
theorem Th27:
  for K be Field, V be VectSp of K for X be Subspace of V, fi be
linear-Functional of X, v be Vector of V, y be Vector of X + Lin {v} st v = y &
  not v in X for r be Element of K ex psi be linear-Functional of X + Lin{v} st
  psi|the carrier of X=fi & psi.y = r
proof
  let K be Field, V be VectSp of K, X be Subspace of V, fi be
  linear-Functional of X, v be Vector of V, y be Vector of X + Lin{v};
  assume that
A1: v = y and
A2: not v in X;
  reconsider W1 = X as Subspace of X + Lin{v} by VECTSP_5:7;
  let r be Element of K;
  defpred P[Element of (X + Lin{v}), Element of K] means for x be Vector of X,
s be Element of K st ($1 |-- (W1,Lin{y}))`1 = x & ($1 |-- (W1,Lin{y}))`2 = s*v
  holds $2 = fi.x + s*r;
A3: for e be Element of (X + Lin{v}) ex a be Element of K st P[e,a]
  proof
    let e be Element of (X + Lin{v});
    consider x be Vector of X, s be Element of K such that
A4: e |-- (W1,Lin{y}) = [x,s*v] by A1,A2,Th18;
    take fi.x + s*r;
    let x9 be Vector of X, t be Element of K such that
A5: (e |-- (W1,Lin{y}))`1 = x9 and
A6: (e |-- (W1,Lin{y}))`2 = t*v;
    v <> 0.V by A2,VECTSP_4:17;
    then t = s by A4,A6,Th4;
    hence thesis by A4,A5;
  end;
  consider f be Function of the carrier of X + Lin{v},the carrier of K such
  that
A7: for e be Element of X + Lin{v} holds P[e,f.e] from FUNCT_2:sch 3(A3);
A8: now
    let a be object;
    assume a in dom fi;
    then reconsider x = a as Vector of X;
    X is Subspace of X + Lin{v} by VECTSP_5:7;
    then the carrier of X c= the carrier of X + Lin{v} by VECTSP_4:def 2;
    then reconsider v1 = x as Vector of X + Lin{v};
    v1 in X;
    then (v1 |-- (W1,Lin{y})) = [v1,0.V] by A1,A2,Th16
      .= [v1,0.K*v] by Th1;
    then
A9: ( v1 |-- (W1,Lin{y}))`1 = x & (v1 |-- (W1,Lin{y}))`2 = 0.K*v;
    thus fi.a = fi.x + 0.K by RLVECT_1:4
      .= fi.x + 0.K*r
      .= f.a by A7,A9;
  end;
  reconsider f as Functional of X + Lin{v};
A10: y |-- (W1,Lin{y}) = [0.W1,y] by A1,A2,Th15;
  then
A11: (y |-- (W1,Lin{y}))`1 = 0.X;
A12: f is additive
  proof
    let v1,v2 be Vector of X + Lin{v};
    consider x1 be Vector of X, s1 be Element of K such that
A13: v1 |-- (W1,Lin{y}) = [x1,s1*v] by A1,A2,Th18;
A14: (v1 |-- (W1,Lin{y}))`1 = x1 & (v1 |-- (W1,Lin{y}))`2 = s1*v by A13;
    consider x2 be Vector of X, s2 be Element of K such that
A15: v2 |-- (W1,Lin{y}) = [x2,s2*v] by A1,A2,Th18;
A16: (v2 |-- (W1,Lin{y}))`1 = x2 & (v2 |-- (W1,Lin{y}))`2 = s2*v by A15;
    v1 + v2 |-- (W1,Lin{y}) = [x1 +x2,(s1+s2)*v] by A1,A2,A13,A15,Th19;
    then
    (v1 + v2 |-- (W1,Lin{y}))`1 = x1 + x2 & (v1 + v2 |-- (W1,Lin{y}))`2 =
    (s1+ s2)*v;
    hence f.(v1+v2) = fi.(x1 + x2) + (s1 + s2)*r by A7
      .= fi.(x1 + x2) + (s1*r + s2*r) by VECTSP_1:def 3
      .= fi.(x1) + fi.(x2) + (s1*r + s2*r) by VECTSP_1:def 20
      .= fi.(x1) + fi.(x2) + s1*r + s2*r by RLVECT_1:def 3
      .= fi.(x1) + s1*r + fi.(x2) + s2*r by RLVECT_1:def 3
      .= fi.(x1) + s1*r + (fi.(x2) + s2*r) by RLVECT_1:def 3
      .= f.v1 + (fi.(x2) + s2*r) by A7,A14
      .= f.v1+f.v2 by A7,A16;
  end;
  f is homogeneous
  proof
    let v0 be Vector of X + Lin{v}, t be Element of K;
    consider x0 be Vector of X, s0 be Element of K such that
A17: v0 |-- (W1,Lin{y}) = [x0,s0*v] by A1,A2,Th18;
A18: (v0 |-- (W1,Lin{y}))`1 = x0 & (v0 |-- (W1,Lin{y}))`2 = s0*v by A17;
    t*v0 |-- (W1,Lin{y}) = [t*x0,t*s0*v] by A1,A2,A17,Th20;
    then (t*v0 |-- (W1,Lin{y}))`1 = t*x0 & (t*v0 |-- (W1,Lin{y}))`2 = t*s0*v;
    hence f.(t*v0) = fi.(t*x0) + t*s0*r by A7
      .= t*(fi.x0) + t*s0*r by HAHNBAN1:def 8
      .= t*(fi.x0) + t*(s0*r) by GROUP_1:def 3
      .= t*(fi.x0 + s0*r) by VECTSP_1:def 2
      .= t*f.v0 by A7,A18;
  end;
  then reconsider f as linear-Functional of X + Lin{v} by A12;
  take f;
A19: dom fi = the carrier of X by FUNCT_2:def 1;
  dom f = the carrier of X + Lin{v} & X is Subspace of X + Lin{v} by
FUNCT_2:def 1,VECTSP_5:7;
  then dom fi c= dom f by A19,VECTSP_4:def 2;
  then dom fi = dom f /\ the carrier of X by A19,XBOOLE_1:28;
  hence f|the carrier of X=fi by A8,FUNCT_1:46;
  (y |-- (W1,Lin{y}))`2 = y by A10
    .= (1_K)*v by A1;
  hence f.y = fi.(0.X) + (1_K)*r by A7,A11
    .= 0.K + (1_K)*r by HAHNBAN1:def 9
    .= 0.K + r
    .= r by RLVECT_1:4;
end;
