reserve i, k, m, n for Nat,
  r, s for Real,
  rn for Real,
  x, y , z, X for set,
  T, T1, T2 for non empty TopSpace,
  p, q for Point of T,
  A, B, C for Subset of T,
  A9 for non empty Subset of T,
  pq for Element of [:the carrier of T,the carrier of T:],
  pq9 for Point of [:T,T:],
  pmet,pmet1 for Function of [:the carrier of T,the carrier of T:],REAL,
  pmet9,pmet19 for RealMap of [:T,T:] ,
  f,f1 for RealMap of T,
  FS2 for Functional_Sequence of [:the carrier of T,the carrier of T:],REAL,
  seq for Real_Sequence;

theorem Th18:
  for D being non empty set, p,q be FinSequence of D,B be BinOp of
  D st p is one-to-one & q is one-to-one & rng q c= rng p & B is commutative
  associative & (B is having_a_unity or len q>=1 & len p>len q) holds ex r be
FinSequence of D st r is one-to-one & rng r=rng p \rng q & B "**" p =B.(B "**"
  q,B "**" r)
proof
  let D be non empty set, p,q be FinSequence of D,B be BinOp of D such that
A1: p is one-to-one and
A2: q is one-to-one and
A3: rng q c= rng p and
A4: B is commutative associative and
A5: B is having_a_unity or len q>=1 & len p>len q;
A6: card (rng p)=len p by A1,FINSEQ_4:62;
  consider r be FinSequence such that
A7: rng r=rng p \rng q and
A8: r is one-to-one by FINSEQ_4:58;
  reconsider r as FinSequence of D by A7,FINSEQ_1:def 4;
  rng (q^r)=rng q \/(rng p\ rng q ) by A7,FINSEQ_1:31;
  then
A9: rng (q^r)=rng p by A3,XBOOLE_1:45;
  rng r misses rng q by A7,XBOOLE_1:79;
  then
A10: q^r is one-to-one by A2,A8,FINSEQ_3:91;
  then card (rng (q^r))=len (q^r) by FINSEQ_4:62;
  then len q < len q+ len r or B is having_a_unity by A5,A9,A6,FINSEQ_1:22;
  then
A11: 1<=len r& 1<=len q & 1<=len p or B is having_a_unity by A5,NAT_1:19
,XXREAL_0:2;
  ex P be Permutation of dom p st p*P=(q^r) & dom P = dom p & rng P = dom
  p by A1,A10,A9,BHSP_5:1;
  then
A12: B "**" p= B "**" (q^r) by A4,A11,FINSOP_1:7;
  B "**" (q ^ r) = B.(B "**" q,B "**" r) by A4,A11,FINSOP_1:5;
  hence thesis by A7,A8,A12;
end;
