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 Th16:
  not b _|_ a implies ProJ(a,b,x+y) = ProJ(a,b,x) + ProJ(a,b,y)
proof
  set 1F = 1_F;
  set L = (x-ProJ(a,b,x)*b)+(y-ProJ(a,b,y)*b);
A1: L = (((-ProJ(a,b,x)*b)+x)+y)+(-ProJ(a,b,y)*b) by RLVECT_1:def 3
    .= ((x+y)+(-ProJ(a,b,x)*b))+(-ProJ(a,b,y)*b) by RLVECT_1:def 3
    .= (x+y)+((-ProJ(a,b,x)*b)+(-ProJ(a,b,y)*b)) by RLVECT_1:def 3
    .= (x+y)+((1F)*(-ProJ(a,b,x)*b)+(-ProJ(a,b,y)*b))
    .= (x+y)+((1F)*(-ProJ(a,b,x)*b)+(1F)*(-ProJ (a,b,y)*b))
    .= (x+y)+((1F)*((-ProJ(a,b,x))*b)+(1F)*(-ProJ(a,b,y)*b)) by VECTSP_1:21
    .= (x+y)+((1F)*((-ProJ(a,b,x))*b)+(1F)*((-ProJ (a,b,y))*b)) by VECTSP_1:21
    .= (x+y)+(((1F)*(-ProJ(a,b,x)))*b+(1F)*((-ProJ (a,b,y))*b))
    .= (x+y)+(((-ProJ(a,b,x))*(1F))*b+((1F)*(-ProJ(a,b,y)))*b)
    .= (x+y)+((-ProJ(a,b,x))*b+((-ProJ(a,b,y))*(1F))*b)
    .= (x+y)+((-ProJ(a,b,x))*b+(-ProJ(a,b,y))*b)
    .= (x+y)+((-ProJ(a,b,x))+(-ProJ(a,b,y)))*b by VECTSP_1:def 15
    .= (x+y)+(-(ProJ(a,b,x)+ProJ(a,b,y)))*b by RLVECT_1:31
    .= (x+y)-(ProJ(a,b,x)+ProJ(a,b,y))*b by VECTSP_1:21;
  assume
A2: not b _|_ a;
  then x-ProJ(a,b,x)*b _|_ a & y-ProJ(a,b,y)*b _|_ a by Th14;
  then
A3: L _|_ a by Def1;
  (x+y)-ProJ(a,b,x+y)*b _|_ a by A2,Th14;
  hence thesis by A2,A3,A1,Th12;
end;
