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 Th18:
  not b _|_ a & l <> 0.F implies ProJ(l*a,b,x) = ProJ(a,b,x)
proof
  assume that
A1: not b _|_ a and
A2: l <> 0.F;
  not b _|_ l*a by A1,A2,Th5;
  then x-ProJ(l*a,b,x)*b _|_ l*a by Th14;
  then l*a _|_ x-ProJ(l*a,b,x)*b by Th2;
  then l"*(l*a) _|_ x-ProJ(l*a,b,x)*b by Def1;
  then (l"*l)*a _|_ x-ProJ(l*a,b,x)*b by VECTSP_1:def 16;
  then 1_F*a _|_ x-ProJ(l*a,b,x)*b by A2,VECTSP_1:def 10;
  then a _|_ x-ProJ(l*a,b,x)*b;
  then
A3: x-ProJ(l*a,b,x)*b _|_ a by Th2;
  x-ProJ(a,b,x)*b _|_ a by A1,Th14;
  hence thesis by A1,A3,Th12;
end;
