reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

theorem Th44:
  for A be Matrix of len b1,len b2,K st the_rank_of A = len b1
  holds Mx2Tran(A,b1,b2) is one-to-one
proof
  let A be Matrix of len b1,len b2,K such that
A1: the_rank_of A = len b1;
  set S=Space_of_Solutions_of (A@);
  set M=Mx2Tran(A,b1,b2);
A2: now
    per cases;
    suppose
      len b1=0;
      then dim V1=0 by Th21;
      then
A3:   (Omega).V1=(0).V1 by VECTSP_9:29;
      the carrier of ker M c= the carrier of V1 by VECTSP_4:def 2;
      hence the carrier of ker M c= {0.V1} by A3,VECTSP_4:def 3;
    end;
    suppose
A4:   len b1>0;
A5:   len b1<= width A by A1,MATRIX13:74;
      then
A6:   width (A@) = len A by A4,MATRIX_0:54;
A7:   len A=len b1 by A4,MATRIX_0:23;
A8:   width A=len b2 by A4,MATRIX_0:23;
      thus the carrier of ker M c= {0.V1}
      proof
        let x be object such that
A9:     x in the carrier of ker M;
        the carrier of ker M c= the carrier of V1 by VECTSP_4:def 2;
        then reconsider v=x as Element of V1 by A9;
        dim S = len b1 - the_rank_of (A@) by A4,A7,A6,MATRIX15:68
          .= len b1 - len b1 by A1,MATRIX13:84
          .= 0;
        then
A10:    (Omega).S=(0).S by VECTSP_9:29;
        v in ker M by A9;
        then v|--b1 in S by A4,A8,A5,Th41;
        then v|--b1 in the carrier of (0).S by A10;
        then v|--b1 in the carrier of (0).((len b1)-VectSp_over K) by A7,A6,
VECTSP_4:36;
        then v|--b1 in {0.((len b1)-VectSp_over K)} by VECTSP_4:def 3;
        then v|--b1 = 0.((len b1)-VectSp_over K) by TARSKI:def 1
          .= len b1 |->0.K by MATRIX13:102
          .= 0.V1|-- b1 by Th20;
        then v=0.V1 by MATRLIN:34;
        hence thesis by TARSKI:def 1;
      end;
    end;
  end;
  0.V1 in ker M by RANKNULL:11;
  then 0.V1 in the carrier of ker M;
  then {0.V1} c= the carrier of ker M by ZFMISC_1:31;
  then the carrier of ker M = {0.V1} by A2,XBOOLE_0:def 10
    .= the carrier of (0).V1 by VECTSP_4:def 3;
  then ker M=(0).V1 by VECTSP_4:29;
  hence thesis by Th43;
end;
