
theorem Th58:
  for V being RealUnitarySpace holds LattStr (# Subspaces(V),
    SubJoin(V), SubMeet(V) #) is modular
proof
  let V be RealUnitarySpace;
  set S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #);
  for A,B,C being Element of S st A [= C holds A "\/" (B "/\" C) = (A "\/"
  B) "/\" C
  proof
    let A,B,C be Element of S;
    reconsider W1 = A, W2 = B, W3 = C as strict Subspace of V by Def3;
    assume
A1: A [= C;
    reconsider AB = W1 + W2 as Element of S by Def3;
    reconsider BC = W2 /\ W3 as Element of S by Def3;
    W1 + W3 = SubJoin(V).(A,C) by Def7
      .= A "\/" C by LATTICES:def 1
      .= W3 by A1,LATTICES:def 3;
    then
A2: W1 is Subspace of W3 by Th8;
    thus A "\/" (B "/\" C) = SubJoin(V).(A,B "/\" C) by LATTICES:def 1
      .= SubJoin(V).(A,SubMeet(V).(B,C)) by LATTICES:def 2
      .= SubJoin(V).(A,BC) by Def8
      .= W1 + (W2 /\ W3) by Def7
      .= (W1 + W2) /\ W3 by A2,Th29
      .= SubMeet(V).(AB,C) by Def8
      .= SubMeet(V).(SubJoin(V).(A,B),C) by Def7
      .= SubMeet(V).(A "\/" B,C) by LATTICES:def 1
      .= (A "\/" B) "/\" C by LATTICES:def 2;
  end;
  hence thesis by LATTICES:def 12;
end;
