theorem Th45:
  for f being FinSequence of REAL st W is_weight>=0of G & f =
  RealSequence(pe,W) holds for i st i in dom f holds f.i >= 0
proof
  let f be FinSequence of REAL;
  assume that
A1: W is_weight>=0of G and
A2: f = RealSequence(pe,W);
A3: W is Function of the carrier' of G, Real>=0 by A1;
  let i;
  assume
A4: i in dom f;
A5: W is_weight_of G by A1,Th44;
  then
A6: dom pe = dom f by A2,Def15;
  then f.i=W.(pe.i) by A2,A5,A4,Def15;
  then f.i in Real>=0 by A6,A3,A4,FINSEQ_2:11,FUNCT_2:5;
  then ex r being Real st f.i=r & r >= 0;
  hence thesis;
end;
