theorem
  Sum<* v,- v *> = 0.V & Sum<* - v,v *> = 0.V
proof
A1: v + - v = 0.V by Def10;
  hence Sum<* v,- v *> = 0.V by Th45;
  - v + v = 0.V by A1,Lm1;
  hence thesis by Th45;
end;
