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 Th43:
  for X being Subset of V, l being Linear_Combination of T .: X st
  T|X is one-to-one holds T@(T#l) = l
proof
  let X be Subset of V, l be Linear_Combination of T .: X such that
A1: T|X is one-to-one;
  let w be Element of W;
  set m = T@(T#l);
  per cases;
  suppose
A2: w in Carrier l;
    then
A3: l.w <> 0.F by VECTSP_6:2;
A4: dom (T#l) = [#]V by FUNCT_2:92;
    Carrier l c= T .: X by VECTSP_6:def 4;
    then consider v being object such that
A5: v in dom T and
A6: v in X and
A7: w = T.v by A2,FUNCT_1:def 6;
    reconsider v as Element of V by A5;
    consider B being Subset of V such that
A8: B misses X and
A9: T"{T.v} = {v} \/ B by A1,A6,Th34;
A10: (T#l).v = l.(T.v) by A1,A6,Th42;
A11: (T#l) .: {v} = Im (T#l,v) .= {(T#l).v} by A4,FUNCT_1:59;
A12: m.w = Sum ((T#l) .: T"{T.v}) by A7,Def5
      .= Sum ({l.(T.v)} \/ ((T#l) .: B)) by A9,A10,A11,RELAT_1:120;
    per cases;
    suppose
      B = {};
      then m.w = Sum ({l.(T.v)} \/ {}F) by A12
        .= l.w by A7,RLVECT_2:9;
      hence thesis;
    end;
    suppose
A13:  B <> {};
      Carrier (T#l) c= X by VECTSP_6:def 4;
      then B misses Carrier (T#l) by A8,XBOOLE_1:63;
      then m.w = Sum ({l.(T.v)} \/ {0.F}) by A12,A13,Th35
        .= Sum ({l.(T.v)}) + Sum ({0.F}) by A3,A7,RLVECT_2:12,ZFMISC_1:11
        .= l.(T.v) + Sum ({0.F}) by RLVECT_2:9
        .= l.(T.v) + 0.F by RLVECT_2:9
        .= l.w by A7,RLVECT_1:4;
      hence thesis;
    end;
  end;
  suppose
A14: not w in Carrier l;
    then
A15: l.w = 0.F;
    now
      assume
A16:  m.w <> 0.F;
      then w in Carrier m;
      then T"{w} meets Carrier (T#l) by Th36;
      then consider v being Element of V such that
A17:  v in T"{w} and
A18:  v in Carrier (T#l) by Th3;
      T.v in {w} by A17,FUNCT_1:def 7;
      then
A19:  T.v = w by TARSKI:def 1;
A20:  Carrier (T#l) c= X by VECTSP_6:def 4;
      then T|(Carrier (T#l)) is one-to-one by A1,Th2;
      then m.w = (T#l).v by A18,A19,Th37
        .= 0.F by A1,A15,A18,A19,A20,Th42;
      hence contradiction by A16;
    end;
    hence thesis by A14;
  end;
end;
