reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;
reserve V,W for finite-rank free Z_Module;
reserve T for linear-transformation of V,W;

theorem ZM05Th35:
  for V being finite-rank free Z_Module, A being Subset of V,
  B being linearly-independent Subset of V,
  T being linear-transformation of V,W
  st rank(V) = card(B) & 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 linearly-independent Subset of V,
    T be linear-transformation of V,W such that
    rank(V) = card(B) and
    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,ZMODUL05:27;
    then x1 - x2 in Lin A9 by A10,ZMODUL03:20; then
    A11: x1 in Lin (A9 \/ {x2}) by ZMODUL05:28;
    {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 A9 \/ {x1,x2} c= B by A2,XBOOLE_1:8;
    hence thesis by A7,A11,A12,ZMODUL02:56,ZMODUL05:32;
  end;
