theorem Th116:
  a + b + H c= (a + H) + (b + H)
proof
  let x be object;
  assume x in a + b + H;
  then consider g such that
A1: x = a + b + g and
A2: g in H by Th103;
A3: x = a + 0_G + b + g by A1,Def4
    .= (a + 0_G) + (b + g) by RLVECT_1:def 3;
A4: a + 0_G in a + H by Th46,Th103;
  b + g in b + H by A2,Th103;
  hence thesis by A3,A4;
end;
