
theorem Th22:
  for G being Group for H1, H2 being Subgroup of G for h1, h2
  being Element of lattice G st h1 = H1 & h2 = H2 holds h1 "\/" h2 = H1 "\/" H2
proof
  let G be Group;
  let H1, H2 be Subgroup of G;
  let h1, h2 be Element of lattice G;
A1: h1 "\/" h2 = SubJoin G.(h1,h2) by LATTICES:def 1;
  assume
A2: h1 = H1 & h2 = H2;
  then H1 is strict & H2 is strict by GROUP_3:def 1;
  hence thesis by A2,A1,GROUP_4:def 10;
end;
