theorem Th20:
  len r = i+j implies ex p,q st len p = i & len q = j & r = p^q
proof
  assume
A1: len r = i+j;
  reconsider z=i as Element of NAT by ORDINAL1:def 12;
  reconsider p = r|(Seg z) as FinSequence by FINSEQ_1:15;
  consider q being FinSequence such that
A2: r = p^q by FINSEQ_1:80;
  take p,q;
  i <= len r by A1,NAT_1:11;
  hence len p = i by FINSEQ_1:17;
  then len(p^q) = i + len q by FINSEQ_1:22;
  hence len q = j by A1,A2;
  thus thesis by A2;
end;
