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 Th11:
  a <> 0 & a * b + c * d + e * f = 0 implies b = - (c/a) * d - (e/a) * f
  proof
    assume that
A1: a <> 0 and
A2: a * b + c * d + e * f = 0;
    b = ((- c) * d + (-e) * f) / a by A1,A2,XCMPLX_1:89
     .= (- c) * d / a + (-e) * f / a by XCMPLX_1:62
     .= (- c) / a * d + (-e) * f / a by XCMPLX_1:74
     .= (- c) / a * d + (-e) / a * f by XCMPLX_1:74
     .= (- (c / a)) * d + (-e) / a * f by XCMPLX_1:187
     .= (- (c / a)) * d + (-(e / a)) * f by XCMPLX_1:187;
    hence thesis;
  end;
