reserve F for Field;
reserve VS for strict VectSp of F;
reserve u,e for set;
reserve x for set;

theorem Th7:
  for VS being strict VectSp of F, p,q being Element of lattice VS,
  H1,H2 being Subspace of VS st p=H1 & q=H2 holds p"\/"q = H1+H2
proof
  let VS be strict VectSp of F;
  let p,q be Element of lattice VS;
  let H1,H2 be Subspace of VS;
  consider H1 being strict Subspace of VS such that
A1: H1=p by VECTSP_5:def 3;
  consider H2 being strict Subspace of VS such that
A2: H2=q by VECTSP_5:def 3;
  p"\/"q = SubJoin(VS).(p,q) by LATTICES:def 1
    .= H1 + H2 by A1,A2,VECTSP_5:def 7;
  hence thesis by A1,A2;
end;
