of H3 implies H1 "\/" H2 is Subgroup of H3
proof
  let H3 be strict Subgroup of G;
  assume
A1: H1 is Subgroup of H3 & H2 is Subgroup of H3;
  now
    let a;
    assume a in H1 "\/" H2;
    then consider F,I such that
A2: len F = len I and
A3: rng F c= carr H1 \/ carr H2 and
A4: a = Product(F |^ I) by Th28;
    the carrier of H1 c= the carrier of H3 & the carrier of H2 c= the
    carrier of H3 by A1,GROUP_2:def 5;
    then carr H1 \/ carr H2 c= carr H3 by XBOOLE_1:8;
    then rng F c= carr H3 by A3;
    then a in gr carr H3 by A2,A4,Th28;
    hence a in H3 by Th31;
  end;
  hence thesis by GROUP_2:58;
end;
