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 Th5:
  not b _|_ a & not l=0.F implies not l*b _|_ a & not b _|_ l*a
proof
  set 1F = 1.F;
  assume that
A1: not b _|_ a and
A2: not l=0.F;
A3: now
    consider k such that
A4: k*l=1F by A2,VECTSP_1:def 9;
    assume b _|_ l*a;
    then l*a _|_ b by Th2;
    then k*(l*a) _|_ b by Def1;
    then 1F*a _|_ b by A4,VECTSP_1:def 16;
    then a _|_ b;
    hence contradiction by A1,Th2;
  end;
  now
    consider k such that
A5: k*l=1F by A2,VECTSP_1:def 9;
    assume l*b _|_ a;
    then k*(l*b) _|_ a by Def1;
    then 1F*b _|_ a by A5,VECTSP_1:def 16;
    hence contradiction by A1;
  end;
  hence thesis by A3;
end;
