theorem
 <%x%> +~ (x,y) = <%y%>
proof
A1: dom(<%x%> +~ (x,y)) = dom<%x%> by FUNCT_4:99
      .= Segm 1 by Th30;
  then <%x%> +~ (x,y) is finite by FINSET_1:10;
  then reconsider p = <%x%> +~ (x,y) as XFinSequence by A1,ORDINAL1:def 7;
A2: rng<%x%> = {x} by Th30;
   then rng p c= {x} \ {x} \/ {y} by FUNCT_4:104;
   then rng p c= {} \/ {y} by XBOOLE_1:37;
   then
A3:  rng p c= {y};
      x in rng <%x%> by A2,TARSKI:def 1;
  then y in rng p by FUNCT_4:101;
  hence <%x%> +~ (x,y) = <%y%> by A1,Th30,A3,ZFMISC_1:33;
end;
