reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;

theorem Th4:
  <*p1,p2,p3*>| Seg 1 = <*p1*>
proof
  set f = <*p1,p2,p3*>| Seg 1;
  len<*p1,p2,p3*> = 3 by FINSEQ_1:45;
  then 1 in dom<*p1,p2,p3*> by FINSEQ_3:25;
  then
A1: Seg 1 c= dom<*p1,p2,p3*> by FINSEQ_1:2,ZFMISC_1:31;
A2: dom f = dom<*p1,p2,p3*> /\ Seg 1 by RELAT_1:61
    .= Seg 1 by A1,XBOOLE_1:28; then
  reconsider f as FinSequence by FINSEQ_1:def 2;
  1 in dom f by A2;
  then
A3: f.1 = <*p1,p2,p3*>.1 by FUNCT_1:47
    .= p1;
  len f = 1 by A2,FINSEQ_1:def 3;
  hence thesis by A3,FINSEQ_1:40;
end;
