reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;

theorem Th110:
  for f being FinSequence st k <= n holds (f|n).k = f.k
proof
  let f be FinSequence;
  assume
A1: k<=n;
  per cases by NAT_1:14;
  suppose
A2: k = 0;
    then
A3: not k in dom f by Th25;
    not k in dom(f|n) by A2,Th25;
    hence (f|n).k = {} by FUNCT_1:def 2
      .= f.k by A3,FUNCT_1:def 2;
  end;
  suppose
    1 <= k;
    then k in Seg n by A1,FINSEQ_1:1;
    then (f|Seg n).k=f.k by FUNCT_1:49;
    hence thesis by FINSEQ_1:def 16;
  end;
end;
