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;
