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 Th17:
  g + h = h + g iff -(g + h) = -g + -h
proof
  thus g + h = h + g implies -(g + h) = -g + -h by Th16;
  assume -(g + h) = -g + -h;
  then
A1: (h + g) + -(g + h) = h + g + -g + -h by RLVECT_1:def 3
    .= h + (g + -g) + -h by RLVECT_1:def 3
    .= h + 0_G + -h by Def5
    .= h + -h by Def4
    .= 0_G by Def5;
  (g + h) + -(g + h) = 0_G by Def5;
  hence thesis by A1,Th6;
end;
