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 Th33:
  1_F+1_F<>0.F & not b _|_ a implies (PProJ(a,b,x,y) = 0.F iff y _|_ x)
proof
  set 0F = 0.F;
  assume that
A1: 1_F+1_F<>0.F and
A2: not b _|_ a;
A3: a<>0.S by A2,Th1,Th2;
A4: PProJ(a,b,x,y) = 0.F implies y _|_ x
  proof
    assume
A5: PProJ(a,b,x,y) = 0.F;
A6: now
      given p such that
A7:   not p _|_ a and
A8:   not p _|_ x;
A9:   now
        assume
A10:    ProJ(p,a,x) = 0.F;
        not a _|_ p by A7,Th2;
        then x _|_ p by A10,Th23;
        hence contradiction by A8,Th2;
      end;
      ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) = 0F by A1,A2,A5,A7,A8,Def3;
      then ProJ(a,b,p)*ProJ(p,a,x) = 0F or ProJ(x,p,y) = 0F by VECTSP_1:12;
      then ProJ(a,b,p) = 0.F or ProJ(p,a,x) = 0.F or ProJ(x,p,y) = 0.F by
VECTSP_1:12;
      hence thesis by A2,A7,A8,A9,Th23;
    end;
    now
      assume for p holds p _|_ a or p _|_ x;
      then x = 0.S by A3,Th9;
      hence thesis by Th1,Th2;
    end;
    hence thesis by A6;
  end;
  y _|_ x implies PProJ(a,b,x,y) = 0.F
  proof
    assume
A11: y _|_ x;
A12: now
      assume x<>0.S;
      then consider p such that
A13:  not p _|_ a and
A14:  not p _|_ x by A3,Th9;
      ProJ(x,p,y) = 0F by A11,A14,Th23;
      then ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) = 0.F;
      hence thesis by A1,A2,A13,A14,Def3;
    end;
    now
      assume x = 0.S;
      then for p holds p _|_ a or p _|_ x by Th1,Th2;
      hence thesis by A1,A2,Def3;
    end;
    hence thesis by A12;
  end;
  hence thesis by A4;
end;
