reserve V,W for Z_Module;
reserve T for linear-transformation of V,W;
reserve T for linear-transformation of V,W;

theorem Th22:
  for V be finite-rank free Z_Module,A being Subset of V,
      B being Basis of V,
      T being linear-transformation of V,W
  st A is Basis of ker T & A c= B holds
  T | (B \ A) is one-to-one
  proof
    let V be finite-rank free Z_Module,
    A be Subset of V, B be Basis of V,
    T be linear-transformation of V,W such that
    A1: A is Basis of ker T and
    A2: A c= B;
    reconsider A9 = A as Subset of V;
    set f = T | (B \ A);
    let x1,x2 be object such that
    A3: x1 in dom f and
    A4: x2 in dom f and
    A5: f.x1 = f.x2 and
    A6: x1 <> x2;
    x2 in dom T by A4,RELAT_1:57;
    then reconsider x2 as Element of V;
    x1 in dom T by A3,RELAT_1:57;
    then reconsider x1 as Element of V;
    A7: not x1 in (A9 \/ {x2})
    proof
      assume
      A8: x1 in A9 \/ {x2};
      per cases by A8,XBOOLE_0:def 3;
      suppose
        x1 in A9;
        hence contradiction by A3,XBOOLE_0:def 5;
      end;
      suppose
        x1 in {x2};
        hence contradiction by A6,TARSKI:def 1;
      end;
    end;
    A9: f.x2 = T.x2 by A4,FUNCT_1:49;
    reconsider A as Subset of (ker T) by A1;
    reconsider A as Basis of (ker T) by A1;
    A10: ker T = Lin A by VECTSP_7:def 3;
    f.x1 = T.x1 by A3,FUNCT_1:49;
    then x1 - x2 in ker T by A5,A9,Th17;
    then x1 - x2 in Lin A9 by A10,ZMODUL03:20; then
    A11: x1 in Lin (A9 \/ {x2}) by Th18;
    {x2} \/ {x1} = {x1,x2} by ENUMSET1:1; then
    A12: (A9 \/ {x2}) \/ {x1} = A9 \/ {x1,x2} by XBOOLE_1:4;
    {x1,x2} c= B
    proof
      let z be object such that
      A13: z in {x1,x2};
      per cases by A13,TARSKI:def 2;
      suppose
        z = x1;
        hence thesis by A3,XBOOLE_0:def 5;
      end;
      suppose
        z = x2;
        hence thesis by A4,XBOOLE_0:def 5;
      end;
    end; then
    A14: A9 \/ {x1,x2} c= B by A2,XBOOLE_1:8;
    B is linearly-independent by VECTSP_7:def 3;
    hence thesis by A7,A11,A12,A14,Th21,ZMODUL02:56;
  end;
