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 Th19:
  not b _|_ a & p _|_ a implies ProJ(a,b+p,c) = ProJ(a,b,c) & ProJ
  (a,b,c+p) = ProJ(a,b,c)
proof
  set 0V = 0.S;
  assume that
A1: not b _|_ a and
A2: p _|_ a;
  not b+p _|_ a by A1,A2,Th4;
  then c-ProJ(a,b+p,c)*(b+p) _|_ a by Th14;
  then c-(ProJ(a,b+p,c)*b+ProJ(a,b+p,c)*p) _|_ a by VECTSP_1:def 14;
  then
A3: c-ProJ(a,b+p,c)*b-ProJ(a,b+p,c)*p _|_ a by VECTSP_1:17;
  c+p-ProJ(a,b,c+p)*b _|_ a & -p _|_ a by A1,A2,Th6,Th14;
  then -p+(p+c-ProJ(a,b,c+p)*b) _|_ a by Def1;
  then -p+(p+(c+(-ProJ(a,b,c+p)*b))) _|_ a by RLVECT_1:def 3;
  then (-p+p)+(c+(-ProJ(a,b,c+p)*b)) _|_ a by RLVECT_1:def 3;
  then 0V+(c+(-ProJ(a,b,c+p)*b)) _|_ a by RLVECT_1:5;
  then
A4: c-ProJ(a,b,c+p)*b _|_ a by RLVECT_1:4;
  ProJ(a,b+p,c)*p _|_ a by A2,Def1;
  then c+(-ProJ(a,b+p,c)*b)-ProJ(a,b+p,c)*p+ProJ(a,b+p,c)*p _|_ a by A3,Def1;
  then c+(-ProJ(a,b+p,c)*b)+((-ProJ(a,b+p,c)*p)+ProJ(a,b+p,c)*p) _|_ a by
RLVECT_1:def 3;
  then c+(-ProJ(a,b+p,c)*b)+0V _|_ a by RLVECT_1:5;
  then
A5: c-ProJ(a,b+p,c)*b _|_ a by RLVECT_1:4;
  c-ProJ(a,b,c)*b _|_ a by A1,Th14;
  hence thesis by A1,A5,A4,Th12;
end;
