reserve X for RealUnitarySpace;
reserve x, y, y1, y2 for Point of X;
reserve xd for set;
reserve i, j, n for Nat;

theorem
  for DK, DX be non empty set
  for f be BinOp of DK st f is commutative associative & f is having_a_unity
  for Y1, Y2 be finite Subset of DX st Y1 misses Y2
  for F be Function of DX, DK st Y1 c= dom F & Y2 c= dom F
  for Z be finite Subset of DX st Z = Y1 \/ Y2 holds setopfunc(Z,DX,DK,F,f)
  = f.(setopfunc(Y1,DX,DK,F,f), setopfunc(Y2,DX,DK,F,f))
proof
  let DK, DX be non empty set;
  let f be BinOp of DK such that
A1: f is commutative associative and
A2: f is having_a_unity;
  let Y1, Y2 be finite Subset of DX such that
A3: Y1 misses Y2;
  let F be Function of DX,DK such that
A4: Y1 c= dom F and
A5: Y2 c= dom F;
  let Z be finite Subset of DX;
  assume
A6: Z = Y1 \/ Y2;
  consider p1 be FinSequence of DX such that
A8: p1 is one-to-one and
A9: rng p1 = Y1 and
A10: setopfunc(Y1, DX,DK, F, f) = f "**" Func_Seq(F,p1) by A1,A2,Def5;
  consider p2 be FinSequence of DX such that
A11: p2 is one-to-one and
A12: rng p2 = Y2 and
A13: setopfunc(Y2, DX,DK, F, f) = f "**" Func_Seq(F,p2) by A1,A2,Def5;
  set q = p1^p2;
A14: q is one-to-one by A3,A8,A9,A11,A12,FINSEQ_3:91;
  rng q = Z by A6,A9,A12,FINSEQ_1:31;
  then
A15: setopfunc(Z, DX, DK, F, f) = f "**" Func_Seq(F,q) by A1,A2,A14,Def5;
  ex fp1, fp2 be FinSequence st fp1 = F*p1 & fp2 = F*p2 & F*(p1^p2) = fp1^fp2
  by A4,A5,A6,A9,A12,HILBASIS:1,XBOOLE_1:8;
  hence thesis by A1,A2,A10,A13,A15,FINSOP_1:5;
end;
