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 Th59:
  LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #) is 1_Lattice
proof
  set S = LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #);
  ex C being Element of S st for A being Element of S holds C "\/" A = C &
  A "\/" C = C
  proof
    consider W9 being strict Subspace of M such that
A1: the carrier of W9 = the carrier of (Omega).M;
    reconsider C = W9 as Element of S by Def3;
    take C;
    let A be Element of S;
    consider W being strict Subspace of M such that
A2: W = A by Def3;
A3: C is Subspace of (Omega).M by Lm6;
    thus C "\/" A = SubJoin(M).(C,A) by LATTICES:def 1
      .= W9 + W by A2,Def7
      .= (Omega).M + W by A1,Lm5
      .= the ModuleStr of M by Th11
      .= C by A1,A3,VECTSP_4:31;
    thus A "\/" C = SubJoin(M).(A,C) by LATTICES:def 1
      .= W + W9 by A2,Def7
      .= W + (Omega).M by A1,Lm5
      .= the ModuleStr of M by Th11
      .= C by A1,A3,VECTSP_4:31;
  end;
  hence thesis by Th57,LATTICES:def 14;
end;
