reserve x,y for object,
  R for Ring,
  V for LeftMod of R,
  L for Linear_Combination of V,
  a for Scalar of R,
  v,u for Vector of V,
  F,G for FinSequence of the carrier of V,
  C for finite Subset of V;
reserve X,Y,Z for set,
  A,B for Subset of V,
  T for finite Subset of V,
  l for Linear_Combination of A,
  f,g for Function of the carrier of V,the carrier of R;
reserve R for Skew-Field;
reserve a,b for Scalar of R;
reserve V for LeftMod of R;
reserve v,v1,v2,u for Vector of V;
reserve f for Function of the carrier of V, the carrier of R;

theorem Th19:
  for R being almost_left_invertible non degenerated Ring
  for V being LeftMod of R for A being Subset of V st
  Lin(A) = V holds ex B being Subset of V st B c= A & B is base
proof
  let R be almost_left_invertible non degenerated Ring;
  let V be LeftMod of R;
  let A be Subset of V;
  defpred P[set] means (ex B being Subset of V st B = $1 & B c= A & B is
  linearly-independent);
  assume
A2: Lin(A) = V;
  consider Q being set such that
A3: for Z holds Z in Q iff Z in bool(the carrier of V) & P[Z] from
  XFAMILY:sch 1;
A4: now
    let Z;
    assume that
    Z <> {} and
A5: Z c= Q and
A6: Z is c=-linear;
    set W = union Z;
    W c= the carrier of V
    proof
      let x be object;
      assume x in W;
      then consider X such that
A7:   x in X and
A8:   X in Z by TARSKI:def 4;
      X in bool(the carrier of V) by A3,A5,A8;
      hence thesis by A7;
    end;
    then reconsider W as Subset of V;
A9: W is linearly-independent
    proof
      deffunc F(object)={C where C is Subset of V: $1 in C & C in Z};
      let l be Linear_Combination of W;
      assume that
A10:  Sum(l) = 0.V and
A11:  Carrier(l) <> {};
      consider f being Function such that
A12:  dom f = Carrier(l) and
A13:  for x being object st x in Carrier(l) holds f.x =F(x)
from FUNCT_1:sch 3;
      reconsider M = rng f as non empty set by A11,A12,RELAT_1:42;
      set F = the Choice_Function of M;
      set S = rng F;
A14:  now
        assume {} in M;
        then consider x being object such that
A15:    x in dom f and
A16:    f.x = {} by FUNCT_1:def 3;
        Carrier(l) c= W by VECTSP_6:def 4;
        then consider X such that
A17:    x in X and
A18:    X in Z by A12,A15,TARSKI:def 4;
        reconsider X as Subset of V by A3,A5,A18;
        X in {C where C is Subset of V : x in C & C in Z} by A17,A18;
        hence contradiction by A12,A13,A15,A16;
      end;
      then
A19:  dom F = M by RLVECT_3:28;
      then dom F is finite by A12,FINSET_1:8;
      then
A20:  S is finite by FINSET_1:8;
A21:  now
        let X;
        assume X in S;
        then consider x being object such that
A22:    x in dom F and
A23:    F.x = X by FUNCT_1:def 3;
        consider y being object such that
A24:    y in dom f & f.y = x by A19,A22,FUNCT_1:def 3;
A25:    x = {C where C is Subset of V : y in C & C in Z} by A12,A13,A24;
        reconsider x as set by TARSKI:1;
        X in x by A14,A19,A22,A23,ORDERS_1:89;
        then ex C being Subset of V st C = X & y in C & C in Z by A25;
        hence X in Z;
      end;
A26:  now
        let X,Y;
        assume X in S & Y in S;
        then X in Z & Y in Z by A21;
        then X,Y are_c=-comparable by A6,ORDINAL1:def 8;
        hence X c= Y or Y c= X;
      end;
      S <> {} by A19,RELAT_1:42;
      then union S in S by A26,A20,CARD_2:62;
      then union S in Z by A21;
      then consider B being Subset of V such that
A27:  B = union S and
      B c= A and
A28:  B is linearly-independent by A3,A5;
      Carrier(l) c= union S
      proof
        let x be object;
        set X = f.x;
        assume
A29:    x in Carrier(l);
        then
A30:    f.x = {C where C is Subset of V : x in C & C in Z} by A13;
A31:    f.x in M by A12,A29,FUNCT_1:def 3;
        then F.X in X by A14,ORDERS_1:89;
        then
A32:    ex C being Subset of V st F.X = C & x in C & C in Z by A30;
        F.X in S by A19,A31,FUNCT_1:def 3;
        hence thesis by A32,TARSKI:def 4;
      end;
      then l is Linear_Combination of B by A27,VECTSP_6:def 4;
      hence thesis by A10,A11,A28;
    end;
    W c= A
    proof
      let x be object;
      assume x in W;
      then consider X such that
A33:  x in X and
A34:  X in Z by TARSKI:def 4;
      ex B being Subset of V st B = X & B c= A & B is
      linearly-independent by A3,A5,A34;
      hence thesis by A33;
    end;
    hence union Z in Q by A3,A9;
  end;
  {}(the carrier of V) c= A & {}(the carrier of V) is linearly-independent;
  then Q <> {} by A3;
  then consider X such that
A35: X in Q and
A36: for Z st Z in Q & Z <> X holds not X c= Z by A4,ORDERS_1:67;
  consider B being Subset of V such that
A37: B = X and
A38: B c= A and
A39: B is linearly-independent by A3,A35;
  take B;
  thus B c= A & B is linearly-independent by A38,A39;
  assume
A40: Lin(B) <> the ModuleStr of V;
  now
    assume
A41: for v being Vector of V st v in A holds v in Lin(B);
    now
      reconsider F = the carrier of Lin(B) as Subset of V by VECTSP_4:def 2;
      let v be Vector of V;
      assume v in Lin(A);
      then consider l being Linear_Combination of A such that
A42:  v = Sum(l) by Th4;
      Carrier(l) c= the carrier of Lin(B)
      proof
        let x be object;
        assume
A43:    x in Carrier(l);
        then reconsider a = x as Vector of V;
        Carrier(l) c= A by VECTSP_6:def 4;
        then a in Lin(B) by A41,A43;
        hence thesis by STRUCT_0:def 5;
      end;
      then reconsider l as Linear_Combination of F by VECTSP_6:def 4;
      Sum(l) = v by A42;
      then v in Lin(F) by Th4;
      hence v in Lin(B) by Th8;
    end;
    then Lin(A) is Subspace of Lin(B) by VECTSP_4:28;
    hence contradiction by A2,A40,VECTSP_4:25;
  end;
  then consider v being Vector of V such that
A44: v in A and
A45: not v in Lin(B);
A46: B \/ {v} is linearly-independent
  proof
    let l be Linear_Combination of B \/ {v};
    assume
A47: Sum(l) = 0.V;
    now
      per cases;
      suppose
        v in Carrier(l);
        then l.v <> 0.R by VECTSP_6:2;
        then - l.v <> 0.R by Lm1;
        then
A48:    (- l.v)" * ((- l.v) * v) = 1.R * v by Lm2
          .= v;
        deffunc F(Vector of V) = l.$1;
        consider f being Function of the carrier of V, the carrier of R such
        that
A49:    f.v = 0.R and
A50:    for u being Vector of V st u <> v holds f.u = F(u) from
        FUNCT_2:sch 6;
        reconsider f as Element of Funcs(the carrier of V, the carrier of R)
        by FUNCT_2:8;
        now
          let u be Vector of V;
          assume not u in Carrier(l) \ {v};
          then not u in Carrier(l) or u in {v} by XBOOLE_0:def 5;
          then l.u = 0.R & u <> v or u = v by TARSKI:def 1;
          hence f.u = 0.R by A49,A50;
        end;
        then reconsider f as Linear_Combination of V by VECTSP_6:def 1;
        Carrier(f) c= B
        proof
          let x be object;
A51:      Carrier(l) c= B \/ {v} by VECTSP_6:def 4;
          assume x in Carrier(f);
          then consider u being Vector of V such that
A52:      u = x and
A53:      f.u <> 0.R;
          f.u = l.u by A49,A50,A53;
          then u in Carrier(l) by A53;
          then u in B or u in {v} by A51,XBOOLE_0:def 3;
          hence thesis by A49,A52,A53,TARSKI:def 1;
        end;
        then reconsider f as Linear_Combination of B by VECTSP_6:def 4;
        deffunc F(Vector of V) = 0.R;
        consider g being Function of the carrier of V, the carrier of R such
        that
A54:    g.v = - l.v and
A55:    for u being Vector of V st u <> v holds g.u = F(u) from
        FUNCT_2:sch 6;
        reconsider g as Element of Funcs(the carrier of V, the carrier of R)
        by FUNCT_2:8;
        now
          let u be Vector of V;
          assume not u in {v};
          then u <> v by TARSKI:def 1;
          hence g.u = 0.R by A55;
        end;
        then reconsider g as Linear_Combination of V by VECTSP_6:def 1;
        Carrier(g) c= {v}
        proof
          let x be object;
          assume x in Carrier(g);
          then ex u being Vector of V st x = u & g.u <> 0.R;
          then x = v by A55;
          hence thesis by TARSKI:def 1;
        end;
        then reconsider g as Linear_Combination of {v} by VECTSP_6:def 4;
        f - g = l
        proof
          let u be Vector of V;
          now
            per cases;
            suppose
A56:          v = u;
              thus (f - g).u = f.u - g.u by VECTSP_6:40
                .= 0.R + (- (- l.v)) by A49,A54,A56,RLVECT_1:def 11
                .= l.v + 0.R by RLVECT_1:17
                .= l.u by A56,RLVECT_1:4;
            end;
            suppose
A57:          v <> u;
              thus (f - g).u = f.u - g.u by VECTSP_6:40
                .= l.u - g.u by A50,A57
                .= l.u - 0.R by A55,A57
                .= l.u by RLVECT_1:13;
            end;
          end;
          hence thesis;
        end;
        then
A58:    0.V = Sum(f) - Sum(g) by A47,VECTSP_6:47;
        Sum(g) = (- l.v) * v by A54,VECTSP_6:17;
        then Sum(f) = (- l.v) * v by A58,VECTSP_1:19;
        then (- l.v) * v in Lin(B) by Th4;
        hence thesis by A45,A48,VECTSP_4:21;
      end;
      suppose
A59:    not v in Carrier(l);
        Carrier(l) c= B
        proof
          let x be object;
          assume
A60:      x in Carrier(l);
          Carrier(l) c= B \/ {v} by VECTSP_6:def 4;
          then x in B or x in {v} by A60,XBOOLE_0:def 3;
          hence thesis by A59,A60,TARSKI:def 1;
        end;
        then l is Linear_Combination of B by VECTSP_6:def 4;
        hence thesis by A39,A47;
      end;
    end;
    hence thesis;
  end;
  {v} c= A by A44,ZFMISC_1:31;
  then B \/ {v} c= A by A38,XBOOLE_1:8;
  then
A61: B \/ {v} in Q by A3,A46;
  v in {v} by TARSKI:def 1;
  then
A62: v in B \/ {v} by XBOOLE_0:def 3;
  not v in B by A45,Th5;
  hence contradiction by A36,A37,A62,A61,XBOOLE_1:7;
end;
