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,H2 being strict Subgroup of G holds H1 is Subgroup of H2
  implies H1 "\/" H3 is Subgroup of H2 "\/" H3
proof
  let H3,H2 be strict Subgroup of G;
  assume
A1: H1 is Subgroup of H2;
  (H1 "\/" H3) "\/" (H2 "\/" H3) = H1 "\/" H3 "\/" H2 "\/" H3 by Th57
    .= H1 "\/" (H3 "\/" H2) "\/" H3 by Th57
    .= H1 "\/" (H2 "\/" H3) "\/" H3
    .= H1 "\/" H2 "\/" H3 "\/" H3 by Th57
    .= H2 "\/" H3 "\/" H3 by A1,Th61
    .= H2 "\/" (H3 "\/" H3) by Th57
    .= H2 "\/" H3 by Th31;
  hence thesis by Th61;
end;
