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 Th44:
  for V,W being finite-dimensional VectSp of F, T being
  linear-transformation of V,W holds dim V = rank(T) + nullity(T)
proof
  let V,W be finite-dimensional VectSp of F, T be linear-transformation of V,W;
  set A = the finite Basis of ker T;
  reconsider A9 = A as Subset of V by Th19;
  consider B being Basis of V such that
A1: A c= B by VECTSP_9:13;
  reconsider B as finite Subset of V;
  reconsider X = B \ A9 as finite Subset of B by XBOOLE_1:36;
  reconsider X as finite Subset of V;
A2: B = A \/ X by A1,XBOOLE_1:45;
  reconsider B as finite Basis of V;
  reconsider A as finite Basis of ker T;
  reconsider C = T .: X as finite Subset of W;
A3: T|X is one-to-one by A1,Th22;
A4: C is linearly-independent
  proof
    assume C is linearly-dependent;
    then consider l being Linear_Combination of C such that
A5: Carrier l <> {} and
A6: Sum l = 0.W by Th41;
    ex m being Linear_Combination of X st l = T@m
    proof
      reconsider l9 = l as Linear_Combination of T .: X;
      set m = T#(l9);
      take m;
      thus thesis by A3,Th43;
    end;
    then consider m being Linear_Combination of B \ A9 such that
A7: l = T@m;
    T.(Sum m) = 0.W by A1,A6,A7,Th40;
    then Sum m in ker T by Th10;
    then Sum m in Lin A by VECTSP_7:def 3;
    then Sum m in Lin A9 by VECTSP_9:17;
    then consider n being Linear_Combination of A9 such that
A8: Sum m = Sum n by VECTSP_7:7;
A9: Carrier (m - n) c= (Carrier m) \/ (Carrier n) & (B \ A9) \/ A9 = B by A1,
VECTSP_6:41,XBOOLE_1:45;
A10: Carrier m c= B \ A9 & Carrier n c= A by VECTSP_6:def 4;
    then (Carrier m) \/ (Carrier n) c= (B \ A9) \/ A by XBOOLE_1:13;
    then Carrier (m - n) c= B by A9;
    then reconsider o = m - n as Linear_Combination of B by VECTSP_6:def 4;
A11: B is linearly-independent by VECTSP_7:def 3;
    (Sum m) - (Sum n) = 0.V by A8,VECTSP_1:19;
    then Sum (m - n) = 0.V by VECTSP_6:47;
    then
A12: Carrier o = {} by A11,VECTSP_7:def 1;
    A9 misses B \ A9 by XBOOLE_1:79;
    then Carrier (m - n) = (Carrier m) \/ (Carrier n) by A10,Th32,XBOOLE_1:64;
    then Carrier m = {} by A12;
    then T .: (Carrier m) = {};
    hence thesis by A5,A7,Th30,XBOOLE_1:3;
  end;
  dom T = [#]V by Th7;
  then X c= dom (T|X) by RELAT_1:62;
  then X,(T|X) .: X are_equipotent by A3,CARD_1:33;
  then X,C are_equipotent by RELAT_1:129;
  then
A13: card C = card X by CARD_1:5;
  reconsider C as finite Subset of im T by Th12;
  reconsider L = Lin C as strict Subspace of im T;
  for v being Element of im T holds v in L
  proof
    reconsider A9 = A as Subset of V by Th19;
    let v be Element of im T;
    reconsider v9 = v as Element of W by VECTSP_4:10;
    reconsider C9 = C as Subset of W;
    v in im T;
    then consider u being Element of V such that
A14: T.u = v9 by Th13;
    V is_the_direct_sum_of Lin A9, Lin (B \ A9) by A1,Th33;
    then
A15: (Omega).V = (Lin A9) + (Lin (B \ A9)) by VECTSP_5:def 4;
    u in (Omega).V;
    then consider u1, u2 being Element of V such that
A16: u1 in Lin A9 and
A17: u2 in Lin (B \ A9) and
A18: u = u1 + u2 by A15,VECTSP_5:1;
    consider l being Linear_Combination of B \ A9 such that
A19: u2 = Sum l by A17,VECTSP_7:7;
    Lin A = ker T by VECTSP_7:def 3;
    then u1 in ker T by A16,VECTSP_9:17;
    then
A20: T.u1 = 0.W by Th10;
    T@l is Linear_Combination of T .: (Carrier l) & Carrier l c= B \ A9
    by Th29,VECTSP_6:def 4;
    then reconsider m = T@l as Linear_Combination of C9 by RELAT_1:123
,VECTSP_6:4;
    T.u = T.u1 + T.u2 by A18,VECTSP_1:def 20;
    then
A21: T.u = T.u2 by A20,RLVECT_1:4;
    ex m being Linear_Combination of C9 st v = Sum m
    proof
      take m;
      thus thesis by A1,A14,A21,A19,Th40;
    end;
    then v in Lin C9 by VECTSP_7:7;
    hence thesis by VECTSP_9:17;
  end;
  then
A22: Lin C = im T by VECTSP_4:32;
  reconsider C as linearly-independent Subset of im T by A4,VECTSP_9:12;
  reconsider C as finite Basis of im T by A22,VECTSP_7:def 3;
A23: nullity T = card A & rank T = card C by VECTSP_9:def 1;
  dim V = card B by VECTSP_9:def 1
    .= rank T + nullity T by A2,A13,A23,CARD_2:40,XBOOLE_1:79;
  hence thesis;
end;
