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 Th9:
  not are_Prop p,q & u = a1*p + b1*q & v = a2*p + b2*q & a1*b2 -
a2*b1=0 & p is not zero & q is not zero implies (are_Prop u,v or u = 0.V or v =
  0.V)
proof
  assume that
A1: not are_Prop p,q and
A2: u = a1*p + b1*q and
A3: v = a2*p + b2*q and
A4: a1*b2 - a2*b1=0 and
A5: p is not zero & q is not zero;
  now
    assume that
    u <> 0.V and
    v <> 0.V;
A6: for p,q,u,v,a1,a2,b1,b2 st not are_Prop p,q & u = a1*p + b1*q & v =
a2*p + b2*q & a1*b2 - a2*b1=0 & p is not zero & q is not zero & a1=0 & u <> 0.V
    & v <> 0.V holds are_Prop u,v
    proof
      let p,q,u,v,a1,a2,b1,b2;
      assume that
      not are_Prop p,q and
A7:   u = a1*p + b1*q and
A8:   v = a2*p + b2*q and
A9:   a1*b2 - a2*b1=0 and
      p is not zero and
      q is not zero and
A10:  a1=0 and
A11:  u <> 0.V and
A12:  v <> 0.V;
      0= (-a2)*b1 by A9,A10;
      then
A13:  -a2=0 or b1=0 by XCMPLX_1:6;
A14:  now
        assume b1=0;
        then u=0.V+0*q by A7,A10,RLVECT_1:10
          .= 0.V+0.V by RLVECT_1:10;
        hence contradiction by A11;
      end;
      then
A15:  b1"<>0 by XCMPLX_1:202;
A16:  now
        assume b2*b1"=0;
        then b2=0 by A15,XCMPLX_1:6;
        then v = 0.V + 0*q by A8,A13,A14,RLVECT_1:10
          .= 0.V + 0.V by RLVECT_1:10;
        hence contradiction by A12;
      end;
      u = 0.V + b1*q by A7,A10,RLVECT_1:10;
      then
A17:  u = b1*q;
      v = 0.V + b2*q by A8,A13,A14,RLVECT_1:10;
      then v = b2*q;
      then v = b2*(b1"*u) by A14,A17,ANALOAF:5;
      then v = (b2*b1")*u by RLVECT_1:def 7;
      then 1*v = (b2*b1")*u by RLVECT_1:def 8;
      hence thesis by A16;
    end;
    now
      assume that
A18:  a1<>0 and
A19:  a2<>0;
A20:  now
        a1"<>0 by A18,XCMPLX_1:202;
        then
A21:    a2*a1" <> 0 by A19,XCMPLX_1:6;
        assume
A22:    b1=0;
        then b2=0 by A4,A18,XCMPLX_1:6;
        then v = a2*p + 0.V by A3,RLVECT_1:10;
        then
A23:    v = a2*p;
        u = a1*p + 0.V by A2,A22,RLVECT_1:10;
        then u = a1*p;
        then v = a2*(a1"*u) by A18,A23,ANALOAF:5
          .= (a2*a1")*u by RLVECT_1:def 7;
        then 1*v = (a2*a1")*u by RLVECT_1:def 8;
        hence are_Prop u,v by A21;
      end;
      now
A24:    b2*u = (a1*b2)*p + (b2*b1)*q & b1*v = (a2*b1)*p + (b1*b2)*q by A2,A3
,Lm5;
        assume
A25:    b1<>0;
        then b2 <> 0 by A4,A19,XCMPLX_1:6;
        hence are_Prop u,v by A4,A25,A24;
      end;
      hence thesis by A20;
    end;
    hence thesis by A1,A2,A3,A4,A5,A6;
  end;
  hence thesis;
end;
