
theorem Th1:
  for V be non empty addLoopStr, L be C_Linear_Combination of V, v
  be Element of V holds L.v = 0c iff not v in Carrier L
proof
  let V be non empty addLoopStr, L be C_Linear_Combination of V, v be Element
  of V;
  thus L.v = 0c implies not v in Carrier L
  proof
    assume
A1: L.v = 0c;
    assume not thesis;
    then ex u being Element of V st v=u & L.u <> 0c;
    hence thesis by A1;
  end;
  thus thesis;
end;
