
theorem Lem10:
for f1,f2 be FinSequence, k be Nat st k in Seg (len f1 * len f2) holds
  (k-'1) mod (len f2) + 1 in dom f2 &
  (k-'1) div (len f2) + 1 in dom f1
proof
   let f1,f2 be FinSequence;
   let k be Nat;
   reconsider i1 = (k-'1) div (len f2) + 1 as Nat;
   reconsider i2 = (k-'1) mod (len f2) + 1 as Nat;
   assume B1: k in Seg (len f1 * len f2); then
B2:1 <= k & k <= len f1 * len f2 by FINSEQ_1:1; then
B3:1 <= len f1 * len f2 by XXREAL_0:2;
B4:len f1 <> 0 & len f2 <> 0 by B1; then
   (k-'1) mod (len f2) +1 <= len f2 by NAT_1:13,NAT_D:1;
   hence (k-'1) mod (len f2) + 1 in dom f2 by NAT_1:11,FINSEQ_3:25;
   1 <= len f1 & 1 <= len f2 by B4,NAT_1:14; then
B6:((len f1)*(len f2)-'1) div (len f2)
    = (len f1)*(len f2) div (len f2) -1 by B3,NAT_2:15,NAT_D:def 3
   .= len f1 - 1 by B4,NAT_D:18;
   k-'1 <= len f1 * len f2 -'1 by B2,NAT_D:42; then
   (k-'1) div (len f2) <= ((len f1)*(len f2)-'1) div (len f2)
        by NAT_2:24; then
   i1 <= len f1 -1+1 by B6,XREAL_1:6;
   hence (k-'1) div (len f2) + 1 in dom f1 by NAT_1:11,FINSEQ_3:25;
end;
