theorem Th114:
  a + H = b + H iff -b + a in H
proof
  thus a + H = b + H implies -b + a in H
  proof
    assume
A1: a + H = b + H;
    -b + a + H = -b + (a + H) by Th32
      .= -b + b + H by A1,Th32
      .= 0_G + H by Def5
      .= carr(H) by Th37;
    hence thesis by Th113;
  end;
  assume
A2: -b + a in H;
  thus a + H = 0_G + (a + H) by Th37
    .= 0_G + a + H by Th32
    .= b + -b + a + H by Def5
    .= b + (-b + a) + H by RLVECT_1:def 3
    .= b + ((-b + a) + H) by Th32
    .= b + H by A2,Th113;
end;
