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;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;

theorem
  -{g,h} = {-g,-h}
proof
  thus -{g,h} c= {-g,-h}
  proof
    let x be object;
    assume x in -{g,h};
    then consider a such that
A1: x = -a and
A2: a in {g,h};
    a = g or a = h by A2,TARSKI:def 2;
    hence thesis by A1,TARSKI:def 2;
  end;
  let x be object;
  assume x in {-g,-h};
  then
A3: x = -g or x = -h by TARSKI:def 2;
  g in {g,h} & h in {g,h} by TARSKI:def 2;
  hence thesis by A3;
end;
