reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem LQ2:
  for E,F be RealNormSpace,
        Q be LinearOperator of E,F,
        v be Point of E
  st Q is one-to-one
  holds Q.v = 0.F iff v = 0.E
  proof
    let E,F be RealNormSpace,
          Q be LinearOperator of E,F,
          v be Point of E;
    assume
    A1: Q is one-to-one;
    hereby
      assume
      A2: Q.v = 0.F;
      A3: dom Q = the carrier of E by FUNCT_2:def 1;
      Q.(0.E) = 0.F by LOPBAN_7:3;
      hence v = 0.E by A1,A2,A3,FUNCT_1:def 4;
    end;
    assume v = 0.E;
    hence Q.v = 0.F by LOPBAN_7:3;
  end;
