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 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;
