theorem
  CutLastLoc(p^<%x%>) = p
proof set q = CutLastLoc(p^<%x%>);
A1: len(p^<%x%>) -' 1 = len p + 1 -' 1 by AFINSQ_1:75
     .= len p by NAT_D:34;
A2: dom(p^<%x%>) = len(p^<%x%>)
   .= Segm(len p + 1) by AFINSQ_1:75
   .= Segm len p \/ {len p} by AFINSQ_1:2;
A3: not len p in dom p;
 LastLoc(p^<%x%>) = len(p^<%x%>) -' 1 by AFINSQ_1:70;
 hence
A4: dom q = dom(p^<%x%>) \ {len p} by A1,VALUED_1:36
     .= dom p by A2,A3,ZFMISC_1:117;
 let y be object;
 assume
A5: y in dom q;
A6: p c= p^<%x%> by AFINSQ_1:74;
 thus q.y = (p^<%x%>).y by A5,GRFUNC_1:2
      .= p.y by A5,A4,A6,GRFUNC_1:2;
end;
