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;
