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 Th37:
  for v being Element of V st T|(Carrier l) is one-to-one & v in
  Carrier l holds (T@l).(T.v) = l.v
proof
  let v be Element of V such that
A1: T|(Carrier l) is one-to-one and
A2: v in Carrier l;
  consider X being Subset of V such that
A3: X misses Carrier l and
A4: T"{T.v} = {v} \/ X by A1,A2,Th34;
  per cases;
  suppose
A5: X = {};
A6: dom l = [#]V by FUNCT_2:92;
    l .: {v} = Im (l,v) .= {l.v} by A6,FUNCT_1:59;
    then Sum (l .: T"{T.v}) = l.v by A4,A5,RLVECT_2:9;
    hence thesis by Def5;
  end;
  suppose
A7: X <> {};
A8: l .: X c= {0.F}
    proof
      let y be object;
      assume y in l .: X;
      then consider x being object such that
A9:   x in dom l and
A10:  x in X and
A11:  y = l.x by FUNCT_1:def 6;
A12:  not x in Carrier l by A3,A10,XBOOLE_0:def 4;
      reconsider x as Element of V by A9;
      l.x = 0.F by A12;
      hence thesis by A11,TARSKI:def 1;
    end;
A13: l .: X misses l .: {v}
    proof
      assume l .: X meets l .: {v};
      then consider x being object such that
A14:  x in l .: X and
A15:  x in l .: {v} by XBOOLE_0:3;
A16:  dom l = [#]V by FUNCT_2:92;
      l .: {v} = Im (l,v) .= {l.v} by A16,FUNCT_1:59;
      then
A17:  x = l.v by A15,TARSKI:def 1;
      x = 0.F by A8,A14,TARSKI:def 1;
      hence thesis by A2,A17,VECTSP_6:2;
    end;
A18: dom l = [#]V by FUNCT_2:92;
    {0.F} c= l .: X
    proof
      consider y being object such that
A19:  y in X by A7,XBOOLE_0:def 1;
A20:  not y in Carrier l by A3,A19,XBOOLE_0:def 4;
      reconsider y as Element of V by A19;
      let x be object;
      assume x in {0.F};
      then x = 0.F by TARSKI:def 1;
      then l.y = x by A20;
      hence thesis by A18,A19,FUNCT_1:def 6;
    end;
    then
A21: l .: X = {0.F} by A8;
A22: l .: {v} = Im (l,v) .= {l.v} by A18,FUNCT_1:59;
    l .: T"{T.v} = (l .: {v}) \/ (l .: X) by A4,RELAT_1:120;
    then Sum (l .: T"{T.v}) = (Sum (l .: {v})) + (Sum (l .: X)) by A13,
RLVECT_2:12
      .= l.v + (Sum ({0.F})) by A22,A21,RLVECT_2:9
      .= l.v + 0.F by RLVECT_2:9
      .= l.v by RLVECT_1:4;
    hence thesis by Def5;
  end;
end;
