reserve V,W for Z_Module;
reserve T for linear-transformation of V,W;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;

theorem Th27:
  for R being Ring, V being LeftMod of R
  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 R be Ring, V be LeftMod of R;
    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.R by Th26;
      (l!A).v = l.v by A4,Th25;
      then r.v = l.v + 0.R by A5,VECTSP_6:22
      .= l.v;
      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.R by Th26;
      (l!(B\A)).v = l.v by A6,Th25;
      then r.v = 0.R + l.v by A7,VECTSP_6:22
      .= l.v;
      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.R by Th26;
      (l!A).v = 0.R by A1,A8,Th26;
      then r.v = 0.R + 0.R by A10,VECTSP_6:22
      .= 0.R;
      hence l.v = r.v by A9;
    end;
  end;
