reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;

theorem Th27:
  for A,B being Subset of V, l being Linear_Combination of B st A
  c= B holds l = (l!A) + (l!(B\A))
proof
  let A,B be Subset of V, l be Linear_Combination of B such that
A1: A c= B;
  set r = (l!A) + (l!(B\A));
  let v be Element of V;
A2: v in B implies (v in A or v in B \ A)
  proof
    assume
A3: v in B;
    B = A \/ (B \ A) by A1,XBOOLE_1:45;
    hence thesis by A3,XBOOLE_0:def 3;
  end;
  per cases by A2;
  suppose
A4: v in A;
    then not v in B \ A by XBOOLE_0:def 5;
    then
A5: (l!(B\A)).v = 0.F by Th26;
    (l!A).v = l.v by A4,Th25;
    then r.v = l.v + 0.F by A5,VECTSP_6:22
      .= l.v by RLVECT_1:4;
    hence l.v = r.v;
  end;
  suppose
A6: v in B\A;
    then not v in A by XBOOLE_0:def 5;
    then
A7: (l!A).v = 0.F by Th26;
    (l!(B\A)).v = l.v by A6,Th25;
    then r.v = 0.F + l.v by A7,VECTSP_6:22
      .= l.v by RLVECT_1:4;
    hence l.v = r.v;
  end;
  suppose
A8: not v in B;
    Carrier l c= B by VECTSP_6:def 4;
    then
A9: not v in Carrier l by A8;
    not v in B\A by A8,XBOOLE_0:def 5;
    then
A10: (l!(B\A)).v = 0.F by Th26;
    not v in A by A1,A8;
    then (l!A).v = 0.F by Th26;
    then r.v = 0.F + 0.F by A10,VECTSP_6:22
      .= 0.F by RLVECT_1:4;
    hence l.v = r.v by A9;
  end;
end;
