reserve a,b,c for set;

theorem Th24:
  for D being non empty set,f,CR being File of D holds f is_preposition_of f^CR
proof
  let D be non empty set,f,CR be File of D;
  per cases;
  suppose
A1: len f>0;
    0+1<= len f + len CR by A1,NAT_1:13;
    then
A2: 1<= len (f^CR) by FINSEQ_1:22;
    0+1<= len f by A1,NAT_1:13;
    then mid(f^CR,1,len f)=f by FINSEQ_8:1;
    hence thesis by A2,FINSEQ_8:def 8;
  end;
  suppose
    len f<=0;
    hence thesis by FINSEQ_8:def 8;
  end;
end;
