
theorem Th78: :: minVSexistance
  for G being _finite _Graph for a,b being Vertex of G ex S being
  VertexSeparator of a,b st S is minimal
proof
  let G be _finite _Graph, a,b be Vertex of G;
  set X = the set of all S where S is VertexSeparator of a,b ;
  set s = the VertexSeparator of a,b;
A1: s in X;
  now
    let x be object;
    assume x in X;
    then ex y being VertexSeparator of a,b st x = y;
    hence x in bool the_Vertices_of G;
  end;
  then X c= bool the_Vertices_of G;
  then reconsider X as non empty finite set by A1;
  defpred P[object,object] means
   ex p being VertexSeparator of a,b st $1 = p & $2 = card p;
A2: now
    let x be object;
    assume x in X;
    then consider Y being VertexSeparator of a,b such that
A3: Y = x;
    card Y in REAL by NUMBERS:19;
    hence ex y being object st y in REAL & P[x,y] by A3;
  end;
  consider F being Function of X, REAL such that
A4: for x being object st x in X holds P[x,F.x] from FUNCT_2:sch 1(A2);
  deffunc FF(Element of X) = F/.$1;
  consider Min being Element of X such that
A5: for N being Element of X holds FF(Min) <= FF(N) from PRE_CIRC:sch 5;
  consider M being VertexSeparator of a,b such that
  M = Min and
A6: card M = F.Min by A4;
A7: dom F = X by FUNCT_2:def 1;
  now
    assume not M is minimal;
    then consider T being Subset of M such that
A8: T<>M and
A9: T is VertexSeparator of a,b;
    T in X by A9;
    then reconsider T2=T as Element of X;
    consider Tp being VertexSeparator of a,b such that
A10: Tp=T2 and
A11: card Tp = F.T2 by A4;
    Tp in dom F by A7;
    then F/.Tp = F.Tp by PARTFUN1:def 6;
    then
A12: card M <= card T by A5,A6,A10,A11;
    card T <= card M by NAT_1:43;
    hence contradiction by A8,A12,CARD_2:102,XXREAL_0:1;
  end;
  hence thesis;
end;
