reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;
reserve i,j for natural Number;
reserve F for Function of [:D,D9:],E;
reserve p,q for FinSequence of D,
  p9,q9 for FinSequence of D9;
reserve f,f9 for Function of C,D,
  h for Function of D,E;
reserve T,T1,T2,T3 for Tuple of i,D;
reserve T9 for Tuple of i, D9;
reserve S for Tuple of j, D;
reserve S9 for Tuple of j, D9;

theorem
  F.:(T^S,T9^S9) = (F.:(T,T9))^(F.:(S,S9))
proof
A0: i is Nat & j is Nat by TARSKI:1;
  defpred P[Nat] means for S being Tuple of $1, D, S9 being Tuple
  of $1, D9 holds F.:(T^S,T9^S9) = (F.:(T,T9))^(F.:(S,S9));
  now
    let j such that
A1: for S,S9 holds F.:(T^S,T9^S9) = (F.:(T,T9))^(F.:(S,S9));
    let S be Tuple of j+1, D;
    let S9 be Tuple of j+1, D9;
    consider S1 being Element of j-tuples_on D, d such that
A2: S = S1^<*d*> by FINSEQ_2:117;
A3: T^S = T^S1^<*d*> by A2,FINSEQ_1:32;
    reconsider S1 as Tuple of j, D;
    consider S19 being Element of j-tuples_on D9, d9 such that
A4: S9 = S19^<*d9*> by FINSEQ_2:117;
A5: T9^S9 = T9^S19^<*d9*> by A4,FINSEQ_1:32;
    reconsider S19 as Tuple of j, D9;
    thus F.:(T^S,T9^S9) = (F.:(T^S1,T9^S19))^<*F.(d,d9)*> by A3,A5,Th10
      .= (F.:(T,T9))^(F.:(S1,S19))^<*F.(d,d9)*> by A1
      .= (F.:(T,T9))^((F.:(S1,S19))^<*F.(d,d9)*>) by FINSEQ_1:32
      .= (F.:(T,T9))^(F.:(S,S9)) by A2,A4,Th10;
  end;
  then
A6: for j be Nat st P[j] holds P[j+1];
  now
    let S be Tuple of 0, D;
    let S9 be Tuple of 0, D9;
    S = <*>D;
    then
A7: F.:(S,S9) = <*>E by FINSEQ_2:73;
    T^S = T & T9^S9 = T9 by FINSEQ_2:95;
    hence F.:(T^S,T9^S9) = (F.:(T,T9))^(F.:(S,S9)) by A7,FINSEQ_1:34;
  end;
  then
A8: P[0];
  for j be Nat holds P[j] from NAT_1:sch 2(A8,A6);
  hence thesis by A0;
end;
