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 Th27:
  not a _|_ b & not c _|_ b implies ProJ(c,b,a) = (-ProJ(b,a,c)")* ProJ(a,b,c)
proof
  set 1F = 1_F;
  assume that
A1: not a _|_ b and
A2: not c _|_ b;
A3: ProJ(b,a,c) <> 0.F by A1,A2,Th23;
  then
A4: -ProJ(b,a,c)" <> 0.F by VECTSP_1:25;
  (-1F)*(ProJ(b,a,c)"*ProJ(b,a,c)) = (-1F)*(1F) by A3,VECTSP_1:def 10;
  then ((-1F)*ProJ(b,a,c)")*ProJ(b,a,c) = (-1F)*(1F) by GROUP_1:def 3;
  then ((-1F)*ProJ(b,a,c)")*ProJ(b,a,c) = (-1F);
  then (-(ProJ(b,a,c)"*(1F)))*ProJ(b,a,c) = -1F by VECTSP_1:9;
  then (-ProJ(b,a,c)")*ProJ(b,a,c) = -1F;
  then ProJ(b,a,(-ProJ(b,a,c)")*c) = -1F by A1,Th15;
  then (-ProJ(b,a,c)")*c-(-1F)*a _|_ b by A1,Th14;
  then (-ProJ(b,a,c)")*c-(-a) _|_ b by VECTSP_1:14;
  then (-ProJ(b,a,c)")*c+a _|_ b by RLVECT_1:17;
  then
A5: b _|_ (-ProJ(b,a,c)")*c+a by Th2;
  not b _|_ a by A1,Th2;
  then ProJ(a,b,(-ProJ(b,a,c)")*c) = ProJ((-ProJ (b,a,c)")*c,b,a) by A5,Th26;
  then ProJ(a,b,(-ProJ(b,a,c)")*c) = ProJ(c,b,a) by A2,A4,Th2,Th18;
  hence thesis by A1,Th2,Th15;
end;
