Proving the LOOP function

We must demonstrate that our guess is good, i.e.

   f = [WHILE]
     = ( R < Y --> ident ) | ( R >= Y and Y > 0 --> Q,R := Q+R/Y, R mod Y )

1. show f behaves like identity when the loop boolean is false
2. show dom(f) = dom([WHILE])
3. show f = [IF] o f where IF is made of the loop components
   
   [IF] = ( R >= Y --> Q,R := Q+1,R-Y ) | ( R < Y --> ident ) <<< watch out here
   
   so compute the composition [IF] o f =

      ( ( R >= Y --> Q,R := Q+1,R-Y ) | ( R < Y --> ident ) )
       o ( ( R < Y --> ident ) | ( R >= Y and Y > 0 --> Q,R := Q+R/Y, R mod Y ) )