
theorem Th27:
  for F being Field for V being finite-dimensional VectSp of F for
  m,n being Nat st 1 <= m & m < n & n <= dim V holds GrassmannSpace(V,m,n) is
  with_non_trivial_blocks
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let m,n be Nat such that
A1: 1 <= m and
A2: m < n and
A3: n <= dim V;
  set S=GrassmannSpace(V,m,n);
  let B be Block of S;
  the topology of S is non empty by A3,Th22;
  then consider W being Subspace of V such that
A4: dim W = n and
A5: B = m Subspaces_of W by Def6;
  m+1 <= n by A2,NAT_1:13;
  then consider U being strict Subspace of W such that
A6: dim U = m+1 by A4,VECTSP_9:35;
  set I = the Basis of U;
A7: card I = m+1 by A6,VECTSP_9:def 1;
  reconsider I9=I as finite Subset of U;
  reconsider m as Element of NAT by ORDINAL1:def 12;
  1+1 <= m+1 by A1,XREAL_1:7;
  then Segm 2 c= Segm(m+1) by NAT_1:39;
  then consider i,j being object such that
A8: i in I and
A9: j in I and
A10: i<>j by PENCIL_1:2,A7;
  reconsider I1=I9\{i} as finite Subset of U;
  I1 c= the carrier of Lin I1
  proof
    let a be object;
    assume a in I1;
    then a in Lin I1 by VECTSP_7:8;
    hence thesis;
  end;
  then reconsider I19=I1 as Subset of Lin I1;
A11: I is linearly-independent by VECTSP_7:def 3;
  then I1 is linearly-independent by VECTSP_7:1,XBOOLE_1:36;
  then reconsider II1=I19 as linearly-independent Subset of Lin I1 by
VECTSP_9:12;
  Lin II1 = the ModuleStr of Lin I1 by VECTSP_9:17;
  then
A12: I1 is Basis of Lin I1 by VECTSP_7:def 3;
  reconsider I2=I9\{j} as finite Subset of U;
  I2 c= the carrier of Lin I2
  proof
    let a be object;
    assume a in I2;
    then a in Lin I2 by VECTSP_7:8;
    hence thesis;
  end;
  then reconsider I29=I2 as Subset of Lin I2;
  I2 is linearly-independent by A11,VECTSP_7:1,XBOOLE_1:36;
  then reconsider II2=I29 as linearly-independent Subset of Lin I2 by
VECTSP_9:12;
  Lin II2 = the ModuleStr of Lin I2 by VECTSP_9:17;
  then
A13: I2 is Basis of Lin I2 by VECTSP_7:def 3;
  card I2 = card I9 - card{j} by A9,EULER_1:4
    .= m+1 - 1 by A7,CARD_1:30;
  then
A14: dim Lin I2 = m by A13,VECTSP_9:def 1;
  Lin I2 is strict Subspace of W by VECTSP_4:26;
  then
A15: Lin I2 in B by A5,A14,VECTSP_9:def 2;
  card I1 = card I9 - card{i} by A8,EULER_1:4
    .= m+1 - 1 by A7,CARD_1:30;
  then
A16: dim Lin I1 = m by A12,VECTSP_9:def 1;
  Lin I1 is strict Subspace of W by VECTSP_4:26;
  then
A17: Lin I1 in B by A5,A16,VECTSP_9:def 2;
  not j in {i} by A10,TARSKI:def 1;
  then j in I1 by A9,XBOOLE_0:def 5;
  then
A18: j in Lin I1 by VECTSP_7:8;
  not j in Lin I2 by A11,A9,VECTSP_9:14;
  hence thesis by A17,A15,A18,PENCIL_1:2;
end;
