theorem
  Funcs(<*X,Y*>,Z) = <*Funcs(X,Z), Funcs(Y,Z)*>
proof
A1: Seg len <*X,Y*> = dom <*X,Y*> by FINSEQ_1:def 3;
A2: dom Funcs(<*X,Y*>,Z) = dom <*X,Y*> by FUNCT_6:def 8;
  then reconsider p = Funcs(<*X,Y*>,Z) as FinSequence by A1,FINSEQ_1:def 2;
A3: len <*X,Y*> = 2 by FINSEQ_1:44;
  then
A4: len p = 2 by A2,A1,FINSEQ_1:def 3;
  <*X,Y*>.2 = Y & 2 in Seg 2 by FINSEQ_1:2,TARSKI:def 2;
  then
A5: p.2 = Funcs(Y,Z) by A3,A1,FUNCT_6:def 8;
  <*X,Y*>.1 = X & 1 in Seg 2 by FINSEQ_1:2,TARSKI:def 2;
  then p.1 = Funcs(X,Z) by A3,A1,FUNCT_6:def 8;
  hence thesis by A4,A5,FINSEQ_1:44;
end;
