theorem Th24:
  for W1,W2 be Subspace of V1 st W1/\W2=(0).V1 for b1 be OrdBasis
  of W1,b2 be OrdBasis of W2,b be OrdBasis of W1+W2 st b=b1^b2 for v,v1,v2 be
  Vector of W1+W2, w1 be Vector of W1,w2 be Vector of W2 st v = v1+v2 & v1=w1 &
  v2=w2 holds v|-- b = (w1|--b1)^(w2|-- b2)
proof
  let W1,W2 be Subspace of V1 such that
A1: W1/\W2=(0).V1;
  [#](0).V1 = {0.V1} by VECTSP_4:def 3;
  then
A2: card ([#](0).V1) = 1 by CARD_1:30;
A3: dim W1 + dim W2 = dim(W1 + W2) + dim(W1 /\ W2) by VECTSP_9:32
    .= dim (W1+W2)+0 by A1,A2,RANKNULL:5;
  let b1 be OrdBasis of W1,b2 be OrdBasis of W2,b be OrdBasis of W1+W2 such
  that
A4: b=b1^b2;
  reconsider R=rng b as Basis of W1+W2 by MATRLIN:def 2;
  let v,v1,v2 be Vector of W1+W2,w1 be Vector of W1,w2 be Vector of W2 such
  that
A5: v=v1+v2 & v1=w1 & v2=w2;
  set wb2=w2|--b2;
  consider L2 be Linear_Combination of W2 such that
A6: w2 = Sum(L2) and
A7: Carrier L2 c= rng b2 and
A8: for k st 1<=k & k<=len wb2 holds wb2/.k=L2.(b2/.k) by MATRLIN:def 7;
A9: W2 is Subspace of W1+W2 by VECTSP_5:7;
  then consider K2 be Linear_Combination of W1+W2 such that
A10: Carrier(K2) = Carrier(L2) and
A11: Sum(K2) = Sum (L2) and
A12: K2|the carrier of W2=L2 by Lm4;
  rng b2 c= R by A4,FINSEQ_1:30;
  then
A13: Carrier K2 c= R by A7,A10;
  set wb1=w1|--b1;
  set vb=v|--b;
  consider L1 be Linear_Combination of W1 such that
A14: w1 = Sum(L1) and
A15: Carrier L1 c= rng b1 and
A16: for k st 1<=k & k<=len wb1 holds wb1/.k=L1.(b1/.k) by MATRLIN:def 7;
  consider L be Linear_Combination of W1+W2 such that
A17: v = Sum(L) & Carrier L c= rng b and
A18: for k st 1<=k & k<=len vb holds vb/.k=L.(b/.k) by MATRLIN:def 7;
A19: len vb=len b by MATRLIN:def 7;
  then
A20: dom vb=dom b by FINSEQ_3:29;
A21: len wb2=len b2 by MATRLIN:def 7;
  then
A22: dom wb2=dom b2 by FINSEQ_3:29;
A23: R is linearly-independent by VECTSP_7:def 3;
A24: W1 is Subspace of W1+W2 by VECTSP_5:7;
  then consider K1 be Linear_Combination of W1+W2 such that
A25: Carrier(K1) = Carrier(L1) and
A26: Sum(K1) = Sum (L1) and
A27: K1|the carrier of W1=L1 by Lm4;
A28: len wb1=len b1 by MATRLIN:def 7;
  then
A29: dom wb1=dom b1 by FINSEQ_3:29;
A30: len (wb1^wb2)=len wb1+len wb2 by FINSEQ_1:22;
A31: len b1=dim W1 & len b2=dim W2 by Th21;
A32: len b=dim (W1+W2) by Th21;
  then
A33: dom b=dom (wb1^wb2) by A28,A21,A31,A30,A3,FINSEQ_3:29;
  rng b1 c= R by A4,FINSEQ_1:29;
  then
A34: Carrier K1 c= R by A15,A25;
  then
A35: L=K1+K2 by A5,A14,A26,A6,A11,A17,A13,A23,MATRLIN:6;
  now
    let k such that
A36: 1<=k & k<=len vb;
A37: k in dom (wb1^wb2) by A28,A21,A19,A31,A32,A30,A3,A36,FINSEQ_3:25;
    now
      per cases by A37,FINSEQ_1:25;
      suppose
A38:    k in dom wb1;
        then 1<=k & k<=len wb1 by FINSEQ_3:25;
        then
A39:    L1.(b1/.k) = wb1/.k by A16
          .= wb1.k by A38,PARTFUN1:def 6
          .= (wb1^wb2).k by A38,FINSEQ_1:def 7;
        reconsider b1k=b1/.k as Vector of W1+W2 by A24,VECTSP_4:10;
A40:    K1.(b1/.k)=L1.(b1/.k) by A27,FUNCT_1:49;
        not b1/.k in Carrier K2
        proof
A41:      b1/.k in W1;
          assume
A42:      b1/.k in Carrier K2;
          then b1/.k in W2 by A10;
          then b1/.k in W1/\W2 by A41,VECTSP_5:3;
          then b1/.k in the carrier of (0).V1 by A1;
          then b1/.k in {0.V1} by VECTSP_4:def 3;
          then b1/.k = 0.V1 by TARSKI:def 1
            .= 0.(W1+W2) by VECTSP_4:11;
          hence thesis by A13,A23,A42,VECTSP_7:2;
        end;
        then K2.b1k=0.K;
        then
A43:    L.b1k = K1.b1k+0.K by A35,VECTSP_6:22
          .= (wb1^wb2).k by A39,A40,RLVECT_1:def 4;
        b1k = b1.k by A29,A38,PARTFUN1:def 6
          .= b.k by A4,A29,A38,FINSEQ_1:def 7
          .= b/.k by A33,A37,PARTFUN1:def 6;
        hence (wb1^wb2).k = vb/.k by A18,A36,A43
          .= vb.k by A33,A20,A37,PARTFUN1:def 6;
      end;
      suppose
        ex n st n in dom wb2 & k=len wb1+n;
        then consider n such that
A44:    n in dom wb2 and
A45:    k=len wb1+n;
        1<=n & n<=len wb2 by A44,FINSEQ_3:25;
        then
A46:    L2.(b2/.n) = wb2/.n by A8
          .= wb2.n by A44,PARTFUN1:def 6
          .= (wb1^wb2).k by A44,A45,FINSEQ_1:def 7;
        reconsider b2n=b2/.n as Vector of W1+W2 by A9,VECTSP_4:10;
A47:    K2.(b2/.n)=L2.(b2/.n) by A12,FUNCT_1:49;
        not b2/.n in Carrier K1
        proof
          assume
A48:      b2/.n in Carrier K1;
          then b2/.n in W2 & b2/.n in W1 by A25;
          then b2/.n in W1/\W2 by VECTSP_5:3;
          then b2/.n in the carrier of (0).V1 by A1;
          then b2/.n in {0.V1} by VECTSP_4:def 3;
          then b2/.n = 0.V1 by TARSKI:def 1
            .= 0.(W1+W2) by VECTSP_4:11;
          hence thesis by A34,A23,A48,VECTSP_7:2;
        end;
        then K1.b2n=0.K;
        then
A49:    L.b2n = 0.K+K2.b2n by A35,VECTSP_6:22
          .= (wb1^wb2).k by A46,A47,RLVECT_1:def 4;
        b2n = b2.n by A22,A44,PARTFUN1:def 6
          .= b.k by A4,A28,A22,A44,A45,FINSEQ_1:def 7
          .= b/.k by A33,A37,PARTFUN1:def 6;
        hence (wb1^wb2).k = vb/.k by A18,A36,A49
          .= vb.k by A33,A20,A37,PARTFUN1:def 6;
      end;
    end;
    hence vb.k=(wb1^wb2).k;
  end;
  hence thesis by A28,A21,A19,A31,A30,A3,Th21;
end;
