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 Th29:
  OuterVx(f,n) <> {} & j=Argmin(OuterVx(f,n),f,n) implies j in dom
  f & 1<=j & j<=n & f.j <> -1 & f.(n+j) <> -1
proof
  set IN=OuterVx(f,n);
  assume IN <> {} & j=Argmin(IN,f,n);
  then
  ex i st i=j & i in IN &( for k st k in IN holds f/.(2*n+ i) <= f/.(2*n+k)
  )& for k st k in IN & f/.(2*n+i) = f/.(2*n+k) holds i <= k by Def10;
  then
  ex k st j=k & k in dom f & 1 <= k & k<= n & f.k <> -1 & f.(n+k) <> -1;
  hence thesis;
end;
