(1.5,0) [[$\bullet$]{}]{}; at (1.5,-0.5) [[$e_1$]{}]{}; at (14,1.4) [${{\ensuremath{\widehat F}}}X$]{}; (0,2) to \[out=20,in=0\] (3,0); at (1.4,0.9) [[$X$]{}]{}; (3,1.4) to \[out=20,in=160\] (13,1.4); at (8,3) [$c=\tilde c|_X$]{};

Remembering that pyramids are free and finitely generated, we will be done if we find a pyramid $Y\supseteq X$ which is mapped into ${{\ensuremath{\widehat F}}}Y$ by $\tilde c$:

(0,0) – (0,2) to \[out=20,in=0\] (3,0) – (0,0); (0,0) – (0,4-5/12) – (11-12/5,0) – (0,0); (0,0) – (12,0); (0,0) – (0,5); at (0,1.5) [[$\bullet$]{}]{}; at (-0.5,1.5) [[$e_2$]{}]{}; at (1.5,0) [[$\bullet$]{}]{}; at (1.5,-0.5) [[$e_1$]{}]{}; at (14,1.4) [${{\ensuremath{\widehat F}}}X$]{}; (0,2) to \[out=20,in=0\] (3,0); at (1.4,0.9) [[$X$]{}]{}; (3,1.4) to \[out=20,in=160\] (13,1.4); at (8,3) [$c=\tilde c|_X$]{}; (-1,4) – (11,-1); at (4.2,1) [[$Y$]{}]{}; at (14,3.4) [${{\ensuremath{\widehat F}}}Y$]{}; at (14.1,2.15)

[90]{} $\subseteq$

; (2,3.4) to \[out=20,in=160\] (13,3.4); at (8,5) [$\tilde c|_Y$]{};

This task can be reformulated as follows: For each pyramid $Y_1$ containing $X$ let $P(Y_1)$ be the set of all pyramids $Y_2$ containing $X$, such that $\tilde c(Y_2)\subseteq{{\ensuremath{\widehat F}}}Y_1$. If we find $Y$ with $Y\in P(Y)$, we are done.

Existence of $Y$ can be established by applying a fixed point principle for set-valued maps. The result sufficient for our present level of generality is Kakutani’s generalisation [@kakutani:1941 Corollary] of Brouwers fixed point theorem.

