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;
reserve n for Nat;
reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem
  Funcs(X,<*Y,Z*>) = <*Funcs(X,Y), Funcs(X,Z)*>
proof
A1: Seg len <*Y,Z*> = dom <*Y,Z*> by FINSEQ_1:def 3;
A2: dom Funcs(X,<*Y,Z*>) = dom <*Y,Z*> by FUNCT_6:def 9;
  then reconsider p = Funcs(X,<*Y,Z*>) as FinSequence by A1,FINSEQ_1:def 2;
A3: len <*Y,Z*> = 2 by FINSEQ_1:44;
  then
A4: len p = 2 by A2,A1,FINSEQ_1:def 3;
  <*Y,Z*>.2 = Z & 2 in Seg 2 by FINSEQ_1:2,TARSKI:def 2;
  then
A5: p.2 = Funcs(X,Z) by A3,A1,FUNCT_6:def 9;
  <*Y,Z*>.1 = Y & 1 in Seg 2 by FINSEQ_1:2,TARSKI:def 2;
  then p.1 = Funcs(X,Y) by A3,A1,FUNCT_6:def 9;
  hence thesis by A4,A5,FINSEQ_1:44;
end;
