[an element/elements] [a relation/relations]
[x holds on y and z] [x is symmetric]
[x is transitive] [x is total]
[x is complement to y]

Definition. Let R be a relation.
R is symmetric iff for all elements x, y
if R holds on x and y then R holds on y and x.

Definition. Let R be a relation.
R is transitive iff for all elements x, y, z
if R holds on x and y and holds on y and z then R holds on x and z.

Definition. Let R be a relation.
R is total iff for all elements x, y R holds on x and y.

Definition. Let R, Q be relations.
R and Q are complement iff for all elements x, y
Q holds on x and y or R holds on x and y.

Proposition.
Let R, Q be transitive complement relations.
Assume R is symmetric. Then R is total or Q is total.
