reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;

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;
