
theorem Th24:
  for F being Field for V being finite-dimensional VectSp of F for
  W being Subspace of V for m being Nat st 1<=m & m <= dim V & m Subspaces_of V
  c= m Subspaces_of W holds (Omega).V = (Omega).W
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let W be Subspace of V;
  let m be Nat such that
A1: 1<=m and
A2: m <= dim V and
A3: m Subspaces_of V c= m Subspaces_of W;
  hereby
A4: dim W <= dim V by VECTSP_9:25;
    assume
A5: (Omega).V <> (Omega).W;
    then dim W <> dim V by VECTSP_9:28;
    then
A6: dim W < dim V by A4,XXREAL_0:1;
    per cases by A2,XXREAL_0:1;
    suppose
      m=dim V;
      then m Subspaces_of W = {} by A6,VECTSP_9:37;
      hence contradiction by A2,A3,VECTSP_9:36;
    end;
    suppose
A7:   m < dim V;
A8:   now
        assume
A9:     the carrier of V = the carrier of W;
        (Omega).W is strict Subspace of V by Th4;
        hence contradiction by A5,A9,VECTSP_4:29;
      end;
      the carrier of W c= the carrier of V by VECTSP_4:def 2;
      then not the carrier of V c= the carrier of W by A8;
      then consider x being object such that
A10:  x in the carrier of V and
A11:  not x in the carrier of W;
      reconsider x as Vector of V by A10;
      0.V in W by VECTSP_4:17;
      then x <> 0.V by A11;
      then {x} is linearly-independent by VECTSP_7:3;
      then consider I being Basis of V such that
A12:  {x} c= I by VECTSP_7:19;
      reconsider J=I as finite Subset of V;
      card J = dim V by VECTSP_9:def 1;
      then consider K being Subset of J such that
A13:  card K = m by A7,Lm1;
A14:  I is linearly-independent by VECTSP_7:def 3;
      per cases;
      suppose
A15:    x in K;
        reconsider L=K as finite Subset of V by XBOOLE_1:1;
        L c= the carrier of Lin L
        proof
          let a be object;
          assume a in L;
          then a in Lin L by VECTSP_7:8;
          hence thesis;
        end;
        then reconsider L9=L as Subset of Lin L;
        L is linearly-independent by A14,VECTSP_7:1;
        then reconsider LL1 = L9 as linearly-independent Subset of Lin L by
VECTSP_9:12;
        Lin LL1 = the ModuleStr of Lin L by VECTSP_9:17;
        then L is Basis of Lin L by VECTSP_7:def 3;
        then dim Lin L = m by A13,VECTSP_9:def 1;
        then Lin L in m Subspaces_of V by VECTSP_9:def 2;
        then ex M being strict Subspace of W st M=Lin L & dim M = m by A3,
VECTSP_9:def 2;
        then x in W by A15,VECTSP_4:9,VECTSP_7:8;
        hence contradiction by A11;
      end;
      suppose
A16:    not x in K;
        consider y being object such that
A17:    y in K by A1,A13,CARD_1:27,XBOOLE_0:def 1;
        (K\{y})\/{x} c= the carrier of V
        proof
          let a be object;
          assume a in (K\{y})\/{x};
          then a in (K\{y}) or a in {x} by XBOOLE_0:def 3;
          hence thesis by TARSKI:def 3;
        end;
        then reconsider L=(K\{y})\/{x} as finite Subset of V;
        L c= the carrier of Lin L
        proof
          let a be object;
          assume a in L;
          then a in Lin L by VECTSP_7:8;
          hence thesis;
        end;
        then reconsider L9=L as Subset of Lin L;
        L c= I
        proof
          let a be object;
          assume a in L;
          then a in K\{y} or a in {x} by XBOOLE_0:def 3;
          hence thesis by A12;
        end;
        then L is linearly-independent by A14,VECTSP_7:1;
        then reconsider LL1 = L9 as linearly-independent Subset of Lin L by
VECTSP_9:12;
        Lin LL1 = the ModuleStr of Lin L by VECTSP_9:17;
        then
A18:    L is Basis of Lin L by VECTSP_7:def 3;
        not x in K\{y} by A16,XBOOLE_0:def 5;
        then card L = card(K\{y})+1 by CARD_2:41
          .= card K - card{y} + 1 by A17,EULER_1:4
          .= card K - 1 + 1 by CARD_1:30
          .= m by A13;
        then dim Lin L = m by A18,VECTSP_9:def 1;
        then Lin L in m Subspaces_of V by VECTSP_9:def 2;
        then
A19:    ex M being strict Subspace of W st M=Lin L & dim M = m by A3,
VECTSP_9:def 2;
        x in {x} by TARSKI:def 1;
        then x in L by XBOOLE_0:def 3;
        then x in W by A19,VECTSP_4:9,VECTSP_7:8;
        hence contradiction by A11;
      end;
    end;
  end;
end;
