
theorem
  for f,g being FinSequence
  st 1<=len f & g is_preposition_of f holds g is_substring_of f,1
proof
  let f,g be FinSequence;
  assume that
A1: 1<=len f and
A2: g is_preposition_of f;
  now per cases;
    case
A3:   len g>0;
      mid(f,1,(1-'1)+len g)=mid(f,1,0+len g) by NAT_2:8
        .= g by A2,A3;
      hence thesis by A1;
    end;
    case len g<=0;
      hence thesis;
    end;
  end;
  hence thesis;
end;
