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 Th13:
  f + h = g iff f = g + -h
proof
  g + -h + h = g + (-h + h) by RLVECT_1:def 3
    .= g + 0_G by Def5
    .= g by Def4;
  hence f + h = g implies f = g + -h by Th6;
  assume f = g + -h;
  hence f + h = g + (-h + h) by RLVECT_1:def 3
    .= g + 0_G by Def5
    .= g by Def4;
end;
