Given a rational (total and transitive) relation or preference R, show that the corresponding indifference relation ~ (i.e. ~ iff xRy and yRx) is also transitive.
Suppose R is rational. Then R is transitive; i.e. if xRy and yRz, then xRz. Suppose x~y and y~z, we must show that x~z. But x~y implies xRy and yRx, and y~x implies yRz and zRy. By transitive property of R, xRy and yRz imply xRz. Similarly, zRy and yRx imply zRx. Since both xRz and zRx, x~z as desired. Q.E.D.