reserve x,y,X for set,
  i,j,k,m,n for Nat,
  p for FinSequence of X,
  ii for Integer;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G,
  p,q for oriented Chain of G,
  W for Function,
  U,V,e,ee for set,
  v1,v2,v3,v4 for Vertex of G;

theorem Th11:
  the carrier of G= U \/ V & v1 in U & v2 in V & (for v3,v4 st v3
  in U & v4 in V holds not (ex e st e in the carrier' of G & e orientedly_joins
  v3,v4)) implies not ex p st p is_orientedpath_of v1,v2
proof
  assume that
A1: the carrier of G= U \/ V and
A2: v1 in U and
A3: v2 in V and
A4: for v3,v4 st v3 in U & v4 in V holds not (ex e st e in the carrier'
  of G & e orientedly_joins v3,v4);
  set FS=the Source of G, FT=the Target of G;
  given p such that
A5: p is_orientedpath_of v1,v2;
  p <> {} by A5,GRAPH_5:def 3;
  then
A6: len p >= 1 by FINSEQ_1:20;
  defpred PP[Nat] means $1 >=1 & $1<=len p & FS.(p.$1) in U;
A7: for k be Nat st PP[k] holds k <= len p;
  FS.(p.1) = v1 by A5,GRAPH_5:def 3;
  then
A8: ex k be Nat st PP[k] by A2,A6;
  consider k be Nat such that
A9: PP[k] & for n be Nat st PP[n] holds n <= k from NAT_1:sch 6(A7,A8
  );
  reconsider k as Element of NAT by ORDINAL1:def 12;
  reconsider vx=FS.(p.k) as Vertex of G by A9,Lm3;
A10: p.k in the carrier' of G by A9,Th2;
A11: FT.(p.(len p))= v2 by A5,GRAPH_5:def 3;
  per cases;
  suppose
    k=len p;
    then p.k orientedly_joins vx,v2 by A11,GRAPH_4:def 1;
    hence contradiction by A3,A4,A9,A10;
  end;
  suppose
    k<>len p;
    then
A12: k < len p by A9,XXREAL_0:1;
A13: k < k+1 by NAT_1:13;
A14: now
      assume
A15:  FS.(p.(k+1)) in U;
      k+1 <= len p & 1 <= k+1 by A12,INT_1:7,NAT_1:12;
      hence contradiction by A9,A13,A15;
    end;
    reconsider vy=FT.(p.k) as Vertex of G by A9,Lm3;
A16: p.k orientedly_joins vx,vy by GRAPH_4:def 1;
    FS.(p.(k+1)) = FT.(p.k) by A9,A12,GRAPH_1:def 15;
    then vy in V by A1,A2,A14,XBOOLE_0:def 3;
    hence contradiction by A4,A9,A10,A16;
  end;
end;
