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 Th12:
  h + f = g iff f = -h + g
proof
  h + (-h + g) = h + -h + g by RLVECT_1:def 3
    .= 0_G + g by Def5
    .= g by Def4;
  hence h + f = g implies f = -h + g by Th6;
  assume f = -h + g;
  hence h + f = h + -h + g by RLVECT_1:def 3
    .= 0_G + g by Def5
    .= g by Def4;
end;
