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
  not b _|_ a & l <> 0.F implies ProJ(a,l*b,x) = l"*ProJ(a,b,x)
proof
  assume that
A1: not b _|_ a and
A2: l <> 0.F;
  set L = x-ProJ(a,l*b,x)*(l*b);
  not l*b _|_ a by A1,A2,Th5;
  then
A3: L _|_ a by Th11;
A4: L = x-(ProJ(a,l*b,x)*l)*b by VECTSP_1:def 16;
  x-ProJ(a,b,x)*b _|_ a by A1,Th11;
  then ProJ(a,b,x)*l" = (ProJ(a,l*b,x)*l)*l" by A1,A3,A4,Th8;
  then ProJ(a,b,x)*l" = ProJ(a,l*b,x)*(l*l") by GROUP_1:def 3;
  then l"*ProJ(a,b,x) = ProJ(a,l*b,x)*(1_F) by A2,VECTSP_1:def 10;
  hence thesis;
end;
