reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;
reserve A,B for set;
reserve x,y,i,j,k for object;
reserve x for set;

theorem
  dom ((x --> y)+*(x .-->z)+*(succ x .-->z)) = succ succ x
proof
  thus dom ((x --> y)+*(x .-->z)+*(succ x .-->z))
      = dom ((x --> y)+*(x .-->z)) \/ dom(succ x .-->z) by Def1
     .= dom (x --> y) \/ dom (x .-->z) \/ dom(succ x .-->z) by Def1

    .= x \/ dom (x .-->z) \/ dom(succ x .-->z)
    .= x \/ {x} \/ dom(succ x .-->z)
    .= x \/ {x} \/ {succ x}
    .= succ x \/ {succ x} by ORDINAL1:def 1
    .= succ succ x by ORDINAL1:def 1;
end;
