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 Th49:
  pe = {} & W is_weight_of G implies cost(pe,W) = 0
proof
  assume that
A1: pe = {} and
A2: W is_weight_of G;
  set f=RealSequence(pe,W);
  dom f = dom pe by A2,Def15;
  then len f = len pe by FINSEQ_3:29
    .= 0 by A1;
  then f = <*>REAL;
  hence thesis by RVSUM_1:72;
end;
