theorem Th29:
  0.R <> 1_R implies (A <> {} & A is linearly-closed iff for l
  holds Sum l in A)
proof
  assume
A1: 0.R <> 1_R;
  thus A <> {} & A is linearly-closed implies for l holds Sum(l) in A
  proof
    defpred P[Nat] means for l st card(Carrier(l)) = $1 holds Sum (l) in A;
    assume that
A2: A <> {} and
A3: A is linearly-closed;
    now
      let l;
      assume card(Carrier(l)) = 0;
      then Carrier(l) = {};
      then l = ZeroLC(V) by Def4;
      then Sum(l) = 0.V by Lm3;
      hence Sum(l) in A by A2,A3,RMOD_2:1;
    end;
    then
A4: P[0];
    now
      let k be Nat;
      assume
A5:   for l st card Carrier l = k holds Sum(l) in A;
      let l;
      deffunc Q(Element of V) = l.$1;
      consider F such that
A6:   F is one-to-one and
A7:   rng F = Carrier(l) and
A8:   Sum(l) = Sum(l (#) F) by Def7;
      reconsider G = F | Seg k as FinSequence of V by FINSEQ_1:18;
      assume
A9:   card Carrier l = k + 1;
      then
A10:  len F = k + 1 by A6,A7,FINSEQ_4:62;
      then
A11:  len(l (#) F) = k + 1 by Def6;
A12:  k + 1 in Seg(k + 1) by FINSEQ_1:4;
      then
A13:  k + 1 in dom F by A10,FINSEQ_1:def 3;
      k + 1 in dom F by A10,A12,FINSEQ_1:def 3;
      then reconsider v = F.(k + 1) as Vector of V by FINSEQ_2:11;
      consider f being Function of V, R such that
A14:  f.v = 0.R and
A15:  for u being Element of V st u <> v holds f.u = Q(u) from
      FUNCT_2:sch 6;
      reconsider f as Element of Funcs(the carrier of V, the carrier of R) by
FUNCT_2:8;
A16:  v in Carrier(l) by A7,A13,FUNCT_1:def 3;
      now
        let u;
        assume
A17:    not u in Carrier(l);
        hence f.u = l.u by A16,A15
          .= 0.R by A17;
      end;
      then reconsider f as Linear_Combination of V by Def2;
A18:  A \ {v} c= A by XBOOLE_1:36;
A19:  Carrier(l) c= A by Def5;
      then
A20:  v * l.v in A by A3,A16;
A21:  Carrier(f) = Carrier(l) \ {v}
      proof
        thus Carrier(f) c= Carrier(l) \ {v}
        proof
          let x be object;
          assume x in Carrier(f);
          then consider u such that
A22:      u = x and
A23:      f.u <> 0.R;
          f.u = l.u by A14,A15,A23;
          then
A24:      x in Carrier(l) by A22,A23;
          not x in {v} by A14,A22,A23,TARSKI:def 1;
          hence thesis by A24,XBOOLE_0:def 5;
        end;
        let x be object;
        assume
A25:    x in Carrier(l) \ {v};
        then x in Carrier(l) by XBOOLE_0:def 5;
        then consider u such that
A26:    x = u and
A27:    l.u <> 0.R;
        not x in {v} by A25,XBOOLE_0:def 5;
        then x <> v by TARSKI:def 1;
        then l.u = f.u by A15,A26;
        hence thesis by A26,A27;
      end;
      then Carrier(f) c= A \ {v} by A19,XBOOLE_1:33;
      then Carrier(f) c= A by A18;
      then reconsider f as Linear_Combination of A by Def5;
A28:  len G = k by A10,FINSEQ_3:53;
      then
A29:  len (f (#) G) = k by Def6;
A30:  rng G = Carrier(f)
      proof
        thus rng G c= Carrier f
        proof
          let x be object;
          assume x in rng G;
          then consider y being object such that
A31:      y in dom G and
A32:      G.y = x by FUNCT_1:def 3;
          reconsider y as Nat by A31,FINSEQ_3:23;
A33:      dom G c= dom F & G.y = F.y by A31,FUNCT_1:47,RELAT_1:60;
          now
            assume x = v;
            then
A34:        k + 1 = y by A6,A13,A31,A32,A33;
            y <= k by A28,A31,FINSEQ_3:25;
            hence contradiction by A34,XREAL_1:29;
          end;
          then
A35:      not x in {v} by TARSKI:def 1;
          x in rng F by A31,A32,A33,FUNCT_1:def 3;
          hence thesis by A7,A21,A35,XBOOLE_0:def 5;
        end;
        let x be object;
        assume
A36:    x in Carrier(f);
        then x in rng F by A7,A21,XBOOLE_0:def 5;
        then consider y being object such that
A37:    y in dom F and
A38:    F.y = x by FUNCT_1:def 3;
        reconsider y as Element of NAT by A37,FINSEQ_3:23;
        now
          assume not y in Seg k;
          then y in dom F \ Seg k by A37,XBOOLE_0:def 5;
          then y in Seg(k + 1) \ Seg k by A10,FINSEQ_1:def 3;
          then y in {k + 1} by FINSEQ_3:15;
          then y = k + 1 by TARSKI:def 1;
          then not v in {v} by A21,A36,A38,XBOOLE_0:def 5;
          hence contradiction by TARSKI:def 1;
        end;
        then y in dom F /\ Seg k by A37,XBOOLE_0:def 4;
        then
A39:    y in dom G by RELAT_1:61;
        then G.y = F.y by FUNCT_1:47;
        hence thesis by A38,A39,FUNCT_1:def 3;
      end;
      Seg(k + 1) /\ Seg k = Seg k by FINSEQ_1:7,NAT_1:12
        .= dom(f (#) G) by A29,FINSEQ_1:def 3;
      then
A40:  dom(f (#) G) = dom(l (#) F) /\ Seg k by A11,FINSEQ_1:def 3;
      now
        let x be object;
A41:    rng F c= the carrier of V by FINSEQ_1:def 4;
        assume
A42:    x in dom(f (#) G);
        then reconsider n = x as Nat by FINSEQ_3:23;
        n in dom(l (#) F) by A40,A42,XBOOLE_0:def 4;
        then
A43:    n in dom F by A10,A11,FINSEQ_3:29;
        then F.n in rng F by FUNCT_1:def 3;
        then reconsider w = F.n as Vector of V by A41;
A44:    n in dom G by A28,A29,A42,FINSEQ_3:29;
        then
A45:    G.n in rng G by FUNCT_1:def 3;
        rng G c= the carrier of V by FINSEQ_1:def 4;
        then reconsider u = G.n as Vector of V by A45;
        not u in {v} by A21,A30,A45,XBOOLE_0:def 5;
        then
A46:    u <> v by TARSKI:def 1;
A47:    (f (#) G).n = u * f.u by A44,Th23
          .= u * l.u by A15,A46;
        w = u by A44,FUNCT_1:47;
        hence (f (#) G).x = (l (#) F).x by A47,A43,Th23;
      end;
      then f (#) G = (l (#) F) | Seg k by A40,FUNCT_1:46;
      then
A48:  f (#) G = (l (#) F) | dom (f (#) G) by A29,FINSEQ_1:def 3;
      v in rng F by A13,FUNCT_1:def 3;
      then {v} c= Carrier(l) by A7,ZFMISC_1:31;
      then card(Carrier(f)) = k + 1 - card{v} by A9,A21,CARD_2:44
        .= k + 1 - 1 by CARD_1:30
        .= k by XCMPLX_1:26;
      then
A49:  Sum(f) in A by A5;
      G is one-to-one by A6,FUNCT_1:52;
      then
A50:  Sum(f (#) G) = Sum(f) by A30,Def7;
      (l (#) F).(len F) = v * l.v by A10,A13,Th23;
      then Sum(l (#) F) = Sum (f (#) G) + v * l.v by A10,A11,A29,A48,
RLVECT_1:38;
      hence Sum(l) in A by A3,A8,A20,A50,A49;
    end;
    then
A51: for k be Nat st P[k] holds P[k+1];
    let l;
A52: card Carrier l = card Carrier l;
    for k be Nat holds P[k] from NAT_1:sch 2(A4,A51);
    hence thesis by A52;
  end;
  assume
A53: for l holds Sum(l) in A;
  hence A <> {};
  ZeroLC(V) is Linear_Combination of A & Sum(ZeroLC(V)) = 0.V by Lm3,Th20;
  then
A54: 0.V in A by A53;
A55: for a,v st v in A holds v * a in A
  proof
    let a,v;
    assume
A56: v in A;
    now
      per cases;
      suppose
        a = 0.R;
        hence thesis by A54,VECTSP_2:32;
      end;
      suppose
A57:    a <> 0.R;
        deffunc F(Element of V)=0.R;
        consider f such that
A58:    f.v = a and
A59:    for u being Element 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;
          assume not u in {v};
          then u <> v by TARSKI:def 1;
          hence f.u = 0.R by A59;
        end;
        then reconsider f as Linear_Combination of V by Def2;
A60:    Carrier(f) = {v}
        proof
          thus Carrier(f) c= {v}
          proof
            let x be object;
            assume x in Carrier(f);
            then consider u such that
A61:        x = u and
A62:        f.u <> 0.R;
            u = v by A59,A62;
            hence thesis by A61,TARSKI:def 1;
          end;
          let x be object;
          assume x in {v};
          then x = v by TARSKI:def 1;
          hence thesis by A57,A58;
        end;
        {v} c= A by A56,ZFMISC_1:31;
        then reconsider f as Linear_Combination of A by A60,Def5;
        consider F such that
A63:    F is one-to-one & rng F = Carrier(f) and
A64:    Sum(f (#) F) = Sum(f) by Def7;
        F = <* v *> by A60,A63,FINSEQ_3:97;
        then f (#) F = <* v * f.v *> by Th25;
        then Sum(f) = v * a by A58,A64,RLVECT_1:44;
        hence thesis by A53;
      end;
    end;
    hence thesis;
  end;
  thus for v,u st v in A & u in A holds v + u in A
  proof
    let v,u;
    assume that
A65: v in A and
A66: u in A;
    now
      per cases;
      suppose
        u = v;
        then v + u = v * 1_R + v by VECTSP_2:def 9
          .= v * 1_R + v * 1_R by VECTSP_2:def 9
          .= v * (1_R + 1_R) by VECTSP_2:def 9;
        hence thesis by A55,A65;
      end;
      suppose
A67:    v <> u;
        deffunc F(Element of V)=0.R;
        consider f such that
A68:    f.v = 1_R & f.u = 1_R and
A69:    for w being Element of V st w <> v & w <> u holds f.w = F(w)
        from FUNCT_2:sch 7(A67);
        reconsider f as Element of Funcs(the carrier of V, the carrier of R)
        by FUNCT_2:8;
        now
          let w;
          assume not w in {v,u};
          then w <> v & w <> u by TARSKI:def 2;
          hence f.w = 0.R by A69;
        end;
        then reconsider f as Linear_Combination of V by Def2;
A70:    Carrier(f) = {v,u}
        proof
          thus Carrier(f) c= {v,u}
          proof
            let x be object;
            assume x in Carrier(f);
            then ex w st x = w & f.w <> 0.R;
            then x = v or x = u by A69;
            hence thesis by TARSKI:def 2;
          end;
          let x be object;
          assume x in {v,u};
          then x = v or x = u by TARSKI:def 2;
          hence thesis by A1,A68;
        end;
        then
A71:    Carrier(f) c= A by A65,A66,ZFMISC_1:32;
A72:    u * 1_R = u & v * 1_R = v by VECTSP_2:def 9;
        reconsider f as Linear_Combination of A by A71,Def5;
        consider F such that
A73:    F is one-to-one & rng F = Carrier(f) and
A74:    Sum(f (#) F) = Sum(f) by Def7;
        F = <* v,u *> or F = <* u,v *> by A67,A70,A73,FINSEQ_3:99;
        then f (#) F = <* v * 1_R, u * 1_R *> or f (#) F = <* u * 1_R, v *
        1_R *> by A68,Th26;
        then Sum(f) = v + u by A74,A72,RLVECT_1:45;
        hence thesis by A53;
      end;
    end;
    hence thesis;
  end;
  thus thesis by A55;
end;
