reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
reserve M for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF;
reserve W,W1,W2,W3 for Subspace of M;
reserve u,u1,u2,v,v1,v2 for Element of M;
reserve X,Y for set, x,y,y1,y2 for object;
reserve F for Field;
reserve V for VectSp of F;
reserve W for Subspace of V;
reserve W,W1,W2 for Subspace of V;
reserve W1,W2 for Subspace of M;
reserve W for Subspace of V;
reserve W1,W2 for Subspace of M;
reserve u,u1,u2,v for Element of M;
reserve C1 for Coset of W1;
reserve C2 for Coset of W2;
reserve t1,t2 for Element of [:the carrier of M, the carrier of M:];
reserve W for Subspace of V;
reserve A1,A2,B for Element of Subspaces(M),
  W1,W2 for Subspace of M;

theorem
  for F being Field, V being VectSp of F holds LattStr (# Subspaces(V),
    SubJoin(V), SubMeet(V) #) is C_Lattice
proof
  let F be Field, V be VectSp of F;
  reconsider S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #) as
  01_Lattice by Th60;
  reconsider S0 = S as 0_Lattice;
  reconsider S1 = S as 1_Lattice;
  consider W9 being strict Subspace of V such that
A1: the carrier of W9 = the carrier of (Omega).V;
  reconsider I = W9 as Element of S by Def3;
  reconsider I1 = I as Element of S1;
  reconsider Z = (0).V as Element of S by Def3;
  reconsider Z0 = Z as Element of S0;
  now
    let A be Element of S0;
    consider W being strict Subspace of V such that
A2: W = A by Def3;
    thus A "/\" Z0 = SubMeet(V).(A,Z0) by LATTICES:def 2
      .= W /\ (0).V by A2,Def8
      .= Z0 by Th20;
  end;
  then
A3: Bottom S = Z by RLSUB_2:64;
  now
    let A be Element of S1;
    consider W being strict Subspace of V such that
A4: W = A by Def3;
A5: W9 is Subspace of (Omega).V by Lm6;
    thus A "\/" I1 = SubJoin(V).(A,I1) by LATTICES:def 1
      .= W + W9 by A4,Def7
      .= W + (Omega).V by A1,Lm5
      .= the ModuleStr of V by Th11
      .= W9 by A1,A5,VECTSP_4:31;
  end;
  then
A6: Top S = I by RLSUB_2:65;
  now
A7: I is Subspace of (Omega).V by Lm6;
    let A be Element of S;
    consider W being strict Subspace of V such that
A8: W = A by Def3;
    set L = the Linear_Compl of W;
    consider W99 being strict Subspace of V such that
A9: the carrier of W99 = the carrier of L by Lm4;
    reconsider B9 = W99 as Element of S by Def3;
    take B = B9;
A10: B "/\" A = SubMeet(V).(B,A) by LATTICES:def 2
      .= W99 /\ W by A8,Def8
      .= L /\ W by A9,Lm8
      .= Bottom S by A3,Th40;
    B "\/" A = SubJoin(V).(B,A) by LATTICES:def 1
      .= W99 + W by A8,Def7
      .= L + W by A9,Lm5
      .= the ModuleStr of V by Th39
      .= Top S by A1,A6,A7,VECTSP_4:31;
    hence B is_a_complement_of A by A10,LATTICES:def 18;
  end;
  hence thesis by LATTICES:def 19;
end;
