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 & p _|_ b & p _|_ c implies ProJ(a+p,b,c) = ProJ(a,b,c)
proof
  assume that
A1: not b _|_ a and
A2: p _|_ b and
A3: p _|_ c;
  b _|_ p by A2,Th2;
  then ProJ(a,b,c)*b _|_ p by Def1;
  then
A4: -(ProJ(a,b,c)*b) _|_ p by Th6;
  c _|_ p by A3,Th2;
  then c+(-(ProJ(a,b,c)*b)) _|_ p by A4,Def1;
  then
A5: p _|_ c-ProJ(a,b,c)*b by Th2;
  c-ProJ(a,b,c)*b _|_ a by A1,Th11;
  then a _|_ c-ProJ(a,b,c)*b by Th2;
  then a+p _|_ c-ProJ(a,b,c)*b by A5,Def1;
  then
A6: c-ProJ(a,b,c)*b _|_ a+p by Th2;
  not a _|_ b by A1,Th2;
  then not a+p _|_ b by A2,Th4;
  then
A7: not b _|_ a+p by Th2;
  then c-ProJ(a+p,b,c)*b _|_ a+p by Th11;
  hence thesis by A7,A6,Th8;
end;
