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;
