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
  for s1,s2 be Element of S,n,m st (F|^m).s1 = s2 & (F|^n).s2 = s2 holds
  (F|^(m+i*n)).s1 = s2
proof
  let s1,s2 be Element of S,n,m such that
A1: (F|^m).s1 = s2 and
A2: (F|^n).s2 = s2;
  defpred P[Nat] means (F|^(m+$1*n)).s1 = s2;
A3: for i st P[i] holds P[i+1]
  proof
    let i such that
A4: P[i];
A5: dom (F|^(m+i*n))=the carrier of S by FUNCT_2:52;
    per cases;
    suppose
A6:   the carrier of S<>{};
      thus (F|^(m+(i+1)*n)).s1 = (F|^(n+(m+i*n))).s1
        .= ((F|^n)*(F|^(m+i*n))).s1 by Th20
        .= s2 by A2,A4,A5,A6,FUNCT_1:13;
    end;
    suppose
      the carrier of S={};
      then ( not s1 in dom (F|^(m+(i+1)*n)))& s2={} by SUBSET_1:def 1;
      hence (F|^(m+(i+1)*n)).s1=s2 by FUNCT_1:def 2;
    end;
  end;
A7: P[0] by A1;
  for i holds P[i] from NAT_1:sch 2(A7,A3);
  hence thesis;
end;
