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 Th26:
  not b _|_ a & b _|_ c+a implies ProJ(a,b,c) = ProJ(c,b,a)
proof
  assume that
A1: not b _|_ a and
A2: b _|_ c+a;
  ProJ(a,b,c)*b _|_ c+a by A2,Def1;
  then
A3: -ProJ(a,b,c)*b _|_ c+a by Th6;
  c-ProJ(a,b,c)*b _|_ a by A1,Th14;
  then a _|_ -ProJ(a,b,c)*b+c by Th2;
  then c _|_ a+(-ProJ(a,b,c)*b) by A3,Def1;
  then
A4: a-ProJ(a,b,c)*b _|_ c by Th2;
  ( not a _|_ b)& c+a _|_ b by A1,A2,Th2;
  then not c _|_ b by Th3;
  then
A5: not b _|_ c by Th2;
  then a-ProJ(c,b,a)*b _|_ c by Th14;
  hence thesis by A5,A4,Th12;
end;
