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
  LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #) is M_Lattice
proof
  set S = LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #);
  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;
    assume
A1: A [= C;
    consider W1 being strict Subspace of M such that
A2: W1 = A by Def3;
    consider W3 being strict Subspace of M such that
A3: W3 = C by Def3;
    W1 + W3 = SubJoin(M).(A,C) by A2,A3,Def7
      .= A "\/" C by LATTICES:def 1
      .= W3 by A1,A3,LATTICES:def 3;
    then
A4: W1 is Subspace of W3 by Th8;
    consider W2 being strict Subspace of M such that
A5: W2 = B by Def3;
    reconsider AB = W1 + W2 as Element of S by Def3;
    reconsider BC = W2 /\ W3 as Element of S by Def3;
    thus A "\/" (B "/\" C) = SubJoin(M).(A,B "/\" C) by LATTICES:def 1
      .= SubJoin(M).(A,SubMeet(M).(B,C)) by LATTICES:def 2
      .= SubJoin(M).(A,BC) by A5,A3,Def8
      .= W1 + (W2 /\ W3) by A2,Def7
      .= (W1 + W2) /\ W3 by A4,Th30
      .= SubMeet(M).(AB,C) by A3,Def8
      .= SubMeet(M).(SubJoin(M).(A,B),C) by A2,A5,Def7
      .= SubMeet(M).(A "\/" B,C) by LATTICES:def 1
      .= (A "\/" B) "/\" C by LATTICES:def 2;
  end;
  hence thesis by Th57,LATTICES:def 12;
end;
