let i be Element of NAT ; :: thesis: for x being set
for q being FinSubsequence st q = {[i,x]} holds
Seq q = <*x*>
let x be set ; :: thesis: for q being FinSubsequence st q = {[i,x]} holds
Seq q = <*x*>
let q be FinSubsequence; :: thesis: ( q = {[i,x]} implies Seq q = <*x*> )
assume A1: q = {[i,x]} ; :: thesis: Seq q = <*x*>
then [i,x] in q by TARSKI:def 1;
then A2: i in dom q by RELAT_1:20;
consider k being Nat such that
A3: dom q c= Seg k by FINSEQ_1:def 12;
i >= 0 + 1 by A2, A3, FINSEQ_1:3;
then A4: i > 0 by NAT_1:13;
then reconsider p = {[i,x]} as FinSubsequence by Th1;
A5: Seq q = q * (Sgm (dom q)) by FINSEQ_1:def 14;
dom p = {i} by RELAT_1:23;
then Seq p = {[i,x]} * <*i*> by A1, A4, A5, FINSEQ_3:50
.= <*({[i,x]} . i)*> by A1, A2, BAGORDER:3
.= <*x*> by GRFUNC_1:16 ;
hence Seq q = <*x*> by A1; :: thesis: verum