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 )
A1: rng g = rng (Rev g) by FINSEQ_5:57;
set h = f ^ (Rev g);
assume f ^ g in Permutations n ; :: thesis: f ^ (Rev g) in Permutations n
then A2: f ^ g is Permutation of (Seg n) by MATRIX_2:def 9;
then A3: g is one-to-one by FINSEQ_3:91;
dom (f ^ g) = Seg n by A2, FUNCT_2:52;
then A4: Seg n = Seg ((len f) + (len g)) by FINSEQ_1:def 7;
len g = len (Rev g) by FINSEQ_5:def 3;
then A5: dom (f ^ (Rev g)) = Seg n by A4, FINSEQ_1:def 7;
A6: rng (f ^ g) = (rng f) \/ (rng g) by FINSEQ_1:31
.= rng (f ^ (Rev g)) by A1, FINSEQ_1:31 ;
A7: rng (f ^ g) = Seg n by A2, FUNCT_2:def 3;
then reconsider h = f ^ (Rev g) as FinSequence-like Function of (Seg n),(Seg n) by A6, A5, FUNCT_2:2;
A8: h is onto by A7, A6, FUNCT_2:def 3;
( rng f misses rng g & f is one-to-one ) by A2, FINSEQ_3:91;
then h is one-to-one by A1, A3, FINSEQ_3:91;
hence f ^ (Rev g) in Permutations n by A8, MATRIX_2:def 9; :: thesis: verum