Def Rg(x,y) == z:A*. (g(z@x)) (g(z@y))

Thm* A:Type, R:(A*A*Prop). (EquivRel x,y:A*. x R y) (x,y,z:A*. (x R y) ((z @ x) R (z @ y))) (g:((x,y:A*//(x R y))). Rg (x,y:A*//(x R y))(x,y:A*//(x R y))Prop)

Def z@x == z @ x

Thm* A:Type, R:(A*A*Prop). (EquivRel x,y:A*. x R y) (x,y,z:A*. (x R y) ((z @ x) R (z @ y))) (z:A*, y:x,y:A*//(x R y). z@y x,y:A*//(x R y))

Def b == if b True else False fi

Thm* b:. b Prop

Def P Q == (P Q) & (P Q)

Thm* A,B:Prop. (A B) Prop

Def as @ bs == Case of as; nil bs ; a.as' a.(as' @ bs) (recursive)

Thm* T:Type, as,bs:T*. (as @ bs) T*

Def P Q == Q P

Thm* A,B:Prop. (A B) Prop