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;
