reconsider n9 = n as Element of NAT by ORDINAL1:def 12;
reconsider F = the FacesAssign of T . n9 as Function of (NatEmbSeq . n9),((FuncsSeq the SkeletonSeq of T) . n9) by PBOOLE:def 15;
F . f in (FuncsSeq the SkeletonSeq of T) . n9 by FUNCT_2:5;
then F . f in Funcs (( the SkeletonSeq of T . (n + 1)),( the SkeletonSeq of T . n)) by Def3;
then consider G being Function such that
A2: G = F . f and
A3: dom G = the SkeletonSeq of T . (n + 1) and
A4: rng G c= the SkeletonSeq of T . n by FUNCT_2:def 2;
G . x in rng G by A1, A3, FUNCT_1:def 3;
then reconsider S = G . x as Symplex of T,n by A4;
take S ; :: thesis: for F, G being Function st F = the FacesAssign of T . n & G = F . f holds
S = G . x
let F1, G1 be Function; :: thesis: ( F1 = the FacesAssign of T . n & G1 = F1 . f implies S = G1 . x )
assume that
A5: F1 = the FacesAssign of T . n and
A6: G1 = F1 . f ; :: thesis: S = G1 . x
thus S = G1 . x by A2, A5, A6; :: thesis: verum