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

theorem Th5:
  <*p1,p2,p3*>| Seg 2 = <*p1,p2*>
proof
  set f = <*p1,p2,p3*>| Seg 2;
A1: len<*p1,p2,p3*> = 3 by FINSEQ_1:45;
  then
A2: 2 in dom<*p1,p2,p3*> by FINSEQ_3:25;
  1 in dom<*p1,p2,p3*> by A1,FINSEQ_3:25;
  then
A3: Seg 2 c= dom<*p1,p2,p3*> by A2,FINSEQ_1:2,ZFMISC_1:32;
A4: dom f = dom<*p1,p2,p3*> /\ Seg 2 by RELAT_1:61
    .= Seg 2 by A3,XBOOLE_1:28; then
  reconsider f as FinSequence by FINSEQ_1:def 2;
A5: len f = 2 by A4,FINSEQ_1:def 3;
  then 2 in dom f by FINSEQ_3:25;
  then
A6: f.2 = <*p1,p2,p3*>.2 by FUNCT_1:47
    .= p2;
  1 in dom f by A5,FINSEQ_3:25;
  then f.1 = <*p1,p2,p3*>.1 by FUNCT_1:47
    .= p1;
  hence thesis by A5,A6,FINSEQ_1:44;
end;
