reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;

theorem Th112:
  for A being set, F being FinSequence holds
  Sgm(F"A) ^ Sgm(F"(rng F \ A)) is Permutation of dom F
proof
  let A be set;
  let F be FinSequence;
A1: dom F = Seg len F by FINSEQ_1:def 3;
  set p = Sgm(F"A)^Sgm(F"(rng F \ A));
  A misses (rng F \ A) by XBOOLE_1:79;
  then
A2: A /\ (rng F \ A) = {};
A3: F"A /\ F"(rng F \ A) = F"(A /\ (rng F \ A)) by FUNCT_1:68
    .= {} by A2;
  then
A4: F"A misses F"(rng F \ A);
    F"(rng F \ A) c= dom F by RELAT_1:132;
  then F"(rng F \ A) c= Seg len F by FINSEQ_1:def 3;
  then
a6: F"(rng F \ A) is included_in_Seg by FINSEQ_1:def 13;
  then
A7: Sgm(F"(rng F \ A)) is one-to-one;
F"A c= dom F by RELAT_1:132;
  then F"A c= Seg len F by FINSEQ_1:def 3;
    then
a9: F"A is included_in_Seg by FINSEQ_1:def 13;
  then rng Sgm(F"A) /\ rng Sgm(F"(rng F \ A)) = F"A /\ rng Sgm(F"(rng F \ A))
  by FINSEQ_1:def 14
    .= {} by a6,A3,FINSEQ_1:def 14;
  then
A10: rng Sgm(F"A) misses rng Sgm(F"(rng F \ A));
A11: rng p = rng Sgm(F"A) \/ rng Sgm(F"(rng F \ A)) by FINSEQ_1:31
    .= F"A \/ rng Sgm(F"(rng F \ A)) by a9,FINSEQ_1:def 14
    .= F"A \/ F"(rng F \ A) by a6,FINSEQ_1:def 14
    .= F"(A \/ (rng F \ A)) by RELAT_1:140
    .= F"((rng F) \/ A) by XBOOLE_1:39
    .= F"(rng F) \/ F"A by RELAT_1:140
    .= (dom F) \/ F"A by RELAT_1:134
    .= dom F by RELAT_1:132,XBOOLE_1:12;
ww: F"A \/ F"(rng F \ A) is included_in_Seg by a6,a9;
  Sgm(F"A) is one-to-one by a9;
  then
A12: p is one-to-one by A7,A10,Th89;
  len p = len Sgm(F"A) + len Sgm(F"(rng F \ A)) by FINSEQ_1:22
    .= card (F"A) + len Sgm(F"(rng F \ A)) by a9,Th37
    .= card (F"A) + card (F"(rng F \ A)) by a6,Th37
    .= card (F"A \/ F"(rng F \ A)) by A4,CARD_2:40
    .= len Sgm (F"A \/ F"(rng F \ A)) by ww,Th37
    .= len Sgm (F"(A \/ (rng F \ A))) by RELAT_1:140
    .= len Sgm (F"((rng F) \/ A)) by XBOOLE_1:39
    .= len Sgm (F"(rng F) \/ F"A) by RELAT_1:140
    .= len Sgm (dom F \/ F"A) by RELAT_1:134
    .= len Sgm dom F by RELAT_1:132,XBOOLE_1:12
    .= card Seg len F by A1,Th37
    .= len F by FINSEQ_1:57;
  then dom p = dom F by Th27;
  then p is Function of dom F, dom F by A11,FUNCT_2:1;
  hence thesis by A12,A11,FUNCT_2:57;
end;
