reserve F for Field;
reserve VS for strict VectSp of F;
reserve u,e for set;
reserve x for set;
reserve Z1 for set;

theorem
  for A being strict VectSp of F holds FuncLatt id the carrier of A = id
  the carrier of lattice A
proof
  let A be strict VectSp of F;
  set f = id the carrier of A;
A1: for x being object st x in the carrier of lattice A
holds (FuncLatt f). x = x
  proof
    let x be object;
    assume x in the carrier of lattice A;
    then consider X1 being strict Subspace of A such that
A2: X1 = x by VECTSP_5:def 3;
A3: the carrier of X1 c= f.:the carrier of X1
    proof
      let z be object;
      assume
A4:   z in the carrier of X1;
      the carrier of X1 c= the carrier of A by VECTSP_4:def 2;
      then reconsider z as Element of A by A4;
      f.z = z;
      hence thesis by A4,FUNCT_2:35;
    end;
A5: (FuncLatt f).X1 = Lin(f.:the carrier of X1) by Def7;
    f.:the carrier of X1 c= the carrier of X1
    proof
      let z be object;
      assume
A6:   z in f.:the carrier of X1;
      then reconsider z as Element of A;
      ex Z being Element of A st Z in the carrier of X1 & z = f.Z by A6,
FUNCT_2:65;
      hence thesis;
    end;
    then f.:the carrier of X1 = the carrier of X1 by A3,XBOOLE_0:def 10;
    hence thesis by A2,A5,VECTSP_7:11;
  end;
  dom FuncLatt f = the carrier of lattice A by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:17;
end;
