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;

theorem Th19:
  for G being connected Graph, X being set, v being Vertex of G st
X meets the carrier' of G & not v in G-VSet X ex v9 being Vertex of G, e being
Element of the carrier' of G st v9 in G-VSet X & not e in X & (v9 = (the Target
  of G).e or v9 = (the Source of G).e)
proof
  let G be connected Graph, X be set, v be Vertex of G;
  assume that
A1: X meets the carrier' of G and
A2: not v in G-VSet X;
  ex e being object st e in X & e in the carrier' of G by A1,XBOOLE_0:3;
  then G-VSet X is non empty by Th17;
  then consider vv being object such that
A3: vv in G-VSet X;
  reconsider vv as Vertex of G by A3;
  consider c being Chain of G, vs being FinSequence of the carrier of G such
  that
A4: c is non empty and
A5: vs is_vertex_seq_of c and
A6: vs.1 = vv and
A7: vs.len vs = v by A2,A3,Th18;
  defpred P[Nat] means 1 <= $1 & $1 <= len c & not vs.($1+1) in G-VSet X;
A8: len vs = len c +1 by A5;
  1+0 <= len c by A4,NAT_1:13;
  then
A9: ex n being Nat st P[n] by A2,A7,A8;
  consider k being Nat such that
A10: P[k] and
A11: for n being Nat st P[n] holds k <= n from NAT_1:sch 5(A9);
  len c <= len c +1 by NAT_1:11;
  then k <= len vs by A8,A10,XXREAL_0:2;
  then k in dom vs by A10,FINSEQ_3:25;
  then reconsider v9 = vs.k as Vertex of G by FINSEQ_2:11;
  reconsider c as FinSequence of the carrier' of G by MSSCYC_1:def 1;
A12: rng c c= the carrier' of G by FINSEQ_1:def 4;
  k in dom c by A10,FINSEQ_3:25;
  then c.k in rng c by FUNCT_1:def 3;
  then reconsider e = c.k as Element of the carrier' of G by A12;
  take v9;
  take e;
  hereby
    per cases by A10,XXREAL_0:1;
    suppose
      k = 1;
      hence v9 in G-VSet X by A3,A6;
    end;
    suppose
A13:  1 < k;
      assume
A14:  not v9 in G-VSet X;
      consider p being Element of NAT such that
A15:  k=1+p and
A16:  1 <= p by A13,FINSEQ_4:84;
      p <= k by A15,NAT_1:11;
      then p <= len c by A10,XXREAL_0:2;
      then k <= p by A11,A15,A16,A14;
      hence contradiction by A15,NAT_1:13;
    end;
  end;
  hereby
    1 <= k+1 & k+1 <= len vs by A8,A10,NAT_1:11,XREAL_1:6;
    then k+1 in dom vs by FINSEQ_3:25;
    then reconsider v99 = vs.(k+1) as Vertex of G by FINSEQ_2:11;
    assume
A17: e in X;
    v99 = (the Target of G).e or v99 = (the Source of G).e by A5,A10,Lm3;
    hence contradiction by A10,A17;
  end;
  thus thesis by A5,A10,Lm3;
end;
