reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;

theorem Th19:
  g + h = h + g iff g + -h = -h + g
proof
  thus g + h = h + g implies g + -h = -h + g
  proof
    assume
A1: g + h = h + g;
    (g + -h) + (-g + h) = g + -h + -g + h by RLVECT_1:def 3
      .= g + (-h + -g) + h by RLVECT_1:def 3
      .= g + (-g + -h) + h by A1,Th18
      .= g + -g + -h + h by RLVECT_1:def 3
      .= 0_G + -h + h by Def5
      .= -h + h by Def4
      .= 0_G by Def5;
    then g + -h = -(-g + h) by Th11
      .= -h + - -g by Th16;
    hence thesis;
  end;
  assume g + -h = -h + g;
  then g + (-h + h) = -h + g + h by RLVECT_1:def 3;
  then g + 0_G = -h + g + h by Def5;
  then g = -h + g + h by Def4;
  then g = -h + (g + h) by RLVECT_1:def 3;
  then h + g = h + -h + (g + h) by RLVECT_1:def 3;
  then h + g = 0_G + (g + h) by Def5;
  hence thesis by Def4;
end;
