
theorem TT2:
for f1,f2 be disjoint_valued FinSequence
 ex f be disjoint_valued FinSequence st
   Union f1 /\ Union f2 = Union f &
   dom f = Seg (len f1 * len f2) &
   for i be Nat st i in dom f holds
     f.i = f1.((i-'1) div (len f2) + 1) /\ f2.((i-'1) mod (len f2) + 1)
proof
   let f1,f2 be disjoint_valued FinSequence;
   consider f be FinSequence such that
A1: Union f1 /\ Union f2 = Union f &
    dom f = Seg (len f1 * len f2) &
    for i be Nat st i in dom f holds
      f.i = f1.((i-'1) div (len f2) + 1) /\ f2.((i-'1) mod (len f2) + 1)
        by TT1;
   now let i,j be object;
    assume A2: i <> j;
    per cases;
    suppose A3: i in dom f & j in dom f; then
     reconsider i1=i, j1=j as Nat;
     f.i = f1.((i1-'1) div (len f2) + 1) /\ f2.((i1-'1) mod (len f2) + 1) &
     f.j = f1.((j1-'1) div (len f2) + 1) /\ f2.((j1-'1) mod (len f2) + 1)
        by A1,A3; then
B1:  f.i c= f1.((i1-'1) div (len f2) + 1) &
     f.j c= f1.((j1-'1) div (len f2) + 1) &
     f.i c= f2.((i1-'1) mod (len f2) + 1) &
     f.j c= f2.((j1-'1) mod (len f2) + 1) by XBOOLE_1:17;
A5:  0 < len f1 & 0 < len f2 or 0 > len f1 & 0 > len f2 by A1,A3;
A6:  now assume
A7:   (i1-'1) div (len f2) + 1 = (j1-'1) div (len f2) + 1 &
      (i1-'1) mod (len f2) + 1 = (j1-'1) mod (len f2) + 1;
      i1-'1 = (len f2) * ((j1-'1) div (len f2)) + ((i1-'1) mod (len f2))
         by A7,A5,NAT_D:2
      .= j1-'1 by A5,A7,NAT_D:2; then
      (1 > i1 or i1 >= j1) & (1 > j1 or j1 >= i1) by NAT_D:56;
      hence contradiction by A2,A1,A3,FINSEQ_1:1,XXREAL_0:1;
     end;
     per cases by A6;
     suppose (i1-'1) div (len f2) + 1 <> (j1-'1) div (len f2) + 1;
      hence f.i misses f.j by B1,XBOOLE_1:64,PROB_2:def 2;
     end;
     suppose (i1-'1) mod (len f2) + 1 <> (j1-'1) mod (len f2) + 1;
      hence f.i misses f.j by B1,XBOOLE_1:64,PROB_2:def 2;
     end;
    end;
    suppose not (i in dom f & j in dom f); then
     f.i = {} or f.j = {} by FUNCT_1:def 2;
     hence f.i misses f.j;
    end;
   end; then
   reconsider f as disjoint_valued FinSequence by PROB_2:def 2;
   take f;
   thus thesis by A1;
end;
