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 Th34:
  OuterVx(f,n) <> {} implies ex j st j in OuterVx(f,n) & 1 <= j &
  j <= n & (findmin n).f.j=-1
proof
  set IX=OuterVx(f,n);
  assume IX <> {};
  then consider i such that
A1: i=Argmin(IX,f,n) and
A2: i in IX and
  for k st k in IX holds f/.(2*n+i) <= f/.(2*n+k) and
  for k st k in IX & f/.(2*n+i) = f/.(2*n+k) holds i <= k by Def10;
  take i;
  thus i in IX by A2;
A3: ex k st i=k & k in dom f & 1 <= k & k <= n & f.k <> - 1 & f.(n+k) <> -1
  by A2;
  hence 1 <= i & i <= n;
  thus (findmin n).f.i = ((f,n*n+3*n+1):=(i,-jj)).i by A1,Def11
    .=-1 by A3,Th19;
end;
