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 Th32:
  i in dom f & f.i=-1 & i <> n*n+3*n+1 implies (findmin n).f.i=-1
proof
  set k=Argmin(OuterVx(f,n),f,n), mi=n*n+3*n+1;
  assume that
A1: i in dom f and
A2: f.i=-1 & i <> mi;
A3: (findmin n).f.i = ((f,mi):=(k,-jj)).i by Def11;
  per cases;
  suppose
    i=k;
    hence thesis by A1,A3,Th19;
  end;
  suppose
    i<>k;
    hence thesis by A2,A3,Th18;
  end;
end;
