reserve x,y,X for set,
  i,j,k,m,n for Nat,
  p for FinSequence of X,
  ii for Integer;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G,
  p,q for oriented Chain of G,
  W for Function,
  U,V,e,ee for set,
  v1,v2,v3,v4 for Vertex of G;
reserve G for finite Graph,
  P,Q for oriented Chain of G,
  v1,v2,v3 for Vertex of G;
reserve G for finite oriented Graph,
  P,Q for oriented Chain of G,
  W for Function of (the carrier' of G), Real>=0,
  v1,v2,v3,v4 for Vertex of G;
reserve f,g,h for Element of REAL*,
  r for Real;
reserve G for oriented Graph,
  v1,v2 for Vertex of G,
  W for Function of (the carrier' of G), Real>=0;

theorem Th40:
  ex i st i<=n & OuterVx((repeat(Relax(n)*findmin(n))).i.f,n) = {}
proof
  set R=Relax n, M=findmin n;
  defpred P[Nat] means
   $1<=n implies card UnusedVx((repeat(R*M)).$1.f,n) <= n-$1;
  set nf=(repeat(R*M)).n.f;
  assume
A1: not ex i st i<=n & OuterVx((repeat(R*M)).i.f,n) = {};
A2: for k st P[k] holds P[k+1]
  proof
    let k;
    assume
A3: P[k];
    now
      set fk=UnusedVx((repeat(R*M)).k.f,n), fk1=UnusedVx((repeat(R*M)).(k+1).f
      ,n);
A4:   k <= k+1 by NAT_1:11;
      assume
A5:   k+1 <= n;
      then OuterVx((repeat(R*M)).k.f,n) <> {} by A1,A4,XXREAL_0:2;
      then fk1 c< fk by Th38;
      then card fk1 < n-k by A3,A5,A4,CARD_2:48,XXREAL_0:2;
      then card fk1 + 1 <= n-k by INT_1:7;
      then card fk1 <= n-k-1 by XREAL_1:19;
      hence card fk1 <= n-(k+1);
    end;
    hence thesis;
  end;
A6: P[0]
  proof
    set f0=(repeat(R*M)).0 .f;
    assume 0<=n;
    card UnusedVx(f0,n) <= card Seg n by Th26,NAT_1:43;
    hence thesis by FINSEQ_1:57;
  end;
  for k holds P[k] from NAT_1:sch 2(A6,A2);
  then P[n];
  then
A7: UnusedVx(nf,n) = {};
  OuterVx(nf,n) c= UnusedVx(nf,n) by Th27;
  hence contradiction by A1,A7,XBOOLE_1:3;
end;
