let n be Nat; :: thesis: for f, g being FinSequence st f ^ g in Permutations n holds
f ^ (Rev g) in Permutations n
let f, g be FinSequence; :: thesis: ( f ^ g in Permutations n implies f ^ (Rev g) in Permutations n )
assume f ^ g in Permutations n ; :: thesis: f ^ (Rev g) in Permutations n
then A1: f ^ g is Permutation of (Seg n) by MATRIX_2:def 11;
then A2: rng (f ^ g) = Seg n by FUNCT_2:def 3;
dom (f ^ g) = Seg n by A1, FUNCT_2:67;
then A3: Seg n = Seg ((len f) + (len g)) by FINSEQ_1:def 7;
A4: len g = len (Rev g) by FINSEQ_5:def 3;
A5: ( dom g = dom (Rev g) & rng g = rng (Rev g) ) by FINSEQ_5:60;
A6: rng (f ^ g) = (rng f) \/ (rng g) by FINSEQ_1:44
.= rng (f ^ (Rev g)) by A5, FINSEQ_1:44 ;
A7: ( rng f misses rng g & f is one-to-one & g is one-to-one ) by A1, FINSEQ_3:98;
set h = f ^ (Rev g);
dom (f ^ (Rev g)) = Seg n by A3, A4, FINSEQ_1:def 7;
then reconsider h = f ^ (Rev g) as FinSequence-like Function of (Seg n),(Seg n) by A2, A6, FUNCT_2:4;
A9: h is onto by A2, A6, FUNCT_2:def 3;
h is one-to-one by A5, A7, FINSEQ_3:98;
then h is bijective by A9;
hence f ^ (Rev g) in Permutations n by MATRIX_2:def 11; :: thesis: verum