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 Th6:
  for p being Simple oriented Chain of G st p=pe^qe & len pe >= 1
& len qe >= 1 holds (the Target of G).(p.len p) <> (the Target of G).(pe.len pe
  ) & (the Source of G).(p.1) <> (the Source of G).(qe.1)
proof
  let p be Simple oriented Chain of G;
  set FT=the Target of G, FS=the Source of G;
  assume that
A1: p=pe^qe and
A2: len pe >= 1 and
A3: len qe >= 1;
  consider vs being FinSequence of the carrier of G such that
A4: vs is_oriented_vertex_seq_of p and
A5: for n,m st 1<=n & n<m & m<=len vs & vs.n=vs.m holds n=1 & m=len vs
  by GRAPH_4:def 7;
A6: len vs = len p + 1 by A4,GRAPH_4:def 5;
  then
A7: 1 <= len vs by NAT_1:12;
  len p = len pe + len qe by A1,FINSEQ_1:22;
  then
A8: len p >= len pe+1 by A3,XREAL_1:7;
  then
A9: len pe+1 < len vs by A6,NAT_1:13;
A10: len p > len pe by A8,NAT_1:13;
  then
A11: len p >= 1 by A2,XXREAL_0:2;
  then p.1 orientedly_joins vs/.1, vs/.(1+1) by A4,GRAPH_4:def 5;
  then
A12: FS.(p.1)=vs/.1 by GRAPH_4:def 1
    .=vs.1 by A7,FINSEQ_4:15;
A13: p.len pe orientedly_joins vs/.len pe, vs/.(len pe+1) by A2,A4,A10,
GRAPH_4:def 5;
  p.len p orientedly_joins vs/.len p, vs/.(len p+1) by A4,A11,GRAPH_4:def 5;
  then
A14: FT.(p.len p)=vs/.(len p+1) by GRAPH_4:def 1
    .=vs.len vs by A6,A7,FINSEQ_4:15;
A15: 1 < len pe +1 by A2,NAT_1:13;
  then
A16: p.(len pe+1) orientedly_joins vs/.(len pe+1), vs/.(len pe+1+1) by A4,A8,
GRAPH_4:def 5;
  FT.(pe.len pe) = FT.(p.len pe) by A1,A2,Lm1
    .=vs/.(len pe+1) by A13,GRAPH_4:def 1
    .=vs.(len pe +1) by A15,A9,FINSEQ_4:15;
  hence FT.(p.len p) <> FT.(pe.len pe) by A5,A14,A15,A9;
  assume
A17: FS.(p.1) = FS.(qe.1);
  FS.(qe.1) = FS.(p.(len pe+1)) by A1,A3,Lm2
    .=vs/.(len pe+1) by A16,GRAPH_4:def 1
    .=vs.(len pe +1) by A15,A9,FINSEQ_4:15;
  hence contradiction by A5,A15,A9,A12,A17;
end;
