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;
reserve v,v1,v2,v3 for Element of G;
reserve p,q for oriented Chain of G;

theorem Th46:
  rng qe c= rng pe & W is_weight_of G & i in dom qe implies ex j
  st j in dom pe & RealSequence(pe,W).j = RealSequence(qe,W).i
proof
  assume that
A1: rng qe c= rng pe and
A2: W is_weight_of G and
A3: i in dom qe;
  set g=RealSequence(qe,W);
  consider y being object such that
A4: y in dom pe and
A5: qe.i = pe.y by A1,A3,FUNCT_1:114;
  reconsider j=y as Element of NAT by A4;
  take j;
  thus j in dom pe by A4;
  g.i=W.(qe.i) by A2,A3,Def15;
  hence thesis by A2,A4,A5,Def15;
end;
