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 Th6:
  rng p1 misses rng p2 & vs1 is_vertex_seq_of p1 & vs2
  is_vertex_seq_of p2 & vs1.len vs1 = vs2.1 implies p1^p2 is Path of G
proof
  set c1 = p1, c2 = p2;
  assume that
A1: rng c1 misses rng c2 and
A2: vs1 is_vertex_seq_of c1 & vs2 is_vertex_seq_of c2 & vs1.len vs1 = vs2.1;
  reconsider c = c1^c2 as Chain of G by A2,GRAPH_2:43;
  now
    let n, m be Nat such that
A3: 1 <= n and
A4: n < m and
A5: m <= len c and
A6: c.n = c.m;
    1 <= m by A3,A4,XXREAL_0:2;
    then
A7: m in dom c by A5,FINSEQ_3:25;
    n <= len c by A4,A5,XXREAL_0:2;
    then
A8: n in dom c by A3,FINSEQ_3:25;
    per cases by A8,A7,FINSEQ_1:25;
    suppose
A9:   n in dom c1 & m in dom c1;
      then c1.n = c.n by FINSEQ_1:def 7
        .= c1.m by A6,A9,FINSEQ_1:def 7;
      hence contradiction by A4,A9,FUNCT_1:def 4;
    end;
    suppose
A10:  n in dom c1 & ex m2 being Nat st m2 in dom c2 & m=len c1 + m2;
      then
A11:  c1.n in rng c1 by FUNCT_1:def 3;
      consider m2 being Nat such that
A12:  m2 in dom c2 and
A13:  m=len c1 + m2 by A10;
A14:  c2.m2 in rng c2 by A12,FUNCT_1:def 3;
      c1.n = c.n by A10,FINSEQ_1:def 7
        .= c2.m2 by A6,A12,A13,FINSEQ_1:def 7;
      hence contradiction by A1,A11,A14,XBOOLE_0:3;
    end;
    suppose
A15:  m in dom c1 & ex n2 being Nat st n2 in dom c2 & n=len c1 + n2;
      then consider n2 being Nat such that
      n2 in dom c2 and
A16:  n=len c1 + n2;
      m <= len c1 by A15,FINSEQ_3:25;
      then len c1 + n2 < len c1 by A4,A16,XXREAL_0:2;
      hence contradiction by NAT_1:11;
    end;
    suppose
A17:  (ex n2 being Nat st n2 in dom c2 & n=len c1 + n2) & ex m2 being
      Nat st m2 in dom c2 & m=len c1 + m2;
      then consider n2 being Nat such that
A18:  n2 in dom c2 & n=len c1 + n2;
      consider m2 being Nat such that
A19:  m2 in dom c2 & m=len c1 + m2 by A17;
      c2.n2 = c.n by A18,FINSEQ_1:def 7
        .= c2.m2 by A6,A19,FINSEQ_1:def 7;
      hence contradiction by A4,A19,A18,FUNCT_1:def 4;
    end;
  end;
  hence thesis by GRAPH_1:def 16;
end;
