Let ; let be their homotopy such that and for each a in A. The mapping cylinder for a map is the quotient space of disjoint union obtained by identifying each with . The mapping cylinder deformation retracts to the subspace B by sliding each point (a, t) along the segment to the endpoint . We can see easily that B is a deformation retract of both and (We can have as either F(A,0) or F(A, 1) and g(A) as either F(A,1) or F(A, 0), respectively). Thus and are homotopy equivalent spaces.