reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a for Element of K;
reserve V for non trivial VectSp of K,
  V1,V2 for VectSp of K,
  f for linear-transformation of V1,V1,
  v,w for Vector of V,
  v1 for Vector of V1,
  L for Scalar of K;
reserve S for 1-sorted,
  F for Function of S,S;

theorem Th32:
  f|im (f|^n) is linear-transformation of im (f|^n),im (f|^n)
proof
  set IM=im (f|^n);
  rng(f|IM) c= the carrier of IM
  proof
    let y be object;
    assume y in rng (f|IM);
    then consider x being object such that
A1: x in dom (f|IM) and
A2: (f|IM).x=y by FUNCT_1:def 3;
    x in the carrier of IM by A1,FUNCT_2:def 1;
    then
A3: x in IM;
    then x in V1 by VECTSP_4:9;
    then reconsider v=x as Vector of V1;
    consider w be Vector of V1 such that
A4: (f|^n).w=v by A3,RANKNULL:13;
A5: the carrier of V1 = dom (f|^1) by FUNCT_2:def 1;
A6: the carrier of V1=dom (f|^n) by FUNCT_2:def 1;
    y = f.x by A1,A2,FUNCT_1:47
      .= (f*(f|^n)).w by A4,A6,FUNCT_1:13
      .= ((f|^1)*(f|^n)).w by Th19
      .= (f|^(1+n)).w by Th20
      .= ((f|^n)*(f|^1)).w by Th20
      .= (f|^n).((f|^1).w) by A5,FUNCT_1:13;
    then y in IM by RANKNULL:13;
    hence thesis;
  end;
  hence thesis by Lm1,FUNCT_2:6;
end;
