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

theorem Th21:
  for V be finite-rank free Z_Module,
      A being Subset of V, x being Element of V st x in Lin A &
  not x in A holds A \/ {x} is linearly-dependent
  proof
    let V be finite-rank free Z_Module,
    A be Subset of V, x be Element of V such that
    A1: x in Lin A and
    A2: not x in A;
    per cases;
    suppose
      A3: A is linearly-independent;
      reconsider X = {x} as Subset of Lin A by A1,SUBSET_1:41;
      reconsider A9 = A as Basis of Lin A by A3,ThLin7;
      reconsider B = A9 \/ X as Subset of Lin A;
      X misses A9
      proof
        assume X meets A9;
        then ex y being object st y in X & y in A9 by XBOOLE_0:3;
        hence contradiction by A2,TARSKI:def 1;
      end;
      then B is linearly-dependent by ZMODUL03:18;
      hence thesis by ZMODUL03:16;
    end;
    suppose
      A is linearly-dependent;
      hence thesis by ZMODUL02:56,XBOOLE_1:7;
    end;
  end;
