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

theorem Th12:
  not b _|_ a implies ProJ(a,b,l*x) = l*ProJ(a,b,x)
proof
  set L = x-ProJ(a,b,x)*b;
A1: l*L = l*x-l*(ProJ(a,b,x)*b) by VECTSP_1:23
    .= l*x-(l*ProJ(a,b,x))*b by VECTSP_1:def 16;
  assume
A2: not b _|_ a;
  then
A3: l*x - ProJ(a,b,l*x)*b _|_ a by Th11;
  L _|_ a by A2,Th11;
  then l*x-(l*ProJ(a,b,x))*b _|_ a by A1,Def1;
  hence thesis by A2,A3,Th8;
end;
