Thm* hyp,M,N:Formula List, q,r:Formula. |= < q.hyp,r.(M @ N) > |= < hyp,M @ (qr.N) > formula_imp_right_sound

Thm* concl,M,N:Formula List, q,r:Formula. |= < M @ N,q.concl > |= < r.(M @ N),concl > |= < M @ (qr.N),concl > formula_imp_left_sound

Thm* concl,M,N:Formula List, q,r:Formula. |= < q.(M @ N),concl > |= < r.(M @ N),concl > |= < M @ (qr.N),concl > formula_or_left_sound

Thm* hyp,M,N:Formula List, q,r:Formula. |= < hyp,[q; r/ M @ N] > |= < hyp,M @ (qr.N) > formula_or_right_sound

Thm* hyp,M,N:Formula List, q,r:Formula. |= < hyp,q.(M @ N) > |= < hyp,r.(M @ N) > |= < hyp,M @ (qr.N) > formula_and_right_sound

Thm* concl,M,N:Formula List, q,r:Formula. |= < [q; r/ M @ N],concl > |= < M @ (qr.N),concl > formula_and_left_sound

Thm* hyp,M,N:Formula List, f2:Formula. |= < f2.hyp,M @ N > |= < hyp,M @ (f2.N) > formula_not_right_sound

Thm* concl,M,N:Formula List, f2:Formula. |= < M @ N,f2.concl > |= < M @ (f2.N),concl > formula_not_left_sound

In prior sections: sequent valid