reserve x,y,y1,y2 for object;
reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr,
  V,X,Y for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF;
reserve a for Element of GF;
reserve u,u1,u2,v,v1,v2 for Element of V;
reserve W,W1,W2 for Subspace of V;
reserve V1 for Subset of V;
reserve w,w1,w2 for Element of W;

theorem Th34:
  V1 <> {} & V1 is linearly-closed implies ex W being strict
  Subspace of V st V1 = the carrier of W
proof
  assume that
A1: V1 <> {} and
A2: V1 is linearly-closed;
  reconsider D = V1 as non empty set by A1;
  reconsider d = 0.V as Element of D by A2,Th1;
  set VV = the carrier of V;
  set C = (comp V) | D;
  dom(comp V) = VV by FUNCT_2:def 1;
  then
A3: dom C = D by RELAT_1:62;
A4: rng C c= D
  proof
    let x be object;
    assume x in rng C;
    then consider y being object such that
A5: y in dom C and
A6: C.y = x by FUNCT_1:def 3;
    reconsider y as Element of V by A3,A5;
    x = (comp V).y by A5,A6,FUNCT_1:47
      .= - y by VECTSP_1:def 13;
    hence thesis by A2,A3,A5,Th2;
  end;
  set M = (the lmult of V) |([:the carrier of GF,D:] qua set);
  dom(the lmult of V) = [:the carrier of GF,VV:] by FUNCT_2:def 1;
  then
A7: dom M = [:the carrier of GF,D:] by RELAT_1:62,ZFMISC_1:96;
A8: rng M c= D
  proof
    let x be object;
    assume x in rng M;
    then consider y being object such that
A9: y in dom M and
A10: M.y = x by FUNCT_1:def 3;
    consider y1,y2 being object such that
A11: [y1,y2] = y by A7,A9,RELAT_1:def 1;
    reconsider y1 as Element of GF by A7,A9,A11,ZFMISC_1:87;
A12: y2 in V1 by A7,A9,A11,ZFMISC_1:87;
    then reconsider y2 as Element of V;
    x = y1 * y2 by A9,A10,A11,FUNCT_1:47;
    hence thesis by A2,A12;
  end;
  set A = (the addF of V)||D;
  dom(the addF of V) = [:VV,VV:] by FUNCT_2:def 1;
  then
A13: dom A = [:D,D:] by RELAT_1:62,ZFMISC_1:96;
A14: rng A c= D
  proof
    let x be object;
    assume x in rng A;
    then consider y being object such that
A15: y in dom A and
A16: A.y = x by FUNCT_1:def 3;
    consider y1,y2 being object such that
A17: [y1,y2] = y by A13,A15,RELAT_1:def 1;
A18: y1 in D & y2 in D by A13,A15,A17,ZFMISC_1:87;
    then reconsider y1,y2 as Element of V;
    x = y1 + y2 by A15,A16,A17,FUNCT_1:47;
    hence thesis by A2,A18;
  end;
  reconsider M as Function of [:the carrier of GF,D:], D by A7,A8,FUNCT_2:def 1
,RELSET_1:4;
  reconsider C as UnOp of D by A3,A4,FUNCT_2:def 1,RELSET_1:4;
  reconsider A as BinOp of D by A13,A14,FUNCT_2:def 1,RELSET_1:4;
  set W = ModuleStr (# D,A,d,M #);
A19: for a,b be Element of W for x,y be Element of V st x = a & b = y holds
  a + b = x + y
  proof
    let a,b be Element of W;
    let x,y be Element of V such that
A20: x = a & b = y;
    thus a + b = A.[a,b] .= x + y by A13,A20,FUNCT_1:47;
  end;
A21: W is Abelian add-associative right_zeroed right_complementable
  proof
    thus W is Abelian
    proof
      let a,b be Element of W;
      reconsider x = a, y = b as Element of V by TARSKI:def 3;
      thus a + b = y + x by A19
        .= b + a by A19;
    end;
    hereby
      let a,b,c be Element of W;
      reconsider x = a, y = b, z = c as Element of V by TARSKI:def 3;
A22:  b + c = y + z by A19;
      a + b = x + y by A19;
      hence a + b + c = x + y + z by A19
        .= x + (y + z) by RLVECT_1:def 3
        .= a + (b + c) by A19,A22;
    end;
    hereby
      let a be Element of W;
      reconsider x = a as Element of V by TARSKI:def 3;
      thus a + 0.W = x + 0.V by A19
        .= a by RLVECT_1:4;
    end;
    let a be Element of W;
    reconsider x = a as Element of V by TARSKI:def 3;
    reconsider a9 = a as Element of D;
    reconsider b = C.a9 as Element of D;
    reconsider b as Element of W;
    take b;
    thus a + b = x + (comp V).x by A3,A19,FUNCT_1:47
      .= x + - x by VECTSP_1:def 13
      .= 0.W by RLVECT_1:5;
  end;
A23: W is vector-distributive
  proof
    let a be Element of GF;
    let v,w be Element of W;
    reconsider x = v, y = w as Element of V by TARSKI:def 3;
A24: now
      let a be Element of GF;
      let x be Element of W;
      let y be Element of V;
      assume
A25:  y = x;
      [a,x] in dom M by A7;
      hence a * x = a * y by A25,FUNCT_1:47;
    end;
    then
A26: a * v = a * x;
A27: a * w = a * y by A24;
    v + w = x + y by A19;
    hence a * (v + w) = a * (x + y) by A24
      .= a * x + a * y by VECTSP_1:def 14
      .= a * v + a * w by A19,A26,A27;
  end;
A28: W is scalar-distributive
  proof
    let a,b be Element of GF;
    let v be Element of W;
    reconsider x = v as Element of V by TARSKI:def 3;
A29: now
      let a be Element of GF;
      let x be Element of W;
      let y be Element of V;
      assume
A30:  y = x;
      [a,x] in dom M by A7;
      hence a * x = a * y by A30,FUNCT_1:47;
    end;
    then
A31: a * v = a * x;
A32: b * v = b * x by A29;
    thus (a + b) * v = (a + b) * x by A29
      .= a * x + b * x by VECTSP_1:def 15
      .= a * v + b * v by A19,A32,A31;
  end;
A33: W is scalar-associative
  proof
    let a,b be Element of GF;
    let v be Element of W;
    reconsider x = v as Element of V by TARSKI:def 3;
A34: now
      let a be Element of GF;
      let x be Element of W;
      let y be Element of V;
      assume
A35:  y = x;
      [a,x] in dom M by A7;
      hence a * x = a * y by A35,FUNCT_1:47;
    end;
then A36: b * v = b * x;
    thus a * b * v = a * b * x by A34
      .= a * (b * x) by VECTSP_1:def 16
      .= a * (b * v) by A34,A36;
  end;
  W is scalar-unital
  proof
    let v be Element of W;
    reconsider x = v as Element of V by TARSKI:def 3;
 now
      let a be Element of GF;
      let x be Element of W;
      let y be Element of V;
      assume
A37:  y = x;
      [a,x] in dom M by A7;
      hence a * x = a * y by A37,FUNCT_1:47;
    end;
    hence 1.GF * v = 1_GF * x
      .= v;
  end;
  then reconsider
  W as Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF by A21,A23,A28,A33;
  0.W = 0.V;
  then reconsider W as strict Subspace of V by Def2;
  take W;
  thus thesis;
end;
