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 ThX7:
  A <> {} iff -A <> {}
proof
  set x = the Element of -A;
  thus A <> {} implies -A <> {}
  proof
    set x = the Element of A;
    assume
A1: A <> {};
    then reconsider x as Element of G by Lm1;
    -x in -A by A1;
    hence thesis;
  end;
  assume -A <> {};
  then ex a st x = -a & a in A by Th2;
  hence thesis;
end;
