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 Th23:
  not b _|_ a & b _|_ c+a implies ProJ(a,b,c) = -ProJ(c,b,a)
proof
  set 1F = 1_F;
  assume that
A1: not b _|_ a and
A2: b _|_ c+a;
  c-ProJ(a,b,c)*b _|_ a by A1,Th11;
  then (-ProJ(a,b,c))*b+c _|_ a by VECTSP_1:21;
  then (-1F)*((-ProJ(a,b,c))*b+c) _|_ a by Def1;
  then (-1F)*((-ProJ(a,b,c))*b)+(-1F)*c _|_ a by VECTSP_1:def 14;
  then ((-1F)*(-ProJ(a,b,c)))*b+(-1F)*c _|_ a by VECTSP_1:def 16;
  then (ProJ(a,b,c)*(1F))*b+(-1F)*c _|_ a by VECTSP_1:10;
  then ProJ(a,b,c)*b+(-1F)*c _|_ a;
  then ProJ(a,b,c)*b-c _|_ a by VECTSP_1:14;
  then a _|_ ProJ(a,b,c)*b-c by Th2;
  then
A3: -a _|_ ProJ(a,b,c)*b-c by Th6;
  ProJ(a,b,c)*b _|_ c+a by A2,Def1;
  then ProJ(a,b,c)*b _|_ c-(-a) by RLVECT_1:17;
  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
A5: not c _|_ b by Th3;
  then
A6: not b _|_ c by Th2;
  then -a-ProJ(c,b,-a)*b _|_ c by Th11;
  then ProJ(a,b,c) = ProJ(c,b,-a) by A6,A4,Th8
    .= ProJ(c,b,(-1F)*a) by VECTSP_1:14
    .= (-1F)*ProJ(c,b,a) by A5,Th2,Th12
    .= -(ProJ(c,b,a)*(1F)) by VECTSP_1:9;
  hence thesis;
end;
