reserve F for Field;
reserve S for SymSp of F;
reserve a,b,c,d,a9,b9,p,q,r,s,x,y,z for Element of S;
reserve k,l for Element of F;

theorem Th12:
  not b _|_ a & x-k*b _|_ a & x-l*b _|_ a implies k = l
proof
  assume that
A1: not b _|_ a and
A2: x-k*b _|_ a & x-l*b _|_ a;
  set 1F=1_F;
  k*b-l*b _|_ a by A2,Th11;
  then k*b+((-1F)*(l*b)) _|_ a by VECTSP_1:14;
  then k*b+(((-1F)*l)*b) _|_ a by VECTSP_1:def 16;
  then k*b+(-(l*(1F)))*b _|_ a by VECTSP_1:9;
  then k*b+(-l)*b _|_ a;
  then (k+(-l))*b _|_ a by VECTSP_1:def 15;
  then (k+(-l))"*((k+(-l))*b) _|_ a by Def1;
  then
A3: ((k+(-l))"*(k+(-l)))*b _|_ a by VECTSP_1:def 16;
  assume not thesis;
  then k-l<>0.F by RLVECT_1:21;
  then (1F)*b _|_ a by A3,VECTSP_1:def 10;
  hence contradiction by A1;
end;
