reserve x,y,z, X,Y,Z for set,
  n for Element of NAT;
reserve A for set,
  D for non empty set,
  a,b,c,l,r for Element of D,
  o,o9 for BinOp of D,
  f,g,h for Function of A,D;
reserve G for non empty multMagma;
reserve A for non empty set,
  a for Element of A,
  p for FinSequence of A,
  m1,m2 for Multiset of A;

theorem Th34:
  dom ({x}|`(p^<*x*>)) = dom ({x}|`p) \/ {len p+1}
proof
  thus dom ({x}|`(p^<*x*>)) c= dom ({x}|`p) \/ {len p+1}
  proof
    let a be object;
A1: len <*x*> = 1 by FINSEQ_1:40;
A2: Seg len p = dom p by FINSEQ_1:def 3;
    assume
A3: a in dom ({x}|`(p^<*x*>));
    then
A4: a in dom (p^<*x*>) by FUNCT_1:54;
    then a in Seg (len p+len <*x*>) by FINSEQ_1:def 7;
    then a in Seg len p \/ {len p+1} by A1,FINSEQ_1:9;
    then
A5: a in dom p or a in {len p+1} by A2,XBOOLE_0:def 3;
A6: (p^<*x*>).a in {x} by A3,FUNCT_1:54;
    reconsider a as Element of NAT by A4;
    a in dom p implies (p^<*x*>).a = p.a by FINSEQ_1:def 7;
    then a in dom p implies a in dom ({x}|`p) by A6,FUNCT_1:54;
    hence thesis by A5,XBOOLE_0:def 3;
  end;
  let a be object;
  len <*x*> = 1 by FINSEQ_1:40;
  then
A7: dom (p^<*x*>) = Seg (len p+1) by FINSEQ_1:def 7;
  assume
A8: a in dom ({x}|`p) \/ {len p+1};
  then a in dom ({x}|`p) or a in {len p+1} by XBOOLE_0:def 3;
  then
A9: a in dom p & p.a in {x} or a in {len p+1} & a = len p+1 & x in {x} by
FUNCT_1:54,TARSKI:def 1;
  Seg len p = dom p & Seg (len p+1) = Seg len p \/ {len p+1} by FINSEQ_1:9
,def 3;
  then
A10: (p^<*x*>).(len p+1) = x & a in dom (p^<*x*>) by A9,A7,FINSEQ_1:42
,XBOOLE_0:def 3;
  reconsider a as Element of NAT by A8;
  a in dom p implies (p^<*x*>).a = p.a by FINSEQ_1:def 7;
  hence thesis by A9,A10,FUNCT_1:54;
end;
