Verify that `x sgnx=|x|` `|x|sgnx=x` `x(sgnx)(sgnx)=x`

Explanation

(i) `x " sgn "x={(x*1",",x gt 0),(0",", x=0),(x*(-1)",",x lt 0):} = {(x",",x gt0),(0",",x=0),(-x",", x lt 0):}=|x|`
(ii) `|x| " sgn "x={(x*1",",x gt 0),(0",", x=0),((-x)*(-1)",",x lt 0):} = {(x",",x gt0),(0",",x=0),(x",", x lt 0):}=x`
(iii) `x ("sgn "x) ("sgn "x)=|x| " sgn " (x)=x`