
theorem Th15:
  for G being _finite _Graph, S being VNumberingSeq of G, n being
Nat, x being set holds (S.n).x <= S.Lifespan() & (x in dom (S.n) implies 1 <= (
  S.n).x)
proof
  let G be _finite _Graph, S be VNumberingSeq of G, n be Nat, x be set;
  set CSN = S.n;
  set VLN = CSN;
A1: now
    per cases;
    suppose
      not x in dom VLN;
      hence VLN.x <= S.Lifespan() by FUNCT_1:def 2;
    end;
    suppose
      x in dom VLN;
      then VLN.x in rng VLN by FUNCT_1:def 3;
      then VLN.x in (Seg S.Lifespan()) \ Seg (S.Lifespan() -'n ) by Th14;
      hence VLN.x <= S.Lifespan() by Th3;
    end;
  end;
  now
    assume x in dom (S.n);
    then VLN.x in rng VLN by FUNCT_1:def 3;
    then VLN.x in (Seg S.Lifespan()) \ Seg (S.Lifespan() -'n ) by Th14;
    then S.Lifespan() -' n < VLN.x by Th3;
    then 0+1 <= VLN.x by NAT_1:13;
    hence 1 <= VLN.x;
  end;
  hence thesis by A1;
end;
