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 Th20:
  not b _|_ a implies ( x _|_ a iff ProJ(a,b,x) = 0.F )
proof
  set 0F = 0.F;
  set 0V = 0.S;
A1: now
    assume that
A2: not b _|_ a and
A3: x _|_ a;
    x+0V _|_ a by A3,RLVECT_1:4;
    then x+(-0V) _|_ a by RLVECT_1:12;
    then
A4: x-0F*b _|_ a by VECTSP_1:14;
    x-ProJ(a,b,x)*b _|_ a by A2,Th11;
    hence ProJ(a,b,x) = 0.F by A2,A4,Th8;
  end;
  now
    assume ( not b _|_ a)& ProJ(a,b,x) = 0.F;
    then x-0F*b _|_ a by Th11;
    then x+(-0V) _|_ a by VECTSP_1:14;
    then x+0V _|_ a by RLVECT_1:12;
    hence x _|_ a by RLVECT_1:4;
  end;
  hence thesis by A1;
end;
