theorem Th36:
  vs9 = vs & vs is_vertex_seq_of c implies vs9 is_vertex_seq_of c
proof
  assume that
A1: vs9 = vs and
A2: vs is_vertex_seq_of c;
  thus len vs9 = len c + 1 by A1,A2;
  let n be Nat;
  set T = the Target of G;
  set S = the Source of G;
  set v = c.n;
  set x = vs/.n;
  set y = vs/.(n+1);
  assume
A3: 1<=n & n<=len c;
  then c.n joins vs/.n, vs/.(n+1) by A2;
  then
A4: S.v = x & T.v = y or S.v = y & T.v = x;
  set G9 = AddNewEdge(v1, v2);
  set S9 = the Source of G9;
  set T9 = the Target of G9;
A5: the carrier of G = the carrier of G9 by Def7;
  c is FinSequence of the carrier' of G by MSSCYC_1:def 1;
  then
A6: rng c c= the carrier' of G by FINSEQ_1:def 4;
  n in dom c by A3,FINSEQ_3:25;
  then c.n in rng c by FUNCT_1:def 3;
  then S9.v = S.v & T.v = T9.v by A6,Th35;
  hence thesis by A1,A5,A4;
end;
