reserve x for set;
reserve a,b,c,d,e,r1,r2,r3,r4,r5,r6 for Real;
reserve V for RealLinearSpace;
reserve u,v,v1,v2,v3,w,w1,w2,w3 for VECTOR of V;
reserve W,W1,W2 for Subspace of V;

theorem Th7:
  u <> v & u <> w & v <> w & {u,v,w} is linearly-independent iff
  for a,b,c st a * u + b * v + c * w = 0.V holds a = 0 & b = 0 & c = 0
proof
  thus u <> v & u <> w & v <> w & {u,v,w} is linearly-independent implies for
  a,b,c st a * u + b * v + c * w = 0.V holds a = 0 & b = 0 & c = 0
  proof
    deffunc F(VECTOR of V)=zz;
    assume that
A1: u <> v and
A2: u <> w and
A3: v <> w and
A4: {u,v,w} is linearly-independent;
    let a,b,c;
     reconsider aa=a, bb=b, cc=c as Element of REAL by XREAL_0:def 1;
    consider f being Function of the carrier of V, REAL such that
A5: f.u = aa & f.v = bb & f.w = cc and
A6: for x being VECTOR of V st x <> u & x <> v & x <> w
     holds f.x = F(x) from LambdaSep3(A1,A2,A3 );
    reconsider f as Element of Funcs(the carrier of V,REAL) by FUNCT_2:8;
    now
      let x be VECTOR of V;
      assume
A7:   not x in {u,v,w};
      then
A8:   x <> w by ENUMSET1:def 1;
      x <> u & x <> v by A7,ENUMSET1:def 1;
      hence f.x = 0 by A6,A8;
    end;
    then reconsider f as Linear_Combination of V by RLVECT_2:def 3;
    Carrier f c= {u,v,w}
    proof
      let x be object;
      assume
A9:   x in Carrier f;
      then f.x <> 0 by RLVECT_2:19;
      then x = u or x = v or x = w by A6,A9;
      hence thesis by ENUMSET1:def 1;
    end;
    then reconsider f as Linear_Combination of {u,v,w} by RLVECT_2:def 6;
    assume a * u + b * v + c * w = 0.V;
    then
A10: 0.V = Sum(f) by A1,A2,A3,A5,Th6;
    then
A11: not u in Carrier f by A4;
    ( not v in Carrier f)& not w in Carrier f by A4,A10;
    hence thesis by A5,A11,RLVECT_2:19;
  end;
  assume
A12: for a,b,c st a * u + b * v + c * w = 0.V holds a = 0 & b = 0 & c = 0;
A13: now
    assume
A14: u = v or u = w or v = w;
    now
      per cases by A14;
      suppose
        u = v;
        then 1 * u + (- 1) * v + 0 * w = u + (- 1) * u + 0 * w by
RLVECT_1:def 8
          .= u + - u + 0 * w by RLVECT_1:16
          .= u + - u + 0.V by RLVECT_1:10
          .= u + - u by RLVECT_1:4
          .= 0.V by RLVECT_1:5;
        hence contradiction by A12;
      end;
      suppose
        u = w;
        then 1 * u + 0 * v + (- 1) * w = u + 0 * v + (- 1) * u by
RLVECT_1:def 8
          .= u + 0.V + (- 1) * u by RLVECT_1:10
          .= u + 0.V + - u by RLVECT_1:16
          .= u + - u by RLVECT_1:4
          .= 0.V by RLVECT_1:5;
        hence contradiction by A12;
      end;
      suppose
        v = w;
        then 0 * u + 1 * v + (- 1) * w = 0 * u + 1 * v + - v by RLVECT_1:16
          .= 0.V + 1 * v + - v by RLVECT_1:10
          .= 0.V + v + - v by RLVECT_1:def 8
          .= v + - v by RLVECT_1:4
          .= 0.V by RLVECT_1:5;
        hence contradiction by A12;
      end;
    end;
    hence contradiction;
  end;
  hence u <> v & u <> w & v <> w;
  let l be Linear_Combination of {u,v,w};
  assume Sum(l) = 0.V;
  then
A15: l.u * u + l.v * v + l.w * w = 0.V by A13,Th6;
  then
A16: l.w = 0 by A12;
A17: l.u = 0 & l.v = 0 by A12,A15;
  thus Carrier l c= {}
  proof
    let x be object;
    assume
A18: x in Carrier l;
    Carrier l c= {u,v,w} by RLVECT_2:def 6;
    then x = u or x = v or x = w by A18,ENUMSET1:def 1;
    hence thesis by A17,A16,A18,RLVECT_2:19;
  end;
  thus thesis;
end;
