
theorem Th28:
  for F being Field for V being finite-dimensional VectSp of F for
  m being Nat st m+1 <= dim V holds GrassmannSpace(V,m,m+1) is
  identifying_close_blocks
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let m be Nat such that
A1: m+1 <= dim V;
  set S = GrassmannSpace(V,m,m+1);
  let K,L be Block of S;
A2: the topology of S is non empty by A1,Th22;
  then consider W1 being Subspace of V such that
A3: dim W1 = m+1 and
A4: K = m Subspaces_of W1 by Def6;
  assume 2 c= card(K/\L);
  then consider x,y being object such that
A5: x in K/\L and
A6: y in K/\L and
A7: x <> y by PENCIL_1:2;
  y in K by A6,XBOOLE_0:def 4;
  then consider Y being strict Subspace of W1 such that
A8: y=Y and
A9: dim Y = m by A4,VECTSP_9:def 2;
  consider W2 being Subspace of V such that
A10: dim W2 = m+1 and
A11: L = m Subspaces_of W2 by A2,Def6;
  y in L by A6,XBOOLE_0:def 4;
  then consider Y9 being strict Subspace of W2 such that
A12: y=Y9 and
  dim Y9 = m by A11,VECTSP_9:def 2;
  x in L by A5,XBOOLE_0:def 4;
  then consider X9 being strict Subspace of W2 such that
A13: x=X9 and
  dim X9 = m by A11,VECTSP_9:def 2;
  x in K by A5,XBOOLE_0:def 4;
  then consider X being strict Subspace of W1 such that
A14: x=X and
A15: dim X = m by A4,VECTSP_9:def 2;
  reconsider x,y as strict Subspace of V by A14,A8,VECTSP_4:26;
A16: now
    reconsider y9=y as strict Subspace of x+y by VECTSP_5:7;
    reconsider x9=x as strict Subspace of x+y by VECTSP_5:7;
    assume
A17: dim(x+y)=m;
    then (Omega).x9 = (Omega).(x+y) by A14,A15,VECTSP_9:28;
    then x = y+x by VECTSP_5:5;
    then
A18: y is Subspace of x by VECTSP_5:8;
    (Omega).y9 = (Omega).(x+y) by A8,A9,A17,VECTSP_9:28;
    then x is Subspace of y by VECTSP_5:8;
    hence contradiction by A7,A18,VECTSP_4:25;
  end;
  x+y is Subspace of W1 by A14,A8,VECTSP_5:34;
  then x is Subspace of x+y & dim (x+y) <= m+1 by A3,VECTSP_5:7,VECTSP_9:25;
  then
A19: dim (x+y) = m+1 by A14,A15,A16,NAT_1:9,VECTSP_9:25;
  X9+Y9=x+y by A13,A12,VECTSP10:12;
  then
A20: (Omega).(X9+Y9) = (Omega).W2 by A10,A19,VECTSP_9:28;
A21: X+Y=x+y by A14,A8,VECTSP10:12;
  then (Omega).(X+Y) = (Omega).W1 by A3,A19,VECTSP_9:28;
  hence thesis by A4,A11,A13,A12,A21,A20,Th23,VECTSP10:12;
end;
