reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;
reserve f for Function of A,B;
reserve f for Function;
reserve x1,x2,x3,x4,x5 for object;
reserve p for FinSequence;
reserve ND for non empty set;
reserve y1,y2,y3,y4,y5 for Element of ND;
reserve X, A for non empty finite set,
  PX for a_partition of X,
  PA1, PA2 for a_partition of A;
reserve f for FinSequence of D;

theorem
  m in dom f implies f/.m = (f|m)/.len(f|m)
proof
  assume that
A1: m in dom f;
  m<=len f by A1,FINSEQ_3:25;
  then
A2: len (f|m) = m by FINSEQ_1:59;
  1<=m by A1,FINSEQ_3:25;
  then m in Seg m;
  hence thesis by A1,A2,Th71;
end;
