reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,a1,a2 for Element of K,
  D for non empty set,
  d,d1,d2 for Element of D,
  M,M1,M2 for (Matrix of D),
  A,A1,A2,B1,B2 for (Matrix of K),
  f,g for FinSequence of NAT;
reserve F,F1,F2 for FinSequence_of_Matrix of D,
  G,G9,G1,G2 for FinSequence_of_Matrix of K;
reserve S,S1,S2 for FinSequence_of_Square-Matrix of D,
  R,R1,R2 for FinSequence_of_Square-Matrix of K;
reserve N for (Matrix of n,K),
  N1 for (Matrix of m,K);

theorem
  1.(K,f|n) = 1.(K,f) |n
proof
  dom 1.(K,f|n) = dom (f|n) by Def8;
  then
A1: len 1.(K,f|n)=len (f|n) by FINSEQ_3:29;
  dom 1.(K,f) = dom f by Def8;
  then
A2: len 1.(K,f)=len f by FINSEQ_3:29;
  per cases;
  suppose
A3: n>len f;
    then f|n=f by FINSEQ_1:58;
    hence thesis by A2,A3,FINSEQ_1:58;
  end;
  suppose
A4: n<=len f;
    f=(f|n)^(f/^n) by RFINSEQ:8;
    then
A5: 1.(K,f)=(1.(K,f|n))^1.(K,f/^n) by Th58;
    len (1.(K,f|n))=n by A1,A4,FINSEQ_1:59;
    hence thesis by A5,FINSEQ_5:23;
  end;
end;
