reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;
reserve i for Nat;
reserve m for Nat,
        D for non empty set;
reserve l for Nat;
reserve M for Nat;
reserve m,n for Nat;
reserve x1,x2,x3,x4 for object;
reserve e,u for object;

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;
