Suppose SPI holds and that holds with , . Iterating: holds, then holds, and continuing this way holds for every . SPI’s conclusion then gives for all , so WPI holds.

Conversely, suppose WPI holds, and let satisfy SPI’s two assumptions. Define a new predicate as “ holds for all “. Applying WPI to shows that holds for all , which implies that holds for all . So SPI holds.

Ordering enters because proving requires for all — effectively every earlier case at once — rather than only the single predecessor statement . This is why SPI is based on the ordering of while WPI needs only the successor structure.