reserve n,m,i,j,k for Nat,
  x,y,e,X,V,U for set,
  W,f,g for Function;
reserve p,q for FinSequence;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G;

theorem Th19:
  rng qe c= rng pe implies vertices qe c= vertices pe
proof
  assume
A1: rng qe c= rng pe;
  let x be object;
  assume x in vertices qe;
  then consider v being Vertex of G such that
A2: x=v and
A3: ex i st i in dom qe & v in vertices(qe/.i);
  consider i such that
A4: i in dom qe and
A5: v in vertices(qe/.i) by A3;
  qe.i in rng qe by A4,FUNCT_1:def 3;
  then consider j being object such that
A6: j in dom pe and
A7: qe.i = pe.j by A1,FUNCT_1:def 3;
  reconsider j as Element of NAT by A6;
  qe/.i=qe.i by A4,PARTFUN1:def 6;
  then pe/.j=qe/.i by A6,A7,PARTFUN1:def 6;
  hence thesis by A2,A5,A6;
end;
