
theorem Th54:
  for V being RealUnitarySpace holds LattStr (# Subspaces(V),
    SubJoin(V), SubMeet(V) #) is Lattice
proof
  let V be RealUnitarySpace;
  set S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #);
A1: for A,B being Element of S holds A "/\" B = B "/\" A
  proof
    let A,B be Element of S;
    reconsider W1 = A, W2 = B as Subspace of V by Def3;
    thus A "/\" B = SubMeet(V).(A,B) by LATTICES:def 2
      .= W1 /\ W2 by Def8
      .= W2 /\ W1 by Th14
      .= SubMeet(V).(B,A) by Def8
      .= B "/\" A by LATTICES:def 2;
  end;
A2: for A,B being Element of S holds (A "/\" B) "\/" B = B
  proof
    let A,B be Element of S;
    reconsider W1 = A, W2 = B as strict Subspace of V by Def3;
    reconsider AB = W1 /\ W2 as Element of S by Def3;
    thus (A "/\" B) "\/" B = SubJoin(V).(A "/\" B,B) by LATTICES:def 1
      .= SubJoin(V).(SubMeet(V).(A,B),B) by LATTICES:def 2
      .= SubJoin(V).(AB,B) by Def8
      .= (W1 /\ W2) + W2 by Def7
      .= B by Lm6,RUSUB_1:24;
  end;
A3: for A,B,C being Element of S holds A "\/" (B "\/" C) = (A "\/" B) "\/" C
  proof
    let A,B,C be Element of S;
    reconsider W1 = A, W2 = B, W3 = C as Subspace of V by Def3;
    reconsider AB = W1 + W2, BC = W2 + W3 as Element of S by Def3;
    thus A "\/" (B "\/" C) = SubJoin(V).(A,B "\/" C) by LATTICES:def 1
      .= SubJoin(V).(A,SubJoin(V).(B,C)) by LATTICES:def 1
      .= SubJoin(V).(A,BC) by Def7
      .= W1 + (W2 + W3) by Def7
      .= (W1 + W2) + W3 by Th6
      .= SubJoin(V).(AB,C) by Def7
      .= SubJoin(V).(SubJoin(V).(A,B),C) by Def7
      .= SubJoin(V).(A "\/" B,C) by LATTICES:def 1
      .= (A "\/" B) "\/" C by LATTICES:def 1;
  end;
A4: for A,B being Element of S holds A "/\" (A "\/" B) = A
  proof
    let A,B be Element of S;
    reconsider W1 = A, W2 = B as strict Subspace of V by Def3;
    reconsider AB = W1 + W2 as Element of S by Def3;
    thus A "/\" (A "\/" B) = SubMeet(V).(A,A "\/" B) by LATTICES:def 2
      .= SubMeet(V).(A,SubJoin(V).(A,B)) by LATTICES:def 1
      .= SubMeet(V).(A,AB) by Def7
      .= W1 /\ (W1 + W2) by Def8
      .= A by Lm7,RUSUB_1:24;
  end;
A5: for A,B,C being Element of S holds A "/\" (B "/\" C) = (A "/\" B) "/\" C
  proof
    let A,B,C be Element of S;
    reconsider W1 = A, W2 = B, W3 = C as Subspace of V by Def3;
    reconsider AB = W1 /\ W2, BC = W2 /\ W3 as Element of S by Def3;
    thus A "/\" (B "/\" C) = SubMeet(V).(A,B "/\" C) by LATTICES:def 2
      .= SubMeet(V).(A,SubMeet(V).(B,C)) by LATTICES:def 2
      .= SubMeet(V).(A,BC) by Def8
      .= W1 /\ (W2 /\ W3) by Def8
      .= (W1 /\ W2) /\ W3 by Th15
      .= SubMeet(V).(AB,C) by Def8
      .= SubMeet(V).(SubMeet(V).(A,B),C) by Def8
      .= SubMeet(V).(A "/\" B,C) by LATTICES:def 2
      .= (A "/\" B) "/\" C by LATTICES:def 2;
  end;
  for A,B being Element of S holds A "\/" B = B "\/" A
  proof
    let A,B be Element of S;
    reconsider W1 = A, W2 = B as Subspace of V by Def3;
    thus A "\/" B = SubJoin(V).(A,B) by LATTICES:def 1
      .= W1 + W2 by Def7
      .= W2 + W1 by Lm1
      .= SubJoin(V).(B,A) by Def7
      .= B "\/" A by LATTICES:def 1;
  end;
  then S is join-commutative join-associative meet-absorbing meet-commutative
  meet-associative join-absorbing by A3,A2,A1,A5,A4,LATTICES:def 4,def 5,def 6
,def 7,def 8,def 9;
  hence thesis;
end;
