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_1:6;
ex k being Nat st dom q c= Seg k by FINSEQ_1:def 12;
then i >= 0 + 1 by A2, FINSEQ_1:1;
then A3: i > 0 by NAT_1:13;
then reconsider p = {[i,x]} as FinSubsequence by Th1;
A4: Seq q = q * (Sgm (dom q)) by FINSEQ_1:def 14;
dom p = {i} by RELAT_1:9;
then Seq p = {[i,x]} * <*i*> by A1, A3, A4, FINSEQ_3:44
.= <*({[i,x]} . i)*> by A1, A2, FINSEQ_2:34
.= <*x*> by GRFUNC_1:6 ;
hence Seq q = <*x*> by A1; :: thesis: verum