
theorem Th5:
  for G being Group for H1, H2 being Subgroup of G for g being
  Element of G holds g in H1 or g in H2 implies g in H1 "\/" H2
proof
  let G be Group;
  let H1, H2 be Subgroup of G;
  let g be Element of G;
  assume
A1: g in H1 or g in H2;
  now
    per cases by A1;
    suppose
A2:   g in H1;
      the carrier of H1 c= the carrier of G & the carrier of H2 c= the
      carrier of G by GROUP_2:def 5;
      then reconsider
      A = (the carrier of H1) \/ the carrier of H2 as Subset of G
      by XBOOLE_1:8;
      g in the carrier of H1 by A2,STRUCT_0:def 5;
      then g in A by XBOOLE_0:def 3;
      then g in gr A by GROUP_4:29;
      hence thesis by Th4;
    end;
    suppose
A3:   g in H2;
      the carrier of H1 c= the carrier of G & the carrier of H2 c= the
      carrier of G by GROUP_2:def 5;
      then reconsider
      A = (the carrier of H1) \/ the carrier of H2 as Subset of G
      by XBOOLE_1:8;
      g in the carrier of H2 by A3,STRUCT_0:def 5;
      then g in A by XBOOLE_0:def 3;
      then g in gr A by GROUP_4:29;
      hence thesis by Th4;
    end;
  end;
  hence thesis;
end;
