reserve p, q for FinSequence,
  e,X for set,
  i, j, k, m, n for Nat,
  G for Graph;
reserve x,y,v,v1,v2,v3,v4 for Element of G;
reserve vs, vs1, vs2 for FinSequence of the carrier of G,
  c, c1, c2 for oriented Chain of G;

theorem
  G-SVSet {} = {} & G-TVSet {} = {}
proof
  set X = {};
A1: now
    assume
A2: G-SVSet X<>{};
    set x = the Element of G-SVSet X;
    x in G-SVSet X by A2;
    then ex v st ( v=x)&( ex e being Element of the carrier' of G st e
    in X & v = (the Source of G).e);
    hence contradiction;
  end;
  now
    assume
A3: G-TVSet X<>{};
    set x = the Element of G-TVSet X;
    x in G-TVSet X by A3;
    then ex v st ( v=x)&( ex e being Element of the carrier' of G st e
    in X & v = (the Target of G).e);
    hence contradiction;
  end;
  hence thesis by A1;
end;
