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 ThB3:
  -{g} = {-g}
proof
  thus -{g} c= {-g}
  proof
    let x be object;
    assume x in -{g};
    then consider h such that
A1: x = -h and
A2: h in {g};
    h = g by A2,TARSKI:def 1;
    hence thesis by A1,TARSKI:def 1;
  end;
  let x be object;
  assume x in {-g};
  then
A3: x = -g by TARSKI:def 1;
  g in {g} by TARSKI:def 1;
  hence thesis by A3;
end;
