reserve x, y, z, v for set,
  n, m, k for Nat;
reserve G, G1, G2, G3 for Graph;
reserve x, y for Element of (the carrier of G);

theorem
  for G being simple Graph st G1 c= G holds G1 is simple
proof
  let G be simple Graph;
  assume G1 c= G;
then A1: G1 is Subgraph of G;
 not ex x being set st x in the carrier' of G1 &
  (the Source of G1).x = (the Target of G1).x
  proof
    given x being set such that
A2: x in the carrier' of G1 and
A3: (the Source of G1).x = (the Target of G1).x;
A4: (the carrier' of G1) c= (the carrier' of G) by A1,Def18;
 (the Source of G).x = (the Target of G1).x by A1,A2,A3,Def18
      .= (the Target of G).x by A1,A2,Def18;
    hence contradiction by A2,A4,Def9;
  end;
  hence thesis;
end;
