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 Th18:
  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;
    hence -g + -h = -(g + h) by Th16
      .= -h + -g by A1,Th17;
  end;
  assume
A2: -g + -h = -h + -g;
  thus g + h = - -(g + h) .= -(-h + -g) by Th16
    .= - -h + - -g by A2,Th16
    .= h + g;
end;
