
theorem Th59:
  for V being RealUnitarySpace holds LattStr (# Subspaces(V),
    SubJoin(V), SubMeet(V) #) is complemented
proof
  let V be RealUnitarySpace;
  reconsider S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #) as
  01_Lattice by Th57;
  reconsider S0 = S as 0_Lattice;
  reconsider S1 = S as 1_Lattice;
  reconsider Z = (0).V, I = (Omega).V as Element of S by Def3;
  reconsider Z0 = Z as Element of S0;
  reconsider I1 = I as Element of S1;
  now
    let A be Element of S0;
    reconsider W = A as Subspace of V by Def3;
    thus A "/\" Z0 = SubMeet(V).(A,Z0) by LATTICES:def 2
      .= W /\ (0).V by Def8
      .= Z0 by Th18;
  end;
  then
A1: Bottom S = Z by RLSUB_2:64;
  now
    let A be Element of S1;
    reconsider W = A as Subspace of V by Def3;
    thus A "\/" I1 = SubJoin(V).(A,I1) by LATTICES:def 1
      .= W + (Omega).V by Def7
      .= the UNITSTR of V by Th11
      .= (Omega).V by RUSUB_1:def 3;
  end;
  then
A2: Top S = I by RLSUB_2:65;
  now
    let A be Element of S;
    reconsider W = A as Subspace of V by Def3;
    set L = the strict Linear_Compl of W;
    reconsider B9 = L as Element of S by Def3;
    take B = B9;
A3: B "/\" A = SubMeet(V).(B,A) by LATTICES:def 2
      .= L /\ W by Def8
      .= Bottom S by A1,Th37;
    B "\/" A = SubJoin(V).(B,A) by LATTICES:def 1
      .= L + W by Def7
      .= the UNITSTR of V by Th36
      .= Top S by A2,RUSUB_1:def 3;
    hence B is_a_complement_of A by A3,LATTICES:def 18;
  end;
  hence thesis by LATTICES:def 19;
end;
