reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);

theorem Th10:
  for u,v,w being Element of V st a <> 0 &
  a * u + b * v + c * w = 0.V holds u = ((-b)/a) * v + ((-c)/a) * w
  proof
    let u,v,w be Element of V;
    assume that
A1: a <> 0 and
A2: a * u + b * v + c * w = 0.V;
    a*u + (b*v+c*w)=0.V by RLVECT_1:def 3,A2; then
A3: -(b*v+c*w) = a*u by RLVECT_1:6;
    a*u = 1 *(-(b*v+c*w)) by A3,RLVECT_1:def 8
       .= (a*(1/a))*(-(b*v+c*w)) by A1,XCMPLX_1:106
       .= a*((1/a)*(-(b*v+c*w))) by RLVECT_1:def 7;
    then u  = (1/a)*(-(b*v+c*w)) by A1,RLVECT_1:36
           .= (1/a) * ((-1) * (b*v+c*w)) by RLVECT_1:16
           .= (1/a) * ((-1) * (b*v) + (-1)* (c*w)) by RLVECT_1:def 5
           .= (1/a) * ((-1) * (b*v)) + (1/a)*((-1)*(c*w)) by RLVECT_1:def 5
           .= ((1/a) * (-1)) * (b*v) + (1/a)*((-1)*(c*w)) by RLVECT_1:def 7
           .= ((-1/a) * b) * v + (1/a)*((-1)*(c*w)) by RLVECT_1:def 7
           .= ((-1) / a)* b * v + (1/a)*((-1)*(c*w)) by XCMPLX_1:187
           .= ((-1) * b /a) * v + (1/a)*((-1)*(c*w)) by XCMPLX_1:74
           .= ((-b) /a) * v + ((1/a)*(-1))*(c*w) by RLVECT_1:def 7
           .= ((-b) /a) * v + ((-1/a)*c)*w by RLVECT_1:def 7
           .= ((-b) /a) * v + (((-1)/a)*c)*w by XCMPLX_1:187
           .= ((-b) /a) * v + ((-1)*c/a)*w by XCMPLX_1:74
           .= ((-b) /a) * v + ((-c)/a)*w;
    hence thesis;
  end;
