Def L-induced Equiv(x,y) == z:A*. L(z @ x) L(z @ y)

Thm* A:Type, L:LangOver(A). L-induced Equiv A*A*Prop

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

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

Def b == if b True else False fi

Thm* b:. b Prop

Thm* T:Type, E:(TTProp). (EquivRel x,y:T. E(x,y)) Prop

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

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

Def LangOver(Alph) == Alph*Prop

Thm* Alph:Type{i}. LangOver(Alph) Type{i'}

Def Trans x,y:T. E(x;y) == a,b,c:T. E(a;b) E(b;c) E(a;c)

Thm* T:Type, E:(TTProp). Trans x,y:T. E(x,y) Prop

Def Sym x,y:T. E(x;y) == a,b:T. E(a;b) E(b;a)

Thm* T:Type, E:(TTProp). Sym x,y:T. E(x,y) Prop

Def Refl(T;x,y.E(x;y)) == a:T. E(a;a)

Thm* T:Type, E:(TTProp). Refl(T;x,y.E(x,y)) Prop

Def P Q == Q P

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