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;

theorem Th71:
  for D being set, f being FinSequence of D, n,m st n in dom f & m in
  Seg n holds m in dom f & (f|n)/.m = f/.m
proof
  let D be set, f be FinSequence of D, n,m;
  assume that
A1: n in dom f and
A2: m in Seg n;
  dom f = Seg len f & n<=len f by A1,FINSEQ_1:def 3,FINSEQ_3:25;
  then
A3: Seg n c= dom f by FINSEQ_1:5;
  hence m in dom f by A2;
A4: Seg n = dom f /\ Seg n by A3,XBOOLE_1:28
    .= dom(f|n) by RELAT_1:61;
  hence (f|n)/.m = (f|n).m by A2,PARTFUN1:def 6
    .= f.m by A2,A4,FUNCT_1:47
    .= f/.m by A2,A3,PARTFUN1:def 6;
end;
