theorem Th57:
  H1 "\/" H2 "\/" H3 = H1 "\/" (H2 "\/" H3)
proof
  H2 "\/" H3 "\/" H1 is Subgroup of H2 "\/" (H3 "\/" H1) & H3 "\/" H1 "\/"
  H2 is Subgroup of H3 "\/" (H1 "\/" H2) by Lm5;
  then
A1: H1 "\/" (H2 "\/" H3) is Subgroup of H3 "\/" (H1 "\/" H2) by GROUP_2:56;
  H1 "\/" H2 "\/" H3 is Subgroup of H1 "\/" (H2 "\/" H3) by Lm5;
  hence thesis by A1,GROUP_2:55;
end;
