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

theorem Th6:
  for VS being strict VectSp of F, p,q being Element of lattice VS,
  H1,H2 being strict Subspace of VS st p=H1 & q=H2 holds p [= q iff the carrier
  of H1 c= the carrier of H2
proof
  let VS be strict VectSp of F;
  let p,q be Element of lattice VS;
  let H1,H2 be strict Subspace of VS;
  consider A1 being strict Subspace of VS such that
A1: A1=p by VECTSP_5:def 3;
  consider A2 being strict Subspace of VS such that
A2: A2=q by VECTSP_5:def 3;
A3: the carrier of A1 c= the carrier of A2 implies p [= q
  proof
    assume the carrier of A1 c= the carrier of A2;
    then (the carrier of A1) /\ the carrier of A2 = the carrier of A1 by
XBOOLE_1:28;
    then
A4: A1 /\ A2 = A1 by VECTSP_5:def 2;
    A1 /\ A2 = (the L_meet of lattice VS).(p,q) by A1,A2,VECTSP_5:def 8
      .= p"/\"q by LATTICES:def 2;
    hence thesis by A1,A4,LATTICES:4;
  end;
  p [= q implies the carrier of A1 c= the carrier of A2
  proof
    assume p [= q;
    then
A5: p"/\"q = p by LATTICES:4;
    p"/\"q= SubMeet(VS).(p,q) by LATTICES:def 2
      .= A1 /\ A2 by A1,A2,VECTSP_5:def 8;
    then
    (the carrier of A1) /\ the carrier of A2 = the carrier of A1 by A1,A5,
VECTSP_5:def 2;
    hence thesis by XBOOLE_1:17;
  end;
  hence thesis by A1,A2,A3;
end;
