theorem Th43:
  for V1,V2 be VectSp of K for f be linear-transformation of V1,V2
  holds f is one-to-one iff ker f = (0).V1
proof
  let V1,V2 be VectSp of K;
  let f be linear-transformation of V1,V2;
  ker f=(0).V1 implies f is one-to-one
  proof
    assume
A1: ker f=(0).V1;
    let x,y be object such that
A2: x in dom f & y in dom f and
A3: f.x=f.y;
    reconsider x9=x,y9=y as Element of V1 by A2,FUNCT_2:def 1;
    x9-y9 in ker f by A3,RANKNULL:17;
    then x9-y9 in the carrier of (0).V1 by A1;
    then x9-y9 in {0.V1} by VECTSP_4:def 3;
    then x9+-y9=0.V1 by TARSKI:def 1;
    hence x = --y9 by VECTSP_1:16
      .= y by RLVECT_1:17;
  end;
  hence thesis by RANKNULL:15;
end;
