
theorem Th22:
  for G being _finite _Graph, S being VNumberingSeq of G, m being
  Nat, v being set st v in dom (S.m) holds S.Lifespan() -' (S.m).v < m & S
  .Lifespan() -' m < (S.m).v
proof
  let G be _finite _Graph, S be VNumberingSeq of G, m be Nat, v be set;
  set VLM = S.m;
  set j = S.Lifespan() -' VLM.v;
  set VLJ = S.j;
  assume
A1: v in dom VLM;
  then
A2: S.PickedAt(j) = v by Th20;
A3: 0 < VLM.v by A1,Th15;
A4: VLM.v <= S.Lifespan() by Th15;
  then j = S.Lifespan() - VLM.v by XREAL_1:233;
  then
A5: j < S.Lifespan() by A3,XREAL_1:44;
A6: now
    per cases;
    suppose
      m <= j;
      then VLM c= VLJ by Th17;
      then dom VLM c= dom VLJ by RELAT_1:11;
      hence S.Lifespan() -' VLM.v < m by A1,A2,A5,Def9;
    end;
    suppose
      m > j;
      hence S.Lifespan() -' VLM.v < m;
    end;
  end;
  now
    per cases;
    suppose
A7:   S.Lifespan() - m >= 0;
      S.Lifespan() - VLM.v < m by A4,A6,XREAL_1:233;
      then S.Lifespan() - VLM.v + VLM.v < m + VLM.v by XREAL_1:6;
      then S.Lifespan() - m < VLM.v + m - m by XREAL_1:9;
      hence S.Lifespan() -'m < VLM.v by A7,XREAL_0:def 2;
    end;
    suppose
      S.Lifespan() - m < 0;
      hence S.Lifespan() -'m < VLM.v by A3,XREAL_0:def 2;
    end;
  end;
  hence thesis by A6;
end;
