reserve V for RealLinearSpace;
reserve p,q,r,u,v,w,y,u1,v1,w1 for Element of V;
reserve a,b,c,d,a1,b1,c1,a2,b2,c2,a3,b3,e,f for Real;

theorem Th6:
  v is not zero & w is not zero & not are_Prop v,w implies (v,w,u
  are_LinDep iff ex a,b st u = a*v + b*w)
proof
  assume that
A1: v is not zero and
A2: w is not zero and
A3: not are_Prop v,w;
A4: w<>0.V by A2;
A5: v<>0.V by A1;
A6: v,w,u are_LinDep implies ex a,b st u = a*v + b*w
  proof
    assume v,w,u are_LinDep;
    then u,v,w are_LinDep by Th5;
    then consider a,b,c such that
A7: a*u + b*v + c*w = 0.V and
A8: a<>0 or b<>0 or c <>0;
    a<>0
    proof
      assume
A9:   a=0;
      then
A10:  0.V = 0.V + b*v + c*w by A7,RLVECT_1:10
        .= b*v + c*w;
A11:  b <> 0
      proof
        assume
A12:    b=0;
        then 0.V = 0.V + c*w by A10,RLVECT_1:10
          .= c*w;
        hence thesis by A4,A8,A9,A12,RLVECT_1:11;
      end;
A13:  c <> 0
      proof
        assume
A14:    c =0;
        then 0.V = b*v + 0.V by A10,RLVECT_1:10
          .= b*v;
        hence thesis by A5,A8,A9,A14,RLVECT_1:11;
      end;
      b*v = (-c)*w by A10,Lm1;
      then b=0 or -c =0 by A3;
      hence contradiction by A11,A13;
    end;
    then u = (-(a"*b))*v + (-(a"*c))*w by A7,Lm2;
    hence thesis;
  end;
  (ex a,b st u = a*v + b*w) implies v,w,u are_LinDep
  proof
    given a,b such that
A15: u = a*v + b*w;
    0.V = a*v + b*w + -u by A15,RLVECT_1:5
      .= a*v + b*w + (-1)*u by RLVECT_1:16;
    hence thesis;
  end;
  hence thesis by A6;
end;
