reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p, p1, p2 for Path of G,
  vs, vs1, vs2 for FinSequence of the carrier of G,
  e, X for set,
  n, m for Nat;
reserve G for finite Graph,
  v for Vertex of G,
  c for Chain of G,
  vs for FinSequence of the carrier of G,
  X1, X2 for set;

theorem Th30:
  Edges_In(v, X) = Edges_In(v, X/\the carrier' of G) & Edges_Out(v
  , X) = Edges_Out(v, X/\the carrier' of G)
proof
  set E = the carrier' of G;
  now
    let x be object;
    hereby
      assume
A1:   x in Edges_In(v, X);
      then x in X by Def1;
      then
A2:   x in X/\E by A1,XBOOLE_0:def 4;
      (the Target of G).x = v by A1,Def1;
      hence x in Edges_In(v, X/\E) by A1,A2,Def1;
    end;
    assume
A3: x in Edges_In(v, X/\E);
    then x in X/\E by Def1;
    then
A4: x in X by XBOOLE_0:def 4;
    (the Target of G).x = v by A3,Def1;
    hence x in Edges_In(v, X) by A3,A4,Def1;
  end;
  hence Edges_In(v, X) = Edges_In(v, X/\the carrier' of G) by TARSKI:2;
  now
    let x be object;
    hereby
      assume
A5:   x in Edges_Out(v, X);
      then x in X by Def2;
      then
A6:   x in X/\E by A5,XBOOLE_0:def 4;
      (the Source of G).x = v by A5,Def2;
      hence x in Edges_Out(v, X/\E) by A5,A6,Def2;
    end;
    assume
A7: x in Edges_Out(v, X/\E);
    then x in X/\E by Def2;
    then
A8: x in X by XBOOLE_0:def 4;
    (the Source of G).x = v by A7,Def2;
    hence x in Edges_Out(v, X) by A7,A8,Def2;
  end;
  hence thesis by TARSKI:2;
end;
