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;
