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
  not u,v,w are_LinDep & a1*u + b1*v + c1*w = a2*u + b2*v + c2*w implies
  a1 = a2 & b1 = b2 & c1 = c2
proof
A1: a1*u + b1*v + c1*w = a2*u + b2*v + c2*w implies (a1-a2)*u + (b1-b2)*v +
  (c1-c2)*w = 0.V
  proof
    assume a1*u + b1*v + c1*w = a2*u + b2*v + c2*w;
    then (c1-c2)*w + (a1*u + b1*v) = a2*u + b2*v by Lm6;
    then ((c1-c2)*w + a1*u) + b1*v = a2*u + b2*v by RLVECT_1:def 3;
    then (b1-b2)*v + ((c1-c2)*w + a1*u) = a2*u by Lm6;
    then ((b1-b2)*v + (c1-c2)*w) + a1*u = a2*u by RLVECT_1:def 3;
    then ((b1-b2)*v + (c1-c2)*w) + a1*u = 0.V + a2*u;
    then (a1-a2)*u + ((b1-b2)*v + (c1-c2)*w) = 0.V by Lm6;
    hence thesis by RLVECT_1:def 3;
  end;
  assume
A2: ( not u,v,w are_LinDep)& a1*u + b1*v + c1*w = a2*u + b2*v + c2*w;
  then
A3: c1 - c2 = 0 by A1;
  a1 - a2 = 0 & b1 - b2 = 0 by A2,A1;
  hence thesis by A3;
end;
